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AN INTRODUCTION TO DIFFERENTIAL GEOMETRY

PRINCETON MATHEMATICAL SERIES
Editors: Marston Morse, H. P. Robertson, A. W. Tucker
1. THE CLASSICAL GROUPS
THEIR INVARIANTS AND REPRESENTATIONS
By Hermann Weyl
2. TOPOLOGICAL GROUPS
By L. Pontrjagin
3. AN INTRODUCTION TO DIFFERENTIAL GEOMETRY
WITH USE OF THE TENSOR CALCULUS
By Luther Pfahler Eisenhart
DIMENSION THEORY
By Witold Hurewicz and Henry Wallman
(In Press)
THE ANALYTICAL FOUNDATIONS
OF CELESTIAL MECHANICS
By Aurel Wintner
(In Preparation)

AN INTRODUCTION TO
DIFFERENTIAL
GEOMETRY
WITH USE OF THE TENSOR
CALCULUS
By LUTHER PFAHLER EISENHART
DOD PROFESSOR OF MATHEMATICS
PRINCETON UNIVERSITY
PRINCETON
PRINCETON UNIVERSITY PRESS
LONDON I HUMPHREY MILFORD : OXFORD UNIVERSITY PRESS
1940

Copyright 1940 by Princeton University Press
All Rights Reserved
Printed in the United States of America

TO
K. S. E.

Preface
Since 1909, when my Differential Geometry of Curves and Surfaces was
published, the tensor calculus, which had previously been invented by
Ricci, was adopted by Einstein in his General Theory of Relativity, and
has been developed further in the study of Riemannian Geometry and
various generalizations of the latter. In the present book the tensor
calculus of euclidean 3-space is developed and then generalized so as to
apply to a Riemannian space of any number of dimensions. The tensor
calculus as here developed is applied in Chapters III and IY to the
study of differential geometry of surfaces in 3-space, the material treated
being equivalent to what appears in general in the first eight chapters
of my former book with such additions as follow from the introduction
of the concept of parallelism of Levi-Civita and the content of the tensor
calculus.
Of the many exercises in the book some involve merely direct appli
cation of the text, but most of them constitute an extension of it.
In the writing of the book I have received valuable assistance and
criticism from Professor H. P. Robertson and from my students, Messrs.
Isaac Battin, Albert J. Coleman, Douglas R. Crosby, John Giese, Donald
C. May, and in particular, Wayne Johnson.
The excellent line drawings and half-tone illustrations were conceived
and executed by Mr. John H. Lewis.
Princeton, September 27, 1940 LUTHER PFAHLEH EISENHART.

Contents
CHAPTER I
CURVES IN SPACE
SECTION PAGE
1. Curves and surfaces. The summation convention 1
2. Length of a curve. Linear element 8
3. Tangent to a curve. Order of contact. Osculating plane 11
4. Curvature. Principal normal. Circle of curvature . 16
5. Binormal. Torsion . . 19
6. The Frenet formulas. The form of a curve in the neighborhood of a
point 25
7. Intrinsic equations of a curve 31
8. Involutes and evolutes of a curve 34
9. The tangent surface of a curve. The polar surface. Osculating sphere . 38
10. Parametric equations of a surface. Coordinates and coordinate curves
in a surface 44
11. Tangent plane to a surface 50
12. Developable surfaces. Envelope of a one-parameter family of surfaces.. 53
CHAPTER II
TRANSFORMATION OF COORDINATES. TENSOR CALCULUS
13. Transformation of coordinates. Curvilinear coordinates 63
14. The fundamental quadratic form of space 70
15. Contravariant vectors. Scalars 74
16. Length of a contravariant vector. Angle between two vectors 80
17. Covariant vectors. Contravariant and covariant components of a
vector 83
18. Tensors. Symmetric and skew symmetric tensors 89
19. Addition, subtraction and multiplication of tensors. Contraction.... 94
20. The Christoffel symbols. The Riemann tensor 98
21. The Frenet formulas in general coordinates 103
22. Covariant differentiation 107
23. Systems of partial differential equations of the first order. Mixed
systems 114
CHAPTER III
INTRINSIC GEOMETRY OF A SURFACE
24. Linear element of a surface. First fundamental quadratic form of a
surface. Vectors in a surface 123
25. Angle of two intersecting curves in a surface. Element of area 129
26. Families of curves in a surface. Principal directions 138
27. The intrinsic geometry of a surface. Isometric surfaces 146
28. The Christoffel symbols for a surface. The Riemannian curvature
tensor. The Gaussian curvature of a surface 149

X CONTENTS
29. Differential parameters 156
30. Isometric orthogonal nets. Isometric coordinates 161
31. Isometric surfaces 166
32. Geodesies 170
33. Geodesic polar coordinates. Geodesic triangles 180
34. Geodesic curvature 186
35. The vector asociate to a given vector with respect to a curve. Paral
lelism of vectors 194
36. Conformal correspondence of two surfaces . 201
37. Geodesiccorrespondenceoftwosurfaces.. 205
CHAPTER IV
SURFACES IN SPACE
38. The second fundamental form of a surface . 212
39. The equation of Gauss and the equations of Codazzi 218
40. Normal curvature of a surface. Principal radii of normal curvature.... 222
41. Lines of curvature of a surface 228
42. Conjugate directions and conjugate nets. Isometric-conjugate nets.... 231
43. Asymptotic directions and asymptotic lines. Mean conjugate directions.
The Dupin indicatrix 237
44. Geodesic curvature and geodesic torsion of a curve 243
45. Parallel vectors in a surface 249
46. Spherical representation of a surface. The Gaussian curvature of a
surface 252
47. Tangential coordinates of a surface 260
48. Surfaces of center of a surface. Parallel surfaces 268
49. Spherical and pseudospherical surfaces 277
50. Minimal surfaces 288
Bibliography 296
Index 298

CHAPTER I
Curves in Space
1. CURVES AND SURFACES. THE SUMMATION
CONVENTION
Consider space referred to a set of rectangular axes. Instead of
denoting, as usual, the coordinates with respect to these axes by x, y, ζ
we use χ1, x, x3, since by using the same letter χ with different super
scripts to distinguish the coordinates we are able often to write equa
tions in a condensed form. Thus we refer to the point of coordinates
χ1, χ2, χ3 as the point x where i takes the values 1, 2, 3. We indicate
a particular point by a subscript as for example x\, and when a point
is a general or representative point we use Xi without a subscript.
When the axes are rectangular, we call the coordinates cartesian.
In this notation parametric equations of the line through the point
x and with direction numbers u, u, u are*
(1.1) χ' = χ[ + υ'ΐ (t = 1,2,3).
This means that (1.1) constitute three equations as i takes the values
1, 2, 3. Here t is a parameter proportional to the distance between the
points x[ and xf, and t is the distance when Ui are direction cosines,
that is, when
(1.2) Σ (.UiY = 1,
i
which we write also at times in the form
= ι.
%
An equation of a plane is
(1.3) aix1 + a^x2 + a3a:8 + a = 0,
where the a's are constants. In order to write this equation in con
densed form we make use of the so-called summation convention that
when the same index appears in a term as a subscript and a superscript
this term stands for the sum of the terms obtained by giving the index
* See C. G., p. 85. A reference of this type is to the author's Coordinate Geom
etry, Ginn and Company, 1939.

2 CURVES IN SPACE [ CH. I
each of its values, in the present case the values 1, 2, 3. By means of
this convention equation (1.3) is written
(1.4) CiiXt + a = 0.
This convention is used throughout the book. At first it may be
troublesome for the reader, but in a short time he will find it to be
preferable to using the summation sign in such cases. A repeated index
indicating summation is called a dummy index. Any letter may be used
as a dummy index, but when a term involves more than one such index
it is necessary to use different dummy indices. Any index which is not
a dummy index and thus appears only once in a term is called a free
index.
Since two intersecting planes meet in a line, the two equations
(1.5) CiiX1 + a = 0, bix' + b = 0,
that is,
aix1 + (hx' + Ci3X3 + a = 0, bix1 + b%x2 + b3x + b = 0,
are equations of a line, provided that the ratios ajbi, az/bz, a3/b3 are
not equal; if these ratios are equal the'planes are coincident or parallel
according as a/b is equal to the above ratios or not.*
An equation (1.4) is an equation of a plane in the sense that it picks
out of space a two dimensional set of points, this set having the property
that every point of a line joining any two points of the set is a point of
the set; this is Euclid's geometric definition of a plane. In like manner
any functional relation between the coordinates, denoted by
(1.6) f{x X2, Xt) = o,
picks out a two dimensional set of points, by which we mean that only
two of the coordinates of a point of the locus may be chosen arbitrarily.
The locus of points whose coordinates satisfy an equation of the form
(1.6) is called a surface. Thus
(1.7) Σ XtXx + 2α,· χ' + b = 0,
i
where the a's and b are constants, is an equation of a sphere with center
at the point — a, and radius r given by
r2 = y. α<α» — b.f
t
* C. G., pp. 100, 101.
t C. G., p. 128, Ex. 14.

§1] CURVES AND SURFACES 3
Whenever throughout this book we consider any function, it is under
stood that the function is considered in a domain within which it is
continuous in all its variables, together with such of its derivatives as
are involved in the discussion.
Since an equation which does not involve one of the coordinates does
not impose any restriction on this coordinate, such an equation is an
equation of a cylinder. Thus
(1.8) f(x χ2) = 0
is an equation of a cylinder whose generators, or elements, are parallel
to the a;3-axis, each generator being determined by a pair of values
satisfying (1.8). If x{, x are two such values, the generator is defined
by the two equations
χ1 = x{, χ2 = x\,
these being a special form of (1.5) in this case. It does not follow that
when all three coordinates enter as in (1.6) that the surface is not a
cylinder, but that if it is a cylinder the generators are not parallel to
one of the coordinate axes. Later (§12) there will be given a means
of determining whether an equation (1.6) is an equation of a cylinder.
Two independent equations
(1.9) /1(2:1, χ2, Xi) = 0, /2(¾1, X2, xs) = 0
define a curve, a one dimensional locus, for, only one of the coordinates
of a point on the locus may be chosen arbitrarily. A line is a curve,
its two equations being linear, as for example in (1.5). A curve, being
one dimensional, may be defined also by three equations involving a
parameter, as
(1.10) *'· = m,
which are called parametric equations of the curve. These are a gen
eralization of equations (1.1). The functions f* in (1.10) are under
stood to be single-valued and such that for no value of t are all the first
derivatives of f equal to zero; the significance of this requirement
appears in §3.
If φ(η) is a single-valued function of u, and one replaces t in equations
(1.10) by ί = <p(u), there is obtained another set of parametric equa
tions of the curve, namely
(1.11) Xi = /V«)) = Au).
Since all the first derivatives — are not to be zero for a value of u,
au

4 CURVES IN SPACE [ CH. I
there is the added condition on φ that ^ ^ 0; this means that the
du
equation t = φ(ν) has a unique inverse.*
Thus the number of sets of parametric equations of a particular curve
is of the order of any function satisfying the above condition. When,
in particular, one of the coordinates, say x, is taken as parameter, the
equations are
(1.12) ζ1 = /(χ3), χ2 = /V), z3 = z3,
the forms of the f's depending, of course, upon the curve. From the
form of (1.12) it follows that the curve is the intersection of the two
cylinders whose respective equations are the first two of (1.12).
When all the points of a curve do not lie in a plane, the curve is said
to be skew or twisted. The condition that a curve with equations (1.10)
be a plane curve, that is, all of its points lie in a plane, is, as follows
from (1.4), that the functions f be such that
(1.13) of + a = 0,
that is
cii/1 + (¾/2 + Qzf3 + α = 0,
where the a's are constants. Differentiating equation (1.13) three
times with respect to t and denoting differentiation by primes, we ob
tain the three equations
(1.14) Oi/'= 0, CLif" = 0, a,f" = 0.
In order that the a's be not all zero, we must havef
/1' f f
r r
CO
Γ' f"
CO
Conversely, we shall show that if three functions f(t) satisfy this
condition, constants a, and a can be found satisfying (1.13); and conse
quently that the curve Xx = f(l) is plane. If (1.15) is satisfied there
exist quantities bi, ordinarily functions of t, such that
(1.16) bif = 0, hf" = 0, &/"" = O.t
* Fine, 1927, 1, p. 55. References of this type are to the Bibliography at the
end of the book.
t C. G., p. 114.
% C. G., p. 116.

§1] CURVES AND SURFACES 5
Assuming that the b's in these equations are functions of t and differ
entiating the first two of these equations with respect to t, the resulting
equations are reducible to
(1.16') b-f = 0, b'if" = 0.
The b's in the first two of (1.16) are proportional respectively to the
cofactors of the elements of the last row in the determinant (1.15).*
The same is true of the b"s in (1.16')· Consequently we have
bJ = bJ = bJ
bi bi bi
If we denote the common value of these ratios by φ'(ί), we have on
integration
bi = α{βν,
where the a's are constants. Substituting in the first of (1.16) and
discarding the factor ev, we obtain
a,·/' = 0.
On integrating this equation with respect to t, we obtain an equation
of the form (1.13). Hence we ha vet
[1.1] A curve with equations (1.10) is a plane curve, if and only if the
functions f satisfy equation (1.15).
When for the curve with the equations
(1.17) χ1 = Cit, χ2 = Cit21 xs = est3
the expressions for x% are substituted in an equation of a plane (1.4),
we obtain a cubic equation in t for each of whose roots the corresponding
point of the curve lies in the given plane. The curve (1.17) is called a
twisted cubic. When a curve meets a general plane in η points, it is
called a twisted curve of the nth order.
* C. G., p. 104.
t In the numbering of an equation or equations, as in (1.3), the number pre
ceding the period is that of the section and the second number specifies the par
ticular equation or equations. The same applies to the number of a theorem but
in this case brackets are used in place of parentheses. This notation is used
throughout the book.

6 CURVES IN SPACE [ CH. I
From (1.17) we have that the projections of the curve upon the
coordinate planes have the respective equations
(1-18)
These curves are shown schematically as follows for positive values of
the c's:
FIG. 1
From (1.18) it is seen that the curve is the intersection of the three
cylinders whose equations are the first of each pair of equations
(1.18), the generators of these cylinders being parallel to the x'-,
x'- and x'-axes respectively.
It will be found that many formulas and equations can be put in
simpler form by means of quantities enk and e/''' defined as follows:
0 when two or three of the indices have the same
values;
1 when the respective indices have the values 1, 2, 3;
2, 3, 1 or 3, 1, 2;
— 1 when the respective indices have the values 1, 3, 2;
3, 2, 1 or 2, 1, 3.
Consider, for example, the two equations
from which it follows that*

for each value of the constant a meets the curve (1.17) in three coincident points
at the point t — a.
4. There pass through a given point x\ in space three planes with equations
of the form of the equation of Ex. 3, and if ai , ai , os denote the corresponding
values of a, we have
is an equation of the plane through three points of the twisted cubic (1.17) with
parameters <i , <2 and ts .
3. The plane
the resulting equations are parametric equations of the projection of the curve
(1.10) on the plane (1.4).
2. The equation
where m is a parameter; as t and u take all values equations (i) give the coordi"
nates of points on the cylinder whose generators pass through points of the curve
and are normal to the plane (1.4); when in equations (i) we put
which is equation (1.15), as one verifies by forming the sum indicated
by the summation convention as each of the indices independently takes
the values 1, 2, 3.
EXERCISES
1. Parametric equations of a line normal to the plane (1.4) and passing through
a point of the curve (1.10) are
(1.14) we obtain
(1.21)
and when these expressions for ai are substituted in the first of equations
Applying this process to the last two of equations (1.14) we have
and consequently (1.20) may be written
§1] CURVES AND SURFACES 7
Thus, denoting by 1/r the factor of proportionality, we have

8 CURVES IN SPACE [ CH. I
from this result and Ex. 2 it follows that an equation of the plane through the
three points in which these three planes meet the cubic (each in three coincident
Doints) is
and the arc of the curve between the points Po and Pa for which the
parametric values are to and t^ respectively. Consider also inter-
(2.1)
2. LENGTH OF A CURVE. LINEAR ELEMENT
Consider a curve with the equations
from this result it follows that
9. Show that
where
and, if the determinant is denoted by a, then
shall be plane; what is the form of the curve?
8. By means of the quantities enk and e"^ one has
is the intersection of a circular cylinder and a hyperbolic paraboloid.
7. Determine f{i) so that the curve
which plane passes through the point xl .
5. Four planes determined by a variable chord of the cubic (1.17) and four
fixed points of the cubic are in constant cross-ratio.
6. The curve

§2] LENGTH OF A CURVE. LINEAR ELEMENT 9
mediate points for which the values of the parameter are
The length h of the chord PhPh+\ is given by
the second expression for 4 following from the mean value theorem of
the differential calculus, where the prime denotes the derivative. As
the number of intermediate points Pk increases indefinitely and each
Ik approaches zero, the limit of the sum of the h's is the definite integral
where
By definition this is the length of the arc PoPa. If then s denotes the
length of the arc from the point of parameter to to a representative
point of parameter t, we have
(2.2)
This gives s as a function of i; we denote it by
(2.3)
where ^ involves ta also.
From (2.2) we have
(2.4)
where As thus expressed ds is called the element of length,
or linear element, of the curve.
As remarked in §1 there is a high degree of arbitrariness in the choice
of a parameter for a curve. In what follows we shall often find that it
adds to the simplicity of a result, if the arc s is taken as parameter.
From (2.2) we have
[2.1] For a curve with equations (2.1) the parameter t is the length of the
curve measured from a given "point, if and only if
(2.5)

10 CURVES IN SPACE [ CH. I
It is evident that s is defined by (2.2) except when
(2.6)
Since it is understood now, and in what follows, that we are considering
only real functions of a real parameter, that is, one assuming only
real values, there is no real solution of (2.6) other than f constant, that
is, the locus is a point. If we admit complex functions of a complex
parameter, the curves for which (2.6) hold are called curves of length
zero, or minimal curves. There are cases in which it is advisable to
consider such curves, but unless otherwise stated they are not involved
in what follows.
Consider a curve defined in terms of the arc s as parameter. Let P
and P of coordinates x' and x be points for which the parameter has
the values s and s + e. By Taylor's theorem we have
(2.7)
Here and in what follows an x with one or more primes means that the
arc s is the parameter and the primes indicate derivatives with respect
to s; if the parameter is other than s, we write and similarly for
higher derivatives.
In this notation (2.5) is
(2.8)
Differentiating this equation with respect to s, we have
(2.9)
If we denote by Z the length of the chord PP, it follows from (2.7),
(2.8) and (2.9) that
(2.10)
From this result it follows that as P approaches P along the curve the
ratio of the chord to the arc e approaches unity as limit.

§3] TANGENT TO A CURVE 11
3. TANGENT TO A CURVE. ORDER OF CONTACT.
OSCULATING PLANE
The quantities χ — xl in (2.10) are direction numbers of the line
through P and P, and (x — xl)/l are its direction cosines. From (2.7)
and (2.10) it follows that
_i % -% i
,. x—x .. x—xe i'
Iim —-— - Iim η = χ .
~p—*p V 6 t
Since by definition the limiting position of the line through P and P
as P approaches P along the curve is the tangent to the curve at P, we
have
[3.1] When for a curve xl are expressed in terms of the length of the arc
from a given point as parameter, the quantities χ1 are direction cosines of
the tangent at a point x
If χ1 for a curve are expressed in terms of a parameter t, the quantities
dx ^
are direction numbers of the tangent. Thus the tangent is not
at
defined if all of these quantities are zero, which possibility was excluded
in §1.
As a result of this theorem we have as parametric equations of the
tangent to a curve at a point x1
(3.1) X1' = Xi + Xt'd,
where Xt are coordinates of a representative point on the tangent and
d is the distance from the point xl to the point X1.* We define positive
sense along the tangent as that for which d in (3.1) is positive, this
means that a half line drawn from the origin parallel to the tangent
makes with the coordinate axes angles whose cosines are χ' . This
same convention applies to any line associated with a curve when direc
tion cosines of such a line are given in terms of quantities defining the
curve.f
* C. G., p. 85.
t Here we define sense by means of direction cosines, which means that a line
has two sets of direction cosines, differing in sign. This is not the convention
adopted in C. G., pp. 77, 78.

12 CURVES IN SPACE [ CH. I
If we denote by a the direction cosines of the tangent, we have from
the above result and (2.2), when the parameter t is any whatever,
(3.2)
The plane through a point of a curve and normal to the tangent at
the point is called the normal plane at the point; an equation of this
plane is
(3.3)
where a are given by (3.2).*
Parametric equations of any line through the point x' of a curve are
(3.4)
where are direction cosines of the line. The square of the distance
of the point P{x'') from this line is given byf
(3.5)
When the point P is a point of the curve, its coordinates being given
by (2.7), the expression (3.5) becomes
(3.6)
Thus d is of the order of e unless the m' are proportional to a;''; since in
the latter case both of these sets of quantities are by theorem [3.1]
direction cosines, it follows that where e is +1 or —1, and
that the distance of a point on the curve nearby x' is of the second, or

§3] ORDER OF CONTACT. OSCULATING PLANE 13
higher, order.The distance is of the third order if also x^" are pro-
portional to , and of the (n + 1)*° order if are
proportional to In general, n = 1 and the contact of the tangent
with the curve is said to be of the first order-, for n > the contact is
of the n*'' order.
If the equations of the curve are in terms of a general parameter t,
as (2.1), we have since t is a function of s
(3.7)
Hence the tangent at a point t for which t satisfies for some value of n
(> 1), if any, the equations
(3.8)
has contact of the order.
By definition the osculating plane of a curve at a point Pix') is the
limiting position of the plane determined by the tangent at P and a
point P of the curve as P approaches P along the curve. Since the
plane passes through P its equation is of the form
(3.9)
where ai being direction numbers of the normal to the plane must be
such that
(3.10)
Equations (3.9) and (3.10) express the condition that the tangent at P
lies in the plane.* Substituting in (3.9) for X' the expressions (2.7)
for x' and making use of (3.10) we obtain
(3.11)
As P approaches P, that is, as o approaches zero, we have in the limit
(3.12)
* C. G., p. 120.

14 CURVES IN SPACE [ CH. I
In order that equations (3.9), (3.10) and (3.12) be satisfied by a's not
all zero, we must have*
(3.13)
which is an equation of the osculating plane at the point x
When the curve is defined by (2.1) in terms of a general parameter t,
it follows from (3.7) that an equation of the osculating plane is
(3.14)
If the tangent at a point has contact of higher order than the first,
equation (3.14) is satisfied identically. In this case from (3.11) and
(3.7) it follows that an equation of the osculating plane at a point for
which the tangent has contact of order n — 1 is
When a curve is plane and its plane is taken for the plane a;' = 0,
equation (3.13) is equivalent to x' = 0, that is, the osculating plane
of a plane curve is the plane of the curve. Conversely, when all the
osculating planes of a curve coincide, the curve is a plane curve since
all of the points of the curve lie in this plane.
EXERCISES
1. The curve
lies on a circular cylinder (see Fig. 2); find the direction cosines of the tangents
to the curve and show that the tangent makes a constant angle with the generators
of the cylinder; the curve is called a circular helix.

§3] OSCULATING PLANE 15
FIG. 2. Circular helix
2. By definition a cylindrical helix is a curve lying on a cylinder and which
meets all the generators under the same angle; if this constant angle is denoted
by e,
are parametric equations of a cylindrical helix; is any cylindrical helix so defined?
3. By definition a conical helix is a curve lying on a cone which meets all the
generators of the cone under the same angle; if this angle is denoted by 9, x® = /"(<)
are equations of a conical helix, if the functions/' satisfy the conditions
where the a's are constants not all of the same sign; is any conical helix so defined?

16 CURVES IN SPACE [ CH. I
4. Find an equation of the osculating plane of the twisted cubic (1.17) and
compare the result with Ex. 3 of §1.
5. The distance of a point P on a curve from the osculating plane at a nearby
point P is of the third order at least in the arc PP; and for any other plane through
P not containing the tangent at P the distance is of the first order; discuss the
case of planes containing the tangent at P.
6. The curve with the equations
xl = (1 - i2)/"«) + - f(t),
X2 = I (1 + t')f'(t) - itf {t) + W),
X3 = </"(<) -
where i = V-Ii /(0 >8 anJr function of t, and primes indicate differentiation
with respect to the parameter t, is a minimal curve, and any minimal curve is so
defined; discuss the case when/(i) = Cii2 + c%t + C3 , where the c's are constants.
7. If at every point of a curve the tangent has contact of the second order
with the curve, the latter is a straight line.
8. In terms of the arc s as parameter equations of the circular helix (Ex. 1) are
s s bs
X1 = a cos —, , X2 = a sin —γ , χ3 = —. ;
V α2 + 62 V α2 + 62 y/a2 + b2
each osculating plane of the helix meets the circular cylinder on which it lies
in an ellipse.
9. The curve
xl = a sin2 t, x2 = a sin t cos t, x3 = a cos i
is a spherical curve, that is, lies on a sphere; its normal planes pass through the
center of the sphere; the curve has a double point at (a, 0, 0), and the tangents
to the curve at this point are perpendicular to one another.
10. Find an equation of the osculating plane of the curve
x1 = a cos t + b sin t, x2 = α sin ί + b cos t, x3 = c sin 21;
find also two equations of the form (1.9) as equations of the curve.
4. CURVATURE. PRINCIPAL NORMAL. CIRCLE OF
CURVATURE
Let P and P be two points on a curve C, As the length of the arc
between these points, and ΔΘ the angle of the tangents at P and P,
that is, the angle between two half-lines through any point and having
Δ0
the positive senses of the two tangents. The limit of — as As ap-
As
proaches zero, measures the rate of change of the direction of the
tangent at P. This limiting value, denoted by κ, is called the curvature
of C at P, and its reciprocal, denoted by p, the radius of curvature; from

§4] CURVATURE. PRINCIPAL NORMAL 17
their definition it follows that κ and ρ are non-negative. When C is a
plane curve, the definition of curvature here given is that usually given
in the differential calculus.
In order to derive in terms of s an expression for κ in terms of the
quantities defining a curve C, we consider the auxiliary curve Γ with
the equations
(4.1) X' = ^,
ds
which in consequence of (2.8) is a curve upon the sphere of unit radius
with center at the origin. The radius of the sphere at any point of Γ
is parallel to the tangent to C at the corresponding point, that is, the
point with this tangent. The curve Γ is called the spherical indicatrix
of the tangents to C. If we denote by σ the arc of Γ, it follows from
(4.1) and (2.4) that
(4.2) iS- Σ ^
,· ds* as2
From the definition of κ we have
κ = Iim
Δβ=0
= Iim
Αθ Ασ
Δσ As
Since Αθ is the length of the arc of the great circle between the points
on the unit sphere corresponding to P and P on C, the limit of — is
Aa
unity in consequence of the result at the close of §2 and the fact that
ΑΘ and Δσ have a common chord. Consequently as follows from (4.2)
/. o . /v fPxi Cl2Xi
(t3) "= Vr d?i?·
For a straight line, with equations (1.1) in which t is the arc, one has
<c = 0, which is evident also geometrically from the fact that the tangent
to a straight line at each point is the line itself. In order to obtain the
expression for κ when the equations of the curve are in terms of a
general parameter t, we observe that from (2.3) we have
(44) dtL = L dIt=-^
K ' ds φ" ds2 φ,3'
where primes denote differentiation with respect to t, and from (2.2)
(4.5) / = Σ/'"/1", φ'φ" = Σ/'"/·",

18 CURVES IN SPACE [ CH. I
the second following from the differentiation of the first. Substituting
in (4.3) from (3.7) and making use of (4.4) and (4.5), we obtain
_ν@ΖΞ
//2
Ψ
(4.6) - - /2
φ2
From (2.9) it follows that the line through the point x% of a curve
<fx*
and with direction numbers —r-r is perpendicular to the tangent at the
as*•
point, and thus is one of an endless number of normals to the curve at
the point. If we define quantities β' by
ιfx*
(4.7) - Λ
it follows from (4.3) that β' are direction cosines of the positive sense
of this normal.* Its equations are
(4.8) Zi = Xi + β1 d,
where d denotes the distance of the point X1 from the point x' of the
curve. This normal is called the principal normal of the curve at the
point. When the expressions (4.8) are substituted in the equation
(3.13) of the osculating plane, the equation is satisfied for all values of d,
in consequence of (4.7), that is, the principal normal at a point lies in
the osculating plane at the point. Hence the osculating plane at a
point of a skew curve is the plane determined by the tangent and
principal normal of the curve at the point.
The circle in the osculating plane with center at the point
(4.9) Zi = Xi + PiSi = Xi + -β{
κ
and of radius ρ is called the circle of curvature of the curve at the point
χ1 and its center the center of curvature of the curve for the point xl.
Evidently this circle and the curve have a common tangent at xl.
EXERCISE
1. When a curve is defined in terms of a general parameter t by (1.10), the
direction cosines β1 of the principal normal are given by
β' = jT3 (<PT" -
φa
where <p(t) is defined by (2.3), and primes denote differentiation with respect to t.
* See the statement about positive sense after equations (3.1).

§5] BINORMAL. TORSION 19
2. For a cylindrical helix, as defined in §3, Ex. 2,
where e is +1 or —1 so that K is positive.
3. Let P be a point of a curve; a circle C with a common tangent to the curve
at P is determined by requiring that it pass through another point Q of the
curve; the limiting circle as Q approaches P along the curve is the circle of curva-
ture of the given curve at P.
4. Find the function <fi{t) so that the curve
shall be a curve of constant curvature.
5. Determine the form of the function ip{t) so that the principal normals to
the curve
are parallel to the xV-pl&ne.
6. The circle of curvature of a curve at a point of the curve has contact of the
second order with the curve; every other circle which lies in the osculating plane
and is tangent to the curve has contact of the first order; accordingly a circle of
curvature is called an osculating circle of the curve.
7. Find equations of the surface consisting of the principal normals of a circu-
lar helix (see §3, Ex. 1), and show that the locus of the center of curvature is a
circular helix.
8. Find the coordinates of the center of curvature of the curve
5. BINORMAL. TORSION
The normal to a curve at a point P which is normal to the osculating
plane at P is called the hinormal at P. Evidently it is perpendicular
to the tangent and to the principal normal at P. From (3,13) it follows
that

20 CURVES IN SPACE [ CH. I
are direction numbers of the binormal. In order to find direction co-
sines of the binormal we make use of the identity
The reader should verify that this is an identity whatever be the quanti-
ties involved without any use of the primes as indicating derivatives.
When in particular the quantities x and have the meaning ascribed
to them in §2, it follows from (2.8), (2.9) and (4.3) that the left-hand
member of (5.1) is equal to K. Hence direction cosines 7' of the bi-
normal and the Dositive sense alona: the latter are defined bv
(5.2)
These expressions may be written in the form
(5.3)
with the understanding that i, j, k take the values 1, 2, 3 cyclically.
Hence equations of the binormal are
(5.4)
The significance of the choice of sign in (5.2) is seen when we observe
that the expressions (3.2), (4.7) and (5.2) for a\ /3', and 7' respectively
are such that
(5.5)
as is readily verified since the right-hand member of (5.1) is equal to K^.
The result (5.5) means that the positive directions of the tangent,
principal normal and binormal of a curve at each point of a curve have
the mutual orientation of the x^-, x^- and x'-axes respectivelj' (Fig. 3).*
* C. G., p. 162.
(5.1)

From the definition of the binormal it follows that the binormals of
a plane curve are the normals to the plane at points of the curve, and
consequently have the same direction at all points of the curve. For a
skew curve the direction of the binormal changes. If Ad is the angle
of the positive directions of the binormals at two points of parameters
s and s + As, the limit as As approaches zero measures the rate
of change of the direction of the binormal at the point of parameter s,
and consequently the rate of change of the orientation of the osculating
as i, j, k take the values 1, 2, 3 cyclically.
and from (5.5) it follows that each element in the determmant equation
(5.5) is equal to its cofactor.* This result may be written
(5.7)
(5.6)
§5] BINORMAL. TORSION 21
From the equations

22 CURVES IN SPACE [ CH. I
plane. This limit is called the torsion of the curve and is denoted by r.
Sometimes the curvature, as defined in §4, and the torsion are called
the first and second curvatures respectively of the curve.
In order to obtain an expression for T, we introduce the spherical
indicatrix of the binormal, that is, the curve defined by
(5.8)
Evidently this is a curve upon the unit sphere with center at the origin,
and such that the radius of the sphere to any point of the curve is
parallel to the positive binormal to the given curve at the point with
the same value of the parameter s. The linear element of the indicatrix
js given by
By an argument similar to that used in §4 we have
(5.9)
we differentiate with respect to In order to find expressions for
s the equations
and obtain
(5.10)
From (4.7) and (3.2) we have
(5.11)
from which and the third of (5.6) it follows that the second of (5.10)
reduces to From this equation and the first of (5.10)
we have* in consequence of the second set of equations (5.7) that
dy*
— is proportional to /3', and from (5.9) that the factor of proportionality
as

§5] BINORMAL. TORSION 23
is T or — T. Thus far T is defined by (5.9) to within sign; we choose the
sign so that we have
(5.12)
We are now in position to obtain an expression for T in terms of the
derivatives of x'. In fact, if we differentiate (5.3) with respect to s,
the result may be written in consequence of (5.12)
If this equation is multiplied by and summed with respect to i, the
result becomes in consequence of the third of (5.6) and (4.7)
(5.13)
From the definition of torsion it follows that T is zero for a plane
curve. Conversely, if T is zero at every point of a curve, the latter is
plane in accordance with theorem [1.1] and equation (5.13). At points
of a curve, if any, for which the determinant in (5.13) is zero the osculat-
ing plane is said to be stationary.
EXERCISES
1. When a curve is defined in terms of a general parameter t by (1.10), the
direction cosines 7' of the binormal are given by
as i, j, k take the values 1, 2, 3 cyclically, where ip(J,) is defined by (2.3) and primes
denote differentiation with respect to t; also the torsion of the curve is given by
2. The curvature and torsion of a circular helix, as defined in §3, Ex. 1, are
constants, namely

from this result it follows that the curves for which K or T is a constant can be
found by quadratures.
are given by
12. The curvature and torsion of the curve
10. A necessary and sufficient condition that the principal normals of a curve
are parallel to a fixed plane is that the curve be a cylindrical helix.
11. If the curve a;*(s) is a cylindrical helix so also is the curve with the equations
6. Find the points of the curve of §3, Ex. 9 at which the torsion is equal to zero.
7. When two curves are symmetric with respect to a point, or a plane, their
curvatures at corresponding points are equal and their torsions differ in sign.
8. A necessary and sufficient condition that the circle of curvature have con-
dK
tact of the third order with the curve at a point is that at the point t = 0, — = 0
da
(see §4, Ex. 6); at such a point the circle is said to superosculaie the curve.
9. If 6 and ip are the angles made with a fixed line in space by the tangent
and binormal respectively of a curve,
are given by
5. The curvature and torsion of the curve
that is, K/T is a constant.
4. Find the curvature and torsion of the curve
from which result and that of §4, Ex. 2 it follows that
24 CURVES IN SPACE [ CH. I
3. For a cylindrical helix, as defined in §3, Ex. 2,

(6.6)
where 5j are Kronecker deltas defined by
(6.5) 1 or 0 according as
By Maclaurin's theorem we have that the coordinates a;' of any point
on the curve are given by
(6.4)
We observed following equation (5.5) that the positive directions of
the tangent, principal normal and binormal at each point of a curve
have the mutual orientation of the coordinate axes. If then we take
for coordinate axes these lines at a point Po of a curve and measure
the arc from the point, we have at Po
(6.3)
Differentiating with respect to s and making use of (6.1), we obtain
(6.2)
On replacing j3' in the second set of equations (6.1) by (see
equations (4.7)), the resulting equations are reducible to
(6.1)
Gathering together this result, (5.11), and (5.12), we have the following
set of equations fundamental in the theory of skew curves and called
the Frenet formulas:
as i, j, k take the values 1, 2, 3 cyclically, are differentiated with respect
to s, the result is reducible by (5.11), (5.12) and (5.7) to
§6] THE FRENET FORMULAS 25
6. THE FRENET. FORMULAS. THE FORM OF A CURVE
IN THE NEIGHBORHOOD OF A POINT
If the second set of equations (5.7), that is,

for suitable values of u and v. We raise the question of determining
u and V as functions of s so that the locus of the points of coordinates
(6.9)
in which /co and to are constants. This curve is a twisted cubic, whose
projections upon the coordinate planes are shown in Fig. 1, the a;'-,
x^-, and x'-axes being respectively the tangent, principal normal, and
binormal of the curve at the point of the given curve which is the origin
of the coordinate system used.
The coordinates X' of a point in the osculating plane to a twisted
curve are given by
(6.8)
It follows from these equations that in the neighborhood of a point,
if any, at which k = 0 the curve approximates a straight line. Also
if K 0 the curve with s increasing crosses the osculating plane at the
point, from the positive to the negative side when t > 0 and vice-versa
when T < 0; in the former case the curve is said to be left-handed and
in the latter right-handed at the point. At a stationary point, that is,
when T = 0, the curve remains on the same side of the osculating plane
in the neighborhood of the point (provided dr/ds 0), since in this
case the sign of x^ does not change with s for sufficiently small values
of s.
These results follow in fact when we consider only the first terms in
each of equations (6.7), that is, the approximate curve
(6.7)
From these results, (6.2), and (6.3) we have from (6.6) for this choice
of coordinate axes
26 CURVES IN SPACE [ CH. I
where a subscript zero indicates the value of the quantity at Po. From
(3.^), (4.7) and (6.4) we have

(i)
and equations of the curve are
EXERCISES
1. For a plane curve the Frenet formulas are
a = 1, 2),
From the theory of linear ordinary differential equations it follows that
the general solution of equation (6.11) may be obtained by quadratures.
When such a solution has been obtained and substituted in the second
of equations (6.10), u is given directly. Hence we have
[6.1] The orthogonal trajectories of the osculating planes of a skew curve
can be obtained by quadratures.
(6.11)
Differentiating the second of equations (6.10) with respect to a and
substituting from the first of (6.10), we obtain
(6.10)
this gives the following conditions to be satisfied:
Since are direction numbers of the tangent to the desired locus,
u and V must be such that are proportional to y\ that is, the co-
efficients of a and /3' must be zero. If we introduce the parameter c,
defined by
§6] THE FORM OF A CURVE 27
Z' given by (6.9) shall be an orthogonal trajectory of the osculating
planes. Differentiating (6.9) with respect to s and making use of
(6.1), we obtain

as i, j, k take the values 1, 2, 3 cyclically, where a is a constant, and 7' are func-
tions of a single parameter such that , are equations of a curve of
torsion 1/a and 7' are direction cosines of the binormal. Does it follow from
where 6* are constants, from which it follows that const., that is, the
curve makes a constant angle with the lines of direction numbers ¥, and hence
is a cylindrical helix.
7. The equations
(see §5, Ex. 3).
6. For a curve for which T/K = c, where c is a constant, it follows from the
Frenet formulas that
where e is +1 or —1 so that K is positive, and
then
If a function <r is defined by
(i)
from equations (i) it follows that <r is the angle which the tangent to the curve
makes with the x'-axis.
2. When all the osculating planes of a curve have a point in common, the
curve is plane.
3. The locus of the centers of curvature of a twisted curve of constant curva-
ture is an orthogonal trajectory of the osculating planes of the curve, and is a
curve of constant curvature.
4. A tangent to the locus of the centers of curvature of a twisted curve C is
perpendicular to the corresponding tangent to C; it coincides with, or is per-
pendicular to, the principal normal to C only at points for which T = 0, or ^ = 0.
as
5. When for a cylindrical helix (see §3, Ex. 2) the generators of the cylinder
are parallel to the x'-axis, then a" = cos 6, and from the Frenet formulas it
follows that
where
28 CURVES IN SPACE [ CH. I

from which and equation (i) of Ex. 10 for i = 1 we have that the locus C is the
a;®-axis. The curve is called the tractrix. In terms of cr, the angle which the
tangent makes with the a'-axis, equations of the curve are
From this result and Ex. 1 we have
the tangents to C are parallel to the corresponding osculating planes of the
given curve.
11. In order that the curve C in Ex. 10 be a straight line it is necessary and
suflScient that t = 0 and , where c is an arbitrary constant. If
we put c = a', we have
where a is a constant, are equations of a curve C whose points are on the tangents
to the curve x'(s) and at the constant distance a from the corresponding points
of contact; the arc s, the direction cosines a* of the tangent, and the curvature
K of C are given by
the curvature is constant.
9. When two twisted curves are in one-to-one correspondence with tangents
at corresponding points parallel, the principal normals at corresponding points
are parallel, and also the binormals; two curves so related are said to be deducible
from one another by a transformation of Combescure.
10. The equations
(i)
§6] THE FRENET FORMULAS 29
this result that any curve on the unit sphere can serve as the spherical indicatrix
of the binormals of a curve of constant torsion?
.8. If C is a curve of constant torsion, for the associated curve with the equa-
tions

where a and 6 are constants, is a Bertrand curve.
17. The binormals of a curve are the binormals of another curve, if and only
if the given curve is plane.
o and 6 being constants different from zero, are equations of a Bertrand curve.
14. A circular helix is a Bertrand curve; it has an infinite number of conju-
gates, each lying on a circular cylinder with the same axis as that of the given
helix.
15. A necessary and sufficient condition that the osculating planes of a Bert-
rand curve and of its conjugate coincide is that the curve be a plane curve; any
curve parallel to the given curve is a conjugate curve.
16. If C is a curve of constant torsion, the curve with equations
where ^(0 is any function of t and
where a and b are constants different from zero, is a Bertrand curve; the equa-
tions (see §5, Ex. 12)
thus e is to be chosen so that K is non-negative.
13. A curve for which
Also K and T for C are given by
Since = 6/S' by hypothesis, where e is -1-1 or —1, it follows that h is a constant;
denoting by w the angle between the osculating planes of C and its conjugate C,
one has a® = cos to a' sin « ; from the Frenet
formulas for C it follows that w is a constant and
30 CURVES IN SPACE [ CH. I
12. A curve whose principal normals are the principal normals of another
curve is called a Bertrand curve; if x' and 5' are the coordinates of corresponding
points on the respective curves C and C and s and s corresponding arcs, one has

§7] INTRINSIC EQUATIONS OF A CURVE 31
18. In order that the principal normals of a curve C be the binomials of a
curve C, it is necessary and sufficient that
(i) κ = α(κ2 + τ1),
where α is a constant; then equations of C are
Xt = Xi + αβ'~,
curves C satisfying the condition (i) can be found by quadratures (see §5, Ex. 12).
7. INTRINSIC EQUATIONS OF A CURVE
9
When equations (6.3) are differentiated successively with respect to
s and in each case the derivatives of α, β and yl are replaced by their
expressions from the Frenet formulas (6.1), we find that each derivative
of Xt is expressible linearly and homogeneously in α, β and γ1, the
coefficients being functions of κ, τ, and their derivatives of various orders
with respect to s. Consequently the coefficients of further terms in
(6.7) as derived from (6.6) involve only the values of κ, τ, and their
derivatives for s = 0, because of the particular values (6.4) at the origin
in the coordinate system used. Hence, if for two curves the functions
κ and τ of s are the same functions, the expressions for Xt for each curve
relative to the axes consisting of the tangent, principal normal, and
binormal of each curve at the point s = 0 are the same. Since either
set of axes can be brought into coincidence with the other by a rigid
motion, we have:
[7.1] Two curves whose curvature and torsion are the same functions re
spectively of the arc are congruent.
From this it follows that a curve is determined to within its position
in space by the expressions for κ and τ in terms of s. Consequently
(7.1) κ = fi(s), τ = Ms)
are equations of the curve. Since they are independent of the coordin
ate system used, they are called intrinsic equations of the curve.
From the manner in which equations (6.7) were obtained, it follows
that κ and τ derived from these equations by means of (4.3) and (5.13)
are power series in s, the coefficients being values of κ, τ, and their
derivatives evaluated for s = 0. Consequently if we have any two
equations (7.1) in which /i(s) is a non-negative function of s, the cor
responding equations (6.7) are equations of the curve for which (7.1)
are the intrinsic equations. Although this method of obtaining Xt as
functions of s gives the equations of a curve for given functions /i and

is an equation in homogeneous coordinates of a non-degenerate
conic in the plane. With respect to this conic
satisfjdng (7.7) are the coordinates of the vertices of a self-
polar triangle, and consequently an endless number of sets of a) satis-
fying (7.7) can be found.* For such a set of a} the quantities
defined by (7.4) are solutions of equations (7.2), that is, we have equa-
tions (6.1), the signs of a} having been chosen so that equation (5.5)
holds.
* Veblen and Young, 1910,1, p. 282.
If the determinant ] c'^ \ is different from zero.
(7.7)
(7.6) or 0 according as
This choice is made in order that a, j3', and y' shall be direction cosines
of three mutualy perpendicular vectors. Substituting from (7.4) in
(7.5) and making use of (7.3) we have
(7.5)
where
where the a's are constants, and seek under what conditions these a's
can be chosen so that
(7.4)
we find by differentiation that in consequence of (7.2) the c's are con-
stants. We define quantities by
(7.3)
If we have three sets of solutidns,of these
equations, which may be denoted by and put
(7.2)
32 CURVES IN SPACE [ CH. I
/j in equations (7.1), it gives these equations as infinite series. We
shall consider another approach to this problem which may in certain
cases lead to finite expressions for x
The three sets of quantities a, and y' as i takes the values 1, 2, 3
are seen from (6.1) to be solutions of the following system of ordinary
differential equations where K and T are given functions of s:

* Goursat, 1924, 1, vol. 2, p. 368.
are solutions of the Riccati equation
constitute a solution of equations (7.2); and any solution of (7.2) is expressed
in terms of three sets of solutions by (7.4) for suitable values of the constants oj .
2. If u, V, W are solutions of equations (7.2) such that the
quantities <r and a defined by
and V and v) given by
for the curve so defined s is the arc and a direction cosines of the tan-
gent. Also from (7.8) and (6.1) we obtain by differentiation equations
(4.7) and (6.2), from which it follows that K and r are the curvature and
torsion of the curve, and |3' and 7' direction cosines of the principal
normal and binormal respectively.
Thus we have shown that three sets of solutions of equations (7.2)
for given functions K and T of s lead by quadratures (7.8) to a curve
for which K and T are the curvature and torsion, provided that the de-
terminant I c^' I is not equal to zero. That there are sets of solutions of
equations (7.2) satisfying this condition follows from the theory of such
sets of equations, namely that there exists a unique solution for a given
set of initial values of u, v and w.* Since c^ are constants, it follows
that one has only to choose the initial values of the three sets of solu-
tions so that it shall follow from (7.3) that the determinant of the c's is
not equal to zero.
EXERCISES
1. A solution of the equation
(7.8)
§7] INTRINSIC EQUATIONS OF A CURVE 33
If then we define x' by

34 CURVES IN SPACE [ CH. I
3. The general integral of an equation of Riccati
dB
rr — L· 2.1/β -f- Νθ2,
as
where L, Μ, N are functions of s, is of the form
em.*±Qt
aB -|- S
where a is an arbitrary constant, and P, Q, B, S are functions of s.
4. From theorem [7.1] and §5 Ex. 2 it follows that a necessary and sufficient
condition that a curve be a circular helix is that its curvature and torsion be
constant; show also by means of Ex. 1 that this condition is sufficient.
5. Establish the statement made about the number of solutions of equations
(7.7) by purely algebraic methods.
8. INVOLUTES AND EVOLUTES OF A CURVE
As shown in §3 the equations
(8.1) Xi = Xi + ua'
are parametric equations of the tangents to the curve C defined by χ'
as functions of the arc s. For a particular tangent u is the distance
between the points x% and X1. If u is replaced in equations (8.1) by a
function of s, the resulting equations are equations of a curve Γ whose
points lie on the tangents to the given curve. Differentiating equation
(8.1) with respect to s, one has in consequence of (3.2) and (6.1)
(8.2) = (l + g) «' + «tt.
dX*
Since -j- are direction numbers of the tangent to Γ, if the latter curve
as
is to be such that its tangent at each point is perpendicular to the
tangent to C through the point, it is necessary and sufficient that
Σ«^ = ο.
i as
Since Σ &βι = 0, this condition is that
t
(8.3) ^ + 1 = 0
as
from which it follows that
(8.4)

§8] INVOLUTES OF A CURVE 35
where c is an arbitrary constant. Hence there is an infinity of such
curves Γ, each defined by
(8.5) Zi = Xi + (c - «)«'
for a particular value of c. They are called the involutes of the given
curve. From (8.4) it follows that the length of the segment of any
tangent to the curve determined by two involutes has the same value,
the difference of the c's of the two involutes.
FIG. 4. Involute of the circular helix of Fig. 2
When a curve C is defined in terms of a general parameter t, the
determination of s requires a quadrature (2.2), and then the involutes
are given directly by (8.5).
An involute when c — s is positive may be described mechanically as
follows: Take a string of length c, fasten one end at the point of the
curve for which s = 0 and bring the string into coincidence with the
curve; when the string is unwound from the curve and is kept taut, the
other end point describes the involute as is seen from (8.5).
From (8.2) and (8.4) we have
(8.6) = (c - s)*0'.
as
Hence we have
[8.1] A curve has an infinity of involutes; a tangent to an involute at a
point X1 is parallel to the principal normal to the curve at the corresponding
point xx.

36 CURVES IN SPACE [ CH. I
If Γ is an involute of a curve C, we say that C is an evolute of Γ, that
is, it is a curve whose tangents are normal to Γ. Suppose then that we
start with a curve of coordinates x and seek its evolutes. Since the
points of an evolute lie in the normal planes to the curve, its equations
are of the form
(8.7) Xi = Xi + Uffi + Vyi,
where u and ν are functions of s to be determined. From these equa
tions in consequence of (3.2) and (6.1) we have by differentiation with
respect to s
(S.S) 4* = (1 - UK)a< + (g + β' + - wr) y*.
Since these quantities are direction numbers of the tangent to an evolute,
they must be proportional to X1 — x that is, to w/3l + vy which as
follows from (8.7) are direction numbers of the line joining the points
X' and Xt. Consequently we must have u = 1 /K and
(du . (dv
When this equation is written in the form
dv du , i ,
u— — υ — = (ω + ν )τ,
ds ds
we see that its integral is
(8.9) - = tan (ω + c),
u
where by definition
(8.10) ω = J τ ds,
and c is an arbitrary constant. Substituting these results in (8.7), we
obtain
(8.11) Xi = Xi + - (β* + tan (ω + c) y*).
κ
For each value of c these are equations of an evolute. Consequently a
curve has an infinity of evolutes. Since the curve is an involute of each
evolute, we have

§8] EVOLUTESOF A CURVE 37
[8.2] A curve C has an infinity of evolutes; the principal normal to an evolute
is parallel to the tangent to C at the corresponding point.
From equations (8.11) and (4.9) it follows that the points of all of
these evolutes corresponding to a given point on the given curve lie on
the line, called the polar line, parallel to the binormal and through the
center of curvature for the given point on the curve. Moreover from
(8.9) it follows that ω + c is the angle which the line joining a point on
the curve to the corresponding point on an evolute makes with the
osculating plane of the curve at this point. Hence we have
i8.3] When each of the normals to a curve C which are tangent to an evolute
s turned through the same angle about the corresponding tangent to C, the
inormals in their new position are tangent to another evolute of C.
From (8.11) and (8.10) it follows that it is possible to choose c so that
the points of the corresponding evolute lie in the osculating planes only
in case ω is a constant, in which case τ = 0, that is, when the curve is
plane. In this exceptional case this evolute is the locus of the centers
of the circle of curvature and is a plane curve, except when the given
curve is a circle, in which case the locus is the center of the circle.
This evolute is the one which in the differential calculus is called the
evolute of the plane curve. However, a plane curve has an infinity of
evolutes, as c in equations (8.11) takes all possible values. From the
form of these equations it follows that these evolutes lie on the cylinder
whose generators are the normals to the plane of the given curve at
points of the evolute in the plane, that is, the evolute for which c = —ω.
Moreover, from the remark preceding theorem [8.3] it follows that the
tangents to each evolute make constant angles with the generators of
this cylinder and consequently these evolutes are cylindrical helices
(see §3 Ex. 2). Hence we have
[8.4] A plane curve other than a circle has an infinity of evolutes, each of
which is a helix of the cylinder whose right section by the plane of the curve
is the plane evolute of the curve.
EXERCISES
1. The involutes of a circular helix (§3, Exs. 1, 8) are plane curves, which
also are involutes of circular sections of the circular cylinder upon which the
helix lies.
2. For an involute (8.5) for which c — s is positive the arc, direction cosines
of the tangent, principal normal, binormal, and the curvature and torsion are

where now u is not the distance of the point from the point as
it is in (9.1).
(9.2)
we obtain a single equation in the X\ Consequently (see §1) the locus
of points on the tangents to a curve is a surface, called the tangent
surface of the curve, and each of the tangent lines is called a generator
of the surface. When the curve is defined in terms of a general param-
eter t equations of the tangent surface are
(9.1)
where e is -f-l or —1 so that k is positive.
9. THE TANGENT SURFACE OF A CURVE. THE POLAR
SURFACE. OSCULATING SPHERE
When for any curve the two parameters s and u are eliminated from
the three parametric equations of its tangents, namely
3. A necessary and sufficient condition that the involutes of a twisted curve
be plane curves is that the curve be a cylindrical helix (see §6, Ex. 6).
4. Find the evolutes of the curves of §5, Exs. 4 and 5.
5. For an evolute (8.11) the arc, direction cosines of the tangent, principal
normal, and binormal, and the curvature and torsion are given by
38 CURVES IN SPACE [ CH. I
given by

§9] TANGENT SURFACE OF A CURVE 39
When s and u in (9.1) are given particular values, equations (9.1)
give the coordinates of a point on the surface. Thus the locus is two
dimensional, which is another proof that the locus is a surface. If in
(9.1) we replace Μ by a function of s, say p(s), the resulting equations
are parametric equations of a curve on the surface. In particular, if
we put M = C — S, where c is a constant, then, as follows from §8, the
curve for each value of c is an involute of the given curve. Consequently
FIG. 5. Tangent surface of the circular helix of Fig. 2
all the involutes of a curve lie on its tangent surface. They are the
curves which intersect the generators of the surface at right angles,
that is, the involutes are the orthogonal trajectories of the generators.
Since there is only one set of orthogonal trajectories of a set of lines, we
have
[9.1] The orthogonal trajectories of the generators of the tangent surface of
a curve are the involutes of the curve.
According as u in equation (9.1) has a positive or negative value the
point lies on the portion of the tangent drawn in the positive direction

40 CURVES IN SPACE [CH. I
from a point on the curve or in the opposite direction. Hence the
surface consists of two parts, or sheets, one part consisting of all the
points for which u ^ 0, the other of the points for which u 5Ξ 0. Thus
the curve forms a common boundary of the two sheets.
In order to get an idea of the form of the surface in the neighborhood
of the curve, we recall from §6 that in the neighborhood of the point
P0 (s = 0) the curve approximates the twisted cubic
(9.3) χ1 = s, χ2 = χ3 = -k^,
where K0 and r0 are the curvature and torsion of the given curve at the
point P0, the tangent, principal normal, and binormal at P0 being the
coordinate axes. Noting that s is the arc of the given curve but not
of the cubic, we have that equations of the tangent surface to the curve
(9.3) are
(9.4) Z1 = s + ut Z2 = κο(^ + SUy Z3 = -K-y(|3 + 4
In this coordinate system the plane x1 = 0 is the plane normal to the
given curve and to the cubic at P0, and cuts this tangent surface in
the curve Γ for which u = —s. From the second and third of equa
tions (9.4) it follows that equations of this plane section are
Z1 = 0, Z2 = Z3 =
On eliminating s from the second and third equations we see that the
curve is a semi-cubical parabola with the negative half of the principal
normal for cuspidal tangent. Since this is the case at every ordinary
point of the curve, that is, every point at which neither κ nor τ is zero,
we have
[9.2] The tangent surface of a curve consists of two sheets which are tangent
to one another along the curve, and thus form a sharp edge, namely the curve.
The curve is called the edge of regression of the surface. An idea of
the form of the surface in the neighborhood of the curve may be had
from Figs. 5 and 6.
The line with the equations
(9.5) Zi = Xi + ρβί + Uyi
for each value of s is the polar line, as defined in §8, corresponding to
the point xl on the curve for this value of s, and the parameter u is the

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"No, no!" exclaimed the Rajah; "I have seen enough, and too
much! I have seen enough to humble me in dust and ashes--to make
me know myself blind, wretched, and guilty! Why," he continued,
bitterly, "why hast thou come to break my peace--to poison my
happiness in the very temple which I have erected to the Deity? Mean
and polluted it may be, but still it hath been raised to his honour."
"Is God then enthroned in this temple?" replied Truth. "It cannot
be, or the presence of divine purity would have purified even it. Self-
deceiver! thou standest now in the centre of the temple--thou
standest at the foot of the shrine; lift up thine eyes and behold the
object of thy worship--behold the Idol which thou hast adored all thy
days!"
Futtey Sing glanced up; the pedestal was lofty--on image of clay
was on the summit. He saw the idol, and he knew it; he saw the
effigy of himself, dressed in the robes of his pride, and he fell on his
face with a cry.
"To exalt self have thy good deeds been performed--to exalt self
has been the motive of thy actions--to exalt self has been the object
of thy life! And will the Supreme see His rightful throne in the heart
given to another? Will not His lightning strike down the idol?"
It seemed as though the words shook the earth beneath their
feet; it shuddered, it reeled, it heaved. It seemed as though the
words awakened the thunders above their heads; they rolled, they
burst, they roared in the sky. The walls trembled, rent, fell with a
fearful crash, as if to bury the sinner beneath their ruins; while a vivid
flash of forked lightning darted from the heavens, struck the idol of
Self, and laid it prostrate in the dust.
"Save me! I perish! I perish!" exclaimed the Rajah; and with that
cry of terror he awoke.

CHAPTER V.
SECRET INFLUENCES.
Mrs. Vernon closed the book. Various comments were made on the
story. The question was discussed whether it were really a translation
from an Eastern composition, or the work of a Christian author, who
had chosen to adopt the peculiarities of the Oriental style. Flora was
decidedly of the latter opinion, and showed so well the grounds upon
which her judgment was formed, that she completely won over her
opponent, Ada, and closed the discussion in triumph.
Was, then, Flora's position so entirely different from that of the
self-righteous Rajah that his story afforded her nothing more than a
field for intellectual exercise?
It was with Flora as it is with many that have been brought up in
what is called "the religious world." She had heard so much, read so
much, talked so much, on spiritual subjects, that she had acquired a
certain amount of theological knowledge, which not only supplied the
place of deep heart-devotion, but blinded her to her own want of it.
Flora never for one moment during her life had felt her soul in danger,
or doubted that she was walking in the narrow path which her
widowed mother so faithfully trod. The warnings which she heard in
sermons she constantly applied to others. She read serious works
rather as a critic than as one anxiously gleaning from them lessons for
the conduct of her own life. Flora earnestly upheld the doctrine of
justification by faith; she owned that through the merits of the
Saviour alone a sinner could find pardon from God. But in the depths

of her heart; unknown to herself, there was a secret lurking feeling
that her own virtue, her benevolence, her gentleness, her filial
obedience, her early piety, deserved the favour of the Almighty. She
did not believe that her slight short-comings merited any severe
condemnation. She was unconsciously going to the marriage-supper
of the King's Son in the garment of her own righteousness. The seed
of the Word, received with so much joy, had fallen on stony ground; it
lacked depth of earth; the leaves were fair to the eye, but the root of
humility was wanting.
On the following day Ada took her departure. Her visit was not
without its effect, both on herself and her cousin; for it is a solemn
consideration that two beings can seldom mix in close and familiar
intercourse without exercising some degree of influence on each other
for evil or for good.
What she had seen and heard at Laurel Bank had rendered Ada
in some degree discontented with herself. She had seen something of
the beauty of a life of holiness and benevolence--at least so it
appeared to her mind; and it sickened her to contrast with it her own
course of selfishness and frivolity. Not, perhaps, that the impression
was a very deep one, or that Ada had the slightest present intention
of following the example which she admired; but she had a vague
hope that a day might come, perhaps when the spring-time of youth
should be over, when she too might be of some use in the world, and
live for some object more noble than to flutter through a round of
gaiety amidst those whose friendship would be as lightly lost as it had
been lightly given.
The effect of Ada's visit was very different upon Flora. It had not
humbled, but rather confirmed her in her false estimate of her own
character. At the same time it had awakened within her bosom a

secret discontent at her own quiet lot, a yearning for the more
brilliant and exciting scenes which her cousin loved to describe. Flora
began to think--though she breathed the thought to no one--that
living in the complete seclusion in which the will of Providence had
placed her, was in truth a serious disadvantage. She cared less for the
beauties of her garden, spent more time at her toilette, and as she
looked at the lovely reflection in her mirror, she turned over in her
mind, as a miser might his treasures, the flattering words of
admiration which she had heard from the lips of Ada. Her school
children seemed to her duller than usual; her thoughts wandered
greatly at prayers; the conversation of the poor old lame captain grew
insufferably tedious; and when Miss Butterfield paid one of her long,
tiresome visits, Flora took care not to appear at all, but left her
mother to entertain the old lady. As Eve, amidst all the charms of
Eden, looked longingly at the one forbidden tree, so Flora, surrounded
by blessings, inwardly repined at the decrees of Providence, yearning
for the one thing denied to her--denied to her by divine wisdom and
love!
Mrs. Vernon had less leisure to observe any change in her
beloved daughter, from being much occupied in making preparations
for the reception of the family from Barbadoes. She studiously
regarded Flora's comfort in all her arrangements, while she quite
neglected her own: but it was impossible to receive so large an
addition to her limited household without making some changes
which necessarily somewhat affected the convenience of her
daughter. Flora felt the petty sacrifices which she was compelled to
make, more than can be readily imagined by those who, from having
been members of large families, have been accustomed from
childhood to submit to them. She had been the object of her mother's

almost undivided attention and care, and had grown a little selfish
without being aware of it.
Nevertheless, Flora had a gentle, kindly heart. The situation of
her sister-in-law touched her compassion; she felt for the young
widow, bowed down by the double trial of poverty and bereavement,
quitting her native land to come amongst those who were strangers
to her. She was sure that she would love the dear helpless little
orphans; and she spoke so sweetly on the subject, seemed to make
so light of difficulties, was so ready to give herself to the congenial
task of comforting the afflicted, that never had her mother more
fervently thanked Heaven for such a child, or visitors left the house
more impressed with the idea that Flora was the impersonification of
every Christian virtue, as she was of every feminine charm!
At length dawned the long-expected day of the arrival. In a place
so retired as the village of Wingsdale, comparatively trifling
occurrences rose to the rank of important events. Flora, who was
imaginative and poetical, had drawn in her mind so touching, a
picture of the pale widow and her golden-haired cherubs, had
rehearsed to herself so often the scene of the meeting, that she had
worked herself into a state of eager impatience. Unable to settle
steadily to anything, she fluttered from room to room, now altering
the arrangement of the flowers with which she had adorned the
widow's pretty boudoir, now bringing some elegant trifle of her own to
add to the beauty of the effect. She pulled up the blind, that the view
might be seen; then drew it down again hastily, lest the glare from
without should fall painfully on the eye of sorrow. She had amused
herself for several evenings by preparing a pretty book of pictures for
the children, and had pleased herself by collecting little toys, with
which she doubted not to find a speedy road to their hearts.

At last, in the quiet village, appeared the unexpected apparition
of a post-chaise and four--an equipage which had never been seen
there since the county member came to canvass in Wingsdale. All the
little rustics ran eagerly to look at the unusual sight, and the cottagers
stood in their doorways as the vehicle rolled past in a cloud of dust,
with a quantity of luggage piled on the top, box upon box, the whole
heap surmounted by a parrot cage with its screaming tenant. But
what most excited the wonder of the rustics was the negro who sat
upon the box. The children stared with open eyes and mouths at his
black face, curly woollen hair, and thick lips, which, parted in a merry,
self-satisfied grin, displayed two rows of shining white teeth.
Through the village, sweeping past the church, up the shady
lane, dashed the post-chaise! The gate of Mrs. Vernon's little
shrubbery stood open, and along the narrow drive rolled the dusty
wheels, the horses' hoofs tearing up the carefully smoothed gravel,
which even the doctor had always respected, tying up his black nag at
the gate. At the sound of the arrival of the vehicle, Mrs. Vernon and
Flora hastened to the entrance, their faces expressive of the welcome
which their lips eagerly pronounced. The carriage-door was opened by
Flora, too impatient to wait till the grinning negro tumbled down from
the box: she stretched out her hands to receive some infant treasure
from the crowded chaise, and a cage with a squirrel, followed by a
bundle of shawls and a broken bonnet-box, were hastily thrust into
them. Before she could disencumber herself of the luggage, three of
the children had been handed, or rather tumbled, out of the carriage--
thin, sickly-looking creatures, wrapped up in gaudy scarfs, which
strangely contrasted with their faded black ribbons and dresses, and
complexions which varied in shades from whity-brown to dingy-yellow.
While Mrs. Vernon stooped to kiss and welcome the little strangers,

the youngest of whom, from some cause unknown, was crying as if
her heart would break, Flora pressed forward to assist the widow to
alight. Instead of the lady, a very fat negress, very gaudily attired,
with gilt bracelets on her wrists and beads in her ears, holding a
screaming baby in her arms, slowly shoved her stout person through
the doorway, and heavily descended to the ground.
"My dear sister!" exclaimed Flora, in a tone of sympathy, again
pressing to the carriage, and leaning forward into it to greet the
afflicted widow.
A very languid voice answered from within, "Just see, do, that
they are careful in taking down the parrot."
It was a chill to the feelings of Flora, and her first glance of her
sister-in-law was little calculated to re-warm them into interest. Emma
Vernon might once have been pretty, at least something in her air
conveyed the idea that she had--and perhaps still considered herself
to be so; but she had a sallow, withered look on her face, an affected
expression in her sleepy black eyes, a languid listlessness in her
manner, which to Flora were almost repulsive. Coldness and
indifference seemed conveyed in the very touch of her thin fingers,
and the cheek which, half hidden by a profusion of black curls, she
turned to receive Flora's kiss. Slowly, very slowly, she descended from
the carriage, leaning heavily on Mrs. Vernon and her daughter, and
moving as though she were scarcely equal to the effort of placing one
foot before the other. With many a pause, and many a sigh, she
reached the lovely little boudoir which Mrs. Vernon, at considerable
personal inconvenience, had appropriated to her use.

Slowly, very slowly, she descended from the
carriage.

Emma sank on the sofa, and in an affected voice exclaimed,
"Take away those flowers--I can't bear them!--take them away or I
shall faint!"
Flora hastened to remove the beautiful bouquet, while Mrs.
Vernon offered a scent-bottle to the languishing lady.
"I hope, dear Emma, that country quiet will soon restore you,"
she said soothingly.
"Do open the window--there's not a breath of air--this room is so
small!" lisped the newly arrived.
"How strange it is," thought Flora, "that she neither asks nor
thinks about her children! Nothing but her own comfort seems to
occupy her mind, and it does not appear very easy to please her. I will
go and look after her little ones."
Flora was followed into the passage by her mother, who looked a
little troubled and anxious.
"My dear child," said Mrs. Vernon, laying her hand on Flora's arm,
"what are we to make of the negro? I never calculated on Emma's
bringing a man-servant with her."
"I'm sure I don't know, mamma. It was very inconsiderate in her,
I think. But everything seems so strange and confusing and
uncomfortable, I am afraid--" she stopped in her sentence.
"We must make the best of everything, my love. Just go and see
that the poor dear children are comfortable in their nursery."
Flora obeyed without reply.
CHAPTER VI.
THE NURSERY.

As Flora approached the nursery, its vicinity was sufficiently indicated
by the sound of loud, passionate crying, and then that of several
sharp slaps; which made her quicken her steps, lest the black nurse,
whose looks she distrusted, should be maltreating any of the children.
The first glance at the interior of the room, however, showed her that
the fat old negress was not the giver, but the recipient of the blows!
Before her was a little boy in a furious tempest of passion, kicking,
striking, and roaring, while Flora's pretty book of pictures lay in a
hundred fragments at his feet!
"Oh! Massa Johnny, Massa Johnny!" exclaimed old Chloe in an
expostulating tone, as he struck her again and again with the ferocity
of a little tiger. Flora sprang forward and caught his hand, but only
turned his passion upon herself. The child clutched at her flowing
locks, and it was not without difficulty and pain that she extricated
her hair from his grasp. He then flung himself down on the floor, and
rolled on it in impotent passion.
"What can be the meaning of all this?" exclaimed Flora, surprised
and ruffled by the unexpected attack.
"Oh! Massa Johnny, he only want to pull de swing 'uns off de
clock; he very angry cause he cannot get em."
"I am afraid that Johnny is a very naughty boy," said Flora,
smoothing down her disordered tresses, and looking down with the
reverse of admiration on the dark little savage before her.
"Oh! Massa Johnny, he have great speerit, he have mighty great
speerit!" was the nonchalant reply, as the negress slowly rose from
her seat to attend to the baby, who had been sleeping in a cradle, but
who, awakened by the noise, now swelled it with his fretful cry.

"I'm sure, if one doesn't want a dozen hands atween them all!"
pursued the old woman, trying to hush the child; "there's Miss Lyddie
now, there's no knowing where she's agone--I've not set eyes on her
this half-hour!"
"Not know where she is!" said Flora, glancing round anxiously.
Emmie, the youngest child but one, was quietly amusing herself in a
corner, breaking off the legs of the wooden animals belonging to an
ark which Mrs. Vernon had provided for her amusement. But no trace
of Miss Lyddie was to be seen. Flora hurried from the room to search
for the little truant.
It was not long before she found her in the dining-room, close to
a small press in which various preserves and other little dainties were
kept. Lyddie was several years older than the other children, and tall
for her age; her lank over-grown form, untidy hair, awkward carriage
and sickly face, conveying to the mind the idea that she was like some
idle weed, which had sprung up uncared-for and untended, She
started slightly on seeing Flora, and hastily closed the door of the
press, which had stood a little ajar.
"O Lyddie!" exclaimed Flora, "it is very wrong indeed to take
sweets without asking leave!"
"I didn't!" said the child, shrinking back from her touch, and
eyeing her with a furtive glance.
"Look there!" cried Flora, pointing very gravely to some
unmistakable crimson stains on the dress and hands of the girl. "It is
still worse to tell an untruth about it."
The girl pouted, and put two fingers into her mouth.
"O Lyddie! has no one taught you who sees our actions, and--"
Flora was commencing a gentle, but very serious reproof, when it was
suddenly cut short by her auditor darting from the room.

"What dreadful children!" said poor Flora to herself; "they seem
more unmanageable, more uncared for, both as regards their physical
and moral condition, than the poorest cottager in the village! We must
speak seriously to their mother about them; it is impossible to let
them go on in this way."
To speak to Mrs. John Vernon was not at that moment
practicable, as she had gone to sleep on the sofa, and was on no
account to be disturbed. The dinner, which had been delayed for
some hours for her arrival, was thus again indefinitely postponed, as
both Mrs. Vernon and her daughter thought it more courteous to take
their meal with their guest, instead of sharing that prepared for the
children. Flora felt irritated and tired, and very little disposed to look
at the bright side of affairs. The noisy voices of the children seemed
never to be silent. They penetrated every part of the house; no room
appeared safe from the intrusion of unwelcome little guests: for a
spirit of active curiosity was a characteristic of Lyddie, and Johnny was
prosecuting a search for his negro Sambo, which carried him to places
where he was unlikely, as well as those where he was likely to find
him.
At length Emma awoke from her siesta, but Mrs. Vernon found
that her politeness towards the lady had been carried to an
unnecessary extent. Emma declined joining the family at table; she
preferred having her dinner carried to her in her boudoir. Flora,
desirous to please her new guest, herself took the refreshment to her,
and had the mortification to find that it consisted of the only thing, as
Emma declared with a sickly smile, that really she could not touch;
while, when pressed to say what she fancied, she named something
which it was difficult, if not impossible to procure!

Flora dined alone with her mother. This was a relief, for she was
weary, and out of spirits and out of patience. She resolved not to
trouble her mother more than she could possibly help with her own
annoyances and perplexities, for Mrs. Vernon looked harassed and
anxious already. When the lady had gone to superintend the sleeping
arrangements of the children, Flora sought the boudoir of her sister-
in-law, having previously rehearsed many times in her mind the
conversation which she thought might take place between them, and
having studied how she could tell painful truths in the most gentle
and least irritating way.
The widow was still reclining on the sofa, her cap put aside on
account of the heat, a fan and scent-bottle beside her; and she
received Flora with the languid, affected smile, which to that young
lady was peculiarly unpleasing.
"I hope that you have now recovered a little from your fatigue,"
said Flora, seating herself beside Emma.
The only reply was a languid sigh, accompanied by a slight
elevation of the eyebrows, and then a closing of the eyes.
Flora paused for awhile, and played with the clasp of her
bracelet, before she ventured to say, "Emma, there was one thing
which I wished to ask you--have you perfect confidence in your black
nurse?"
The lady opened her eyes. "Oh! she's the best creature in the
world!" Here the scent-bottle was in requisition.
"You know, of course, whether she is a Christian?"
"Well--oh! why"--(each word was drawled forth as though to
speak were too fatiguing)--"yes; she has a crucifix and beads; she is a
Christian, I am sure of it."

"And is it possible--" Flora felt herself beginning to warm with her
subject, but with an effort of self-control she commanded herself, and
proceeded in the same gentle tones as those in which she had
commenced.
"Do you think it desirable to trust her so entirely with the
children? I viewed a little scene in the nursery to-day which gave me
an idea that Johnny's temper requires more judicious management."
Emma looked so utterly indifferent, that Flora gave her a more
lively description of the little scene, and of her own unpleasant part in
it, than she had intended to have done.
"Poor dear! he has so much spirit!" was the only observation of
the mother.
"But I have more painful things to tell you," said Flora, feeling
utterly provoked; and without further reserve, she gave an account of
Lyddie's conduct at the press, which would greatly have distressed a
tender and conscientious parent, but which only elicited the words
"Poor dear!" uttered in a more sleepy tone than before.
"But, Emma, this must not be!" exclaimed Flora, with kindling
indignation; "these poor unhappy orphans are not to be left to acquire
habits of dishonesty and untruth--"
She stopped suddenly, for she knew that she had said too much;
she saw it in the malignant expression which lighted for a moment the
sleepy black eye, she felt it in the quick throbbing of her own heart.
Glad was Flora that her mother's entrance gave her an excuse for
quitting the room. She sought her own in a very bitter spirit.
"Mamma," said Flora to Mrs. Vernon, when they were both
retiring to rest, "I fear that we shall have a dreadful time with this
family! There will be no more comfort in the house. Those miserable,
neglected children will be the torments of our lives!"

"We must have patience with them, my love; they will not be
neglected here."
"But Emma will hang as a drag-chain on all our efforts to improve
them. She seems to regard nothing upon earth but her own comfort
and convenience, and listens with that odious smile to things which
should make a mother blush for very shame!"
"My love--" expostulated Mrs. Vernon.
"I do not like that woman, mother, and I am certain that I never
shall. She looks so heartless, and silly, and affected!"
"We must not be hard upon her, my Flora; her own education has
probably been neglected. You have had a day of fatigue and
excitement, and it re-acts on your own spirits, my dear. Go and rest
now, you need it; all may seem more sunshiny in the morning."
Flora sought rest, but that night she was not destined to find it.
In the room next to hers was the baby, and hour after hour his
wailing cry sounded in her ears, driving away sleep; thrice she started
up and hastened to see if nothing could be done to soothe him. As
morning dawned the babe fell asleep, and so did the exhausted Flora,
to dream of the house being attacked by a legion of negroes, till she
was awakened at an earlier hour than usual by the sound of a furious
quarrel between Johnny and Lyddie.
CHAPTER VII.
THE TOUCHSTONE.
And so passed day after day, each appearing to Flora more
unendurable than the last. She at first sought refuge in her favourite

woody haunts, and there reading some pleasant book, or throwing
her own thoughts into verse, she enjoyed brief but delicious respite
from the cares and vexations of her home. But a season of rainy
weather cut off even this source of enjoyment, and she was
imprisoned in the house with four noisy, quarrelsome children, and a
companion whom she disliked and despised.
Mrs. John Vernon was a very weak woman, and had all the
infirmities and follies naturally attendant on characters of such a
stamp. She was, of course, passionately fond of dress--a fondness
which even the necessity for wearing mourning did not subdue. Flora
could scarcely disguise her contempt when she saw how completely
the thoughts and time of this vain, silly woman, were engrossed by
the cares of the toilette; how she squandered money, which she
would perhaps have to borrow, In decking out her person in all the
extravagance of fashion. "As if," Flora thought to herself, "all the silks
and lace in the world could ever give the shadow of beauty to that
insipid, affected face!"
Emma was unpunctual in the extreme; and this, to one
accustomed to the clock-work regularity of a small, well-ordered
household, was a defect of no small magnitude. It interfered with the
comfort of every one, and sorely tried the patience of Flora.
The widow was also absurdly fanciful about her own health. She
was afraid of exertion, afraid of cold; and, by some unfortunate
contrariety, her opinion on the subject of the weather never seemed
to coincide with that of those around her. When Mrs. Vernon felt chilly,
Emma was certain to be longing for open windows and air; if Flora
found the room oppressively warm, Emma languidly suggested a fire.
The widow loved to draw upon herself the attention of all who
approached her; she would rather have been disliked than unnoticed;

she must attract the observation, occupy the thoughts of all, or she
felt herself wronged and neglected.
Gently did Mrs. Vernon bear with the infirmities of one whom it
was difficult to love, impossible to respect. Gradually and quietly she
made suggestions on the management of Emma's family, or the
arrangement of her pecuniary affairs. But tenderly as every hint was
given, it was received either with irritation or peevish distress; and
when, at length, detected instances of dishonesty on the part of both
Sambo and the nurse induced Mrs. Vernon gently, but firmly, to urge
the necessity for dismissing them both, her words occasioned a burst
of passionate tears, succeeded by a long fit of depression.
"Mamma," said Flora one morning to her parent, as they sat
together at breakfast, a meal which was never graced by the
presence of the widow, who kept her own room till noonday--
"Mamma, I do not think that we can endure all this much longer. It is
impossible to please Emma, whatever we do: let her set up house for
herself, and manage as she may!"
Mrs. Vernon looked very grave, and it was some moments before
she replied. "I think, Flora, that you can scarcely have reflected on
what would be the result of such an attempt. You can scarcely have
failed to observe how careless poor Emma is of money, how unable
she is to manage a household, or to keep account of its expenses.
She would certainly involve herself inextricably in debt, while the
consequences to the unhappy orphans must be such as would deeply
distress us both."
"They could scarcely be worse than they are," said Flora, bitterly;
"wild, ignorant, unmanageable little creatures. I have attempted
several times to teach Lyddie, but she has always darted away like a
little wild colt; while Master Johnny, the other day, threw the spelling-

book out of the window. Their footprints are over all my borders, their
fingers cannot be kept off my flowers; Lyddie strewed the walks
yesterday with apple-blossoms, while Emmie managed to get hold of
my paint-box, and has not only mixed all the colours together, but has
left traces of them on a dozen of our books."
"It is very trying; it quite distresses me, my Flora, to see the
annoyance and discomfort which you suffer. Night after night I lie
awake, turning over in my mind by what means I can prevent the
inconvenience from falling upon my child. But no path seems to open
before me. Emma is as unfit to keep house for herself as her own
Lyddie would be; and I feel--I am sure that you feel--that, as long as
we have a home, the orphan grandchildren of your beloved father
should never be denied its shelter!"
Flora pressed her mother's hand fondly to her lips, "Oh, mamma!
you are so good!" she exclaimed; "and what a return do you meet! I
do believe that if we were to give up our house altogether, or only to
remain in it as servants, slaving from morning till night, and denying
ourselves common comforts that Emma might enjoy every luxury, she
would take it quite as a matter of course, think that everything was as
it should be, and feel not one spark of gratitude towards us, whatever
our sacrifices might cost us!"
There was much truth in Flora's remark. In a mind mean and
selfish as the widow's, gratitude has rarely a place. Alas! that in the
world it should be a virtue so rare! Not that I would for a moment
swell with my voice that cry so common, yet often so unjust, which
indiscriminately charges the poor with ingratitude towards their
benefactors. Far from it; in this virtue, as in many others, I believe
that the comparatively rich may often learn a lesson from the poor.
There are perhaps few in the world who have no opportunity of

exercising gratitude, few who lie under no obligations either for
substantial services, or for kindly attentions; watchful care in infancy,
help in difficulty, generous hospitality, or some other of the thousand
acts of benevolence and friendship which so sweeten the cup of
human life. Yes, the many are laid under obligations, but the few have
the candour to acknowledge them; the many are helped, benefited,
and cheered--the few gratefully remember the benefactor. The lepers
in the gospel are still types of human nature. Ten were cleansed, but
where are the nine? Reader! pause a moment; ask your own heart,
do you treasure up the remembrance of benefits--do you carefully
keep up the warm glow which perhaps kindled in your heart when you
first received them? Or has the cold wave of time chilled the generous
warmth of your feelings--or, worse still, did that warmth never exist?
From such observations as I have been enabled to make, it seems to
me to be almost a general rule that the most truly generous are also
the most grateful--that those who most readily do acts of kindness,
most thankfully acknowledge them from others.
Nor let us think want of gratitude a light sin, or one which we
may safely overlook. The same proud, thankless spirit which leads us
to forget our obligations to man, is at the root of our unbelief, our
indifference, our coldness towards Him who is the giver of all good.
We receive our blessings as rights; we think little of the mercy which
bestowed them, or we should scarcely dare to murmur and repine
when the smallest is taken away from us. When Flora accused her
sister-in-law of ingratitude, she little thought how well the charge
might have been retorted on herself. Had she not been loaded with
mercies--granted health, strength, all the comforts of life, opportunity
of benefiting others, and power of pleasing--the love of her friends,

the deep tenderness of a mother--and, above all, innumerable
spiritual blessings, the means of grace, and the hope of glory!
And yet, with all this, the heart which the world deemed so pure,
the heart whose depths she had never yet fathomed, was now filled
with a bitter, almost a rebellious spirit. Flora had worked--was well
pleased to work for God, but it must be in her own way; she could
make sacrifices for religion, but the choice of the sacrifice must be her
own. It was as though the soldier who for years had glittered on
parade, and performed the routine of daily duty with faultless
regularity in time of peace, had started back when the war-trumpet
sounded, had turned from the sterner obligations before him, and
murmured because he was called upon at last to "endure hardness,"
and to face trial in a holy cause.
Flora was still ready to sit by the sick, to visit the dying, and to
teach in the school; she was still willing to give freely to the poor,
looking for a plenteous reward hereafter, and receiving in the present
the interest of human gratitude, admiration, and love. But she was
not ready to be "kind to the unthankful and the evil," to "let patience
have its perfect work," to strive to reclaim wilful and unruly children,
with the prospect of awakening the jealousy of their parent, but never
of rousing her to a sense of obligation. Flora's religion was not "the
love of Christ" which "constraineth," therefore in the time of trial it
failed her.
Consideration for her mother usually restrained Flora from
making audible complaints, though she had not sufficient command
over herself to abstain from them altogether; but she indemnified
herself for her forbearance by writing to Ada full and circumstantial
details of all her petty miseries, with a by no means flattering
description of the family from Barbadoes. It was a letter which Flora

would not willingly have seen in the hands of her revered pastor; she
would never have addressed it to her mother; she had some doubts,
after having finished it, whether it would be well to post it. But it was
really a clever and amusing letter; it eased her heart to write it; she
was glad to have some way of giving vent to the pent-up flood of
bitterness which was beginning to overflow its bounds.
The letter brought a speedy reply, containing an affectionate and
urgent invitation to Flora to join her cousin in London, giving a
glowing description of the amusements which she would enjoy, while
a P.S. entreated her not to delay her visit, lest she should lose all the
May-meetings in Exeter Hall, which Ada was "sure that to one so good
would be a greater pleasure than all the rest."
Flora uttered an involuntary exclamation of delight as she
perused the letter of her cousin. It was as though a caged bird had
suddenly seen the door of his prison open, and the way free to liberty
and to sunshine. She was full of impatience to show Ada's epistle to
her mother, and could scarcely endure the delay occasioned by Mrs.
Vernon's having to examine into the cause of a furious dispute
between Johnny and his elder sister, and then to administer gentle
advice and reproof to each of the little offenders. The interruption
appeared to Flora so vexatious and petty--she would willingly have
ended it at once by sending both the children away to the most
distant part of the house, to settle their disputes by themselves; but
her mother calculated more truly the importance of whatever regards
the training of immortal beings.
At length, however, Johnny and Lyddie were dismissed, having
been, after much trouble, induced to exchange the kiss of
forgiveness; and the door had scarcely closed behind them, when
Flora placed the letter in the hand of her mother. Eagerly she watched

the expression of Mrs. Vernon's countenance as she read it. The lady
perused it to the end before she uttered a word, and then she
glanced up with a smile.
"What do you say to this, Flora?"
"Oh, mamma--it is just as you like--just as you think best--but--"
"This invitation seems to meet a difficulty which has pressed
heavily on my mind. I have grieved to feel how trying to you has been
the change in our family arrangements. You have grown thinner and
paler; your spirits have left you; for the first time in my life it has
pained me to look at my child, It is better, perhaps, that you should
be absent from home till we bring matters into a somewhat better
train."
"It seems almost like deserting you, mamma; and yet--I do not
think that I can help you much--I have not the least influence with the
children."
Why was it that even Flora knew that her absence at this time
would be actually a relief to her mother? How was it that she had
proved a burden rather than a helper? She had never put the question
fairly to her own heart, and was very glad to substitute for it another.
"Does it not seem to you, mamma, as though I might be more
useful in London than I am here? I believe that poor dear Ada really
likes me; I have some influence with her, I believe: she seemed here
to be turning her mind more towards religion than she hitherto had
done; but she has no one now to speak to or consult on serious
subjects. If I were with her she would be induced for my sake to go
to meetings which she would not otherwise attend--it seems to me
that it may really be my duty to go to Ada at this time."
"It will be your pleasure, at least," said Mrs. Vernon, smiling;
"and your happiness is ever near to my heart. I can trust your

principles, my Flora; I believe that you will ever act in my absence as
you would if my eye were upon you. And the Almighty may, and I
trust will, make you, my love, a blessing to others, if you serve Him
with a humble, devoted heart and a single eye to His glory."
CHAPTER VIII.
PLEASURES AND PAINS.
A few days after the conversation recorded in our last chapter took
place, Flora, full of youthful hope and joy, sprang into the carriage
which was to convey her to the station, and waved again and again a
fond farewell to the beloved parent who watched her departure from
the gate.
It was with mixed emotions that the gentle widow beheld
disappearing down the winding lane the carriage which held her
dearest earthly treasure, separated from her for the first time. Mrs.
Vernon took pleasure in her daughter's pleasure, and had, perhaps,
secret pride in the thought that her beauty, talents, and virtues, would
now be more widely known and appreciated. But there was pain also
in parting; pain with which the meek parent reproached herself--that
Flora could be so happy in parting! There was a secret fear, which
Mrs. Vernon thought want of faith, lest the different scenes into which
she was entering should, were it even in the slightest degree, change
one who, in her partial love, she thought could scarcely change for
the better. Mrs. Vernon also suffered from the cares and anxieties of
life, which she now must bear alone; for she was not one to complain,
even to her most intimate friends, of the secret trials of her home.

The peevishness, the selfishness, the heartlessness of Emma, the
wayward passions of her ill-taught children, the loss of the quiet
repose of a well-ordered dwelling, were a cross to Mrs. Vernon as well
as to her daughter; if to the latter it formed the most painful burden,
it was because she murmured, struggled, and chafed under its
weight, while the widow bent meekly under it, remembering the
divine hand that had laid it upon her.
So, quietly and unostentatiously, never dreaming either of merit
or reward, the widow went through her round of daily duties, ordering
her household, teaching the children, caring for her guest, nursing the
infant, and never forgetting the poor. Emma's total indifference on the
subject of religion often grieved Mrs. Vernon; but trusting in God, and
not in herself, the simple Christian would not despair even of a heart
which seemed like the beaten highway, on which the good seed fell
only at once to be carried away. There was no use in lending religious
works to Emma; the volumes lay unopened beside her. She never
considered herself equal to the fatigue of attending service in the
house of God. Mr. Ward and his wife paid her more than one visit. The
good man spoke, as was his wont, out of the abundance of his heart;
while Mrs. Vernon listened meekly with her clasped hands resting on
her knee. But even he was chilled by the affected nod and
meaningless smile with which the daughter-in-law received his words
of holy consolation; while his wife felt uncomfortable under the dark
eye which seemed scrutinizing every article of her simple apparel.
Not every one bore as patiently as Mrs. Ward this scrutinizing
survey of dress from the fashionable and extravagantly-attired young
widow. Miss Butterfield, who had a character and a temper of her
own, was irritated by the close attention paid to her large poke-
bonnet and rusty shawl. She made some observations, more true than

polite, about heads like band-boxes in a milliner's shop, intended to
hold nothing more weighty than quilings and puffings; which brought
an angry tinge to Emma's sallow cheek, and made her bitterly
comment, when the guest had departed, upon the insufferable
vulgarity of Mrs. Vernon's country acquaintance.
"How can I win Emma's attention to anything serious?" such was
Mrs. Vernon's frequent thought, till one day the happy idea struck her
mind of reading to her Flora's manuscript hymns. "If any human
writings can interest her, these will," thought the simple-hearted
mother. "She will listen to them first for the sake of the authoress,
and then their own beauty must touch her heart;--I am sure that it
always does mine!"
Emma could not, of course, refuse her assent to the proposal to
read aloud the verses of her sister. She declared that she would be
charmed to hear them, secretly hoping that the infliction might not be
long, and that her mother-in-law would not think it necessary to go
through the volume from beginning to end. Mrs. Vernon read with
great impressiveness and feeling; every touching sentiment, every
graceful idea, gained added beauty from her earnest expression. She
was pleased and gratified by the profound silence of her listener, and
read on, and on, warming with her subject, till in one favourite hymn,
which described the blessedness of living for eternity, her eyes filled,
and her heart overflowed, and she turned, as she wiped away a
thankful tear, to see if Emma shared her emotions. The widow lay fast
asleep on the sofa!
The only things to which Emma listened with real interest were
portions of Flora's letters from London. These, written to amuse her
mother in her seclusion, and full of lively descriptions given with
freshness and vigour by one to whom everything which she beheld

was new, were, even to a stranger, extremely entertaining. By Mrs.
Vernon the arrival of the post was looked forward to as bringing the
one great treat of the day, and never once was she disappointed of it.
She feasted on the letters of her daughter with unmingled delight; for
she saw in them proofs of the conscientious regard which Flora paid
to her wishes, and of the tender affection with which, in the midst of
her amusements, her heart clung to the parent whom she had left.
All the little circle of friends in the quiet village of Wingsdale
shared in Mrs. Vernon's enjoyment. They listened to accounts of the
first wondrous Crystal Palace, glittering like some fairy structure on
the trodden sward of Hyde Park; the "sermons in stones" preached
from the spoils of old Nineveh; descriptions of the treasures of art, all
the things beautiful, curious, and rare, upon which the eye of Flora
had rested delighted. They heard also personal descriptions of men
with whose names they were already familiar--how Shaftesbury had
spoken, and Guthrie had preached; whilst not least interesting to the
hearts of her rural subjects was a graphic account of our gracious
Queen from the enthusiastic pen of Flora.
But never had the young correspondent expressed herself in such
glowing terms of admiration and pleasure as in her description of Sir
Amery Legrange, whom she spoke of as one of the leading writers of
the day. She had been actually introduced to "the lion," had listened
to the wondrous flow of eloquence which made his conversation an
intellectual treat beyond any which she had ever known before. With
his expansive brow, eagle eye, most poetical cast of countenance, his
were exactly the face and form which a painter would wish to
immortalize on canvas, as representing the beau ideal of a genius;
and Flora could not but imagine that Sir Amery must have drawn
himself in the hero of his famous chef-d'oeuvre, "The Master-Mind."

"'The Master-Mind!'" lisped Emma from the sofa; "Oh! I have
read that--all the world has read it--it is a most charming work! I have
it somewhere in my boxes, I think."
Mrs. Vernon was content to be classed with those not of the
world, for she had never read, nor even heard of the book. She was
pleased, however, that her child should have met one of the literary
celebrities of London, and had thus added another to the pleasant
recollections which she would carry with her from the metropolis.
The next morning's post brought a description of a fête at the
Botanical Gardens; and this sentence occurred in Flora's letter: "We
met our brilliant author just as we were entering the gardens, and he
remained the whole time with our party. He has certainly wonderful
powers of conversation; just such as might be expected from a writer
whose pen seems dipped in the colours of the rainbow, and brightens
whatever it touches. When he speaks, we can do nothing but listen."
"I think, Emma," said Mrs. Vernon, "that you mentioned that you
had a copy of 'The Master-Mind' beside you. I should be obliged, if
you would allow me to read it."
"Oh, certainly, when I can lay hands upon it; but all my luggage
is still one mass of confusion, and whatever I want is certain to be at
the very bottom of the very last box into which I should think of
looking for it. But you really wish to see 'The Master-Mind?' Well,"
added the lady, with an affected laugh, "I should as soon have
dreamed of Mr. Ward's dancing a polka, as of your sitting quietly
down to a novel!"
Emma's promise to look for the work was speedily forgotten by
herself, nor did more than one reminder induce her to take this slight
trouble for one to whom she owed the comfort of a home.

Flora's subsequent letters contained scarcely any mention of Sir
Amery. She occasionally quoted his opinion, or mentioned a brilliant
remark made by him on some subject of general interest; but it was
merely from such passing allusions that her mother gathered that
Flora was not unfrequently in the society of the literary "lion."
Mrs. Vernon had much to occupy her thoughts--much to engage
her anxious attention. She was now patiently listening to the frivolous
complaints of the hypochondriac widow; then quitting her to stand by
the death-bed of a young school-girl--hear her last faint breathings of
devotion--receive her last message of affection to Flora. The next
hour would find Mrs. Vernon seated amongst the children, enduring
their rough play and noisy glee, and ministering to their amusement
as patiently as she had done to the wants of the suffering and the
dying. But the wear upon her spirits and the strain upon her energies
were too much for the strength of Mrs. Vernon. There was no one to
watch her faded cheek, her weary step, her languid eye; no one, at
least in her own home, to think for her, care for her--attend to the
comforts of one who ever attended to the comforts of others. She
might have been ill, she might have been dying, and Emma, absorbed
in her own selfish cares, would never have observed that anything
ailed her. Mrs. Vernon missed Flora each day more and more--longed
more and more to hear her light step on the stair, her sweet song
from the garden--to look again into those soft and loving eyes which
were wont to rest on her so tenderly, so fondly. But Mrs. Vernon
would not abridge a visit which afforded so much enjoyment to her
daughter; she would not, even by the slightest allusion to her own
failing health and spirits, throw a shadow over that enjoyment. She
was content if Flora was happy; and if she herself needed comfort

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Introduction To Differential Geometry Luther Pfahler Eisenhart

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