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Full text of "A course in mathematical analysis"

STAT, 



"VRT 



A COURSE IN 



BY 

EDOUARD GOURSAT 

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PARIS 
TRANSLATED BY 

EARLE RAYMOND HEDRICK 

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MISSOURI 



VOL. I 

DERIVATIVES AND DIFFERENTIALS 

DEFINITE INTEGRALS EXPANSION IN SERIES 

APPLICATIONS TO GEOMETRY 



GINN AND COMPANY 

BOSTON NEW YORK CHICAGO LONDON 
ATLANTA DALLAS COLUMBUS SAN FKANCISCO 






STAT. 

LIBRARY 




ENTERED AT STATIONERS 1 HALL 



COPYRIGHT, 1904, BY 
EARLE RAYMOND HEDRICK 



ALL RIGHTS RESERVED 

PRINTED IN THE UNITED STATES OF AMERICA 
426.6 





jgregg 



GINN AND COMPANY PRO 
PRIETORS BOSTON U.S.A. 



AUTHOR S PREFACE 



This book contains, with slight variations, the material given in 
my course at the University of Paris. I have modified somewhat 
the order followed in the lectures for the sake of uniting in a single 
volume all that has to do with functions of real variables, except 
the theory of differential equations. The differential notation not 
being treated in the " Classe de Mathematiques speciales," * I have 
treated this notation from the beginning, and have presupposed only 
a knowledge of the formal rules for calculating derivatives. 

Since mathematical analysis is essentially the science of the con 
tinuum, it would seem that every course in analysis should begin, 
logically, with the study of irrational numbers. I have supposed, 
however, that the student is already familiar with that subject. The 
theory of incommensurable numbers is treated in so many excellent 
well-known works f that I have thought it useless to enter upon such 
a discussion. As for the other fundamental notions which lie at the 
basis of analysis, such as the upper limit, the definite integral, the 
double integral, etc., I have endeavored to treat them with all 
desirable rigor, seeking to retain the elementary character of the 
work, and to avoid generalizations which would be superfluous in a 
book intended for purposes of instruction. 

Certain paragraphs which are printed in smaller type than the 
body of the book contain either problems solved in detail or else 

*An interesting account of French methods of instruction in mathematics will 
be found in an article by Pierpont, Bulletin Amer. Math. Society, Vol. VI, 2d series 
(1900), p. 225. TRANS. 

t Such books are not common in English. The reader is referred to Pierpont, 
Theory of Functions of Real Variables, Ginn & Company, Boston, 1905; Tannery, 
Lemons d arithiiietique, 1900, and other foreign works on arithmetic and on real 
functions. 

iii 



7814G2 



iv AUTHOR S PREFACE 

supplementary matter which the reader may omit at the first read 
ing without inconvenience. Each chapter is followed by a list of 
examples which are directly illustrative of the methods treated in 
the chapter. Most of these examples have been set in examina 
tions. Certain others, which are designated by an asterisk, are 
somewhat more difficult. The latter are taken, for the most part, 
from original memoirs to which references are made. 

Two of my old students at the Ecole Normale, M. Emile Cotton 
and M. Jean Clairin, have kindly assisted in the correction of proofs ; 
I take this occasion to tender them my hearty thanks. 

E. GOURSAT 
JANUARY 27, 1902 



TRANSLATOR S PREFACE 

The translation of this Course was undertaken at the suggestion 
of Professor W. F. Osgood, whose review of the original appeared 
in the July number of the Bulletin of the American Mathematical 
Society in 1903. The lack of standard texts on mathematical sub 
jects in the English language is too well known to require insistence. 
I earnestly hope that this book will help to fill the need so generally 
felt throughout the American mathematical world. It may be used 
conveniently in our system of instruction as a text for a second course 
in calculus, and as a book of reference it will be found valuable to 
an American student throughout his work. 

Few alterations have been made from the French text. Slight 
changes of notation have been introduced occasionally for conven 
ience, and several changes and .additions have been made at the sug 
gestion of Professor Goursat, who has very kindly interested himself 
in the work of translation. To him is due all the additional matter 
not to be found in the French text, except the footnotes which are 
signed, and even these, though not of his initiative, were always 
edited by him. I take this opportunity to express my gratitude to 
the author for the permission to translate the work and for the 
sympathetic attitude which he has consistently assumed. I am also 
indebted to Professor Osgood for counsel as the work progressed 
and for aid in doubtful matters pertaining to the translation. 

The publishers, Messrs. Ginn & Company, have spared no pains to 
make the typography excellent. Their spirit has been far from com 
mercial in the whole enterprise, and it is their hope, as it is mine, 
that the publication of this book will contribute to the advance of 
mathematics in America. E R HEDRICK 

AUGUST, 1904 



CONTENTS 

CHAPTER PAGE 

I. DERIVATIVES AND DIFFERENTIALS 1 

I. Functions of a Single Variable 1 

II. Functions of Several Variables 11 

III. The Differential Notation 19 

II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGE 

OF VARIABLE 35 

I. Implicit Functions ........ 35 

II. Functional Determinants ...... 52 

III. Transformations ... .... 61 

III. TAYLOR S SERIES. ELEMENTARY APPLICATIONS. MAXIMA 

AND MINIMA ........ 89 

I. Taylor s Series with a Remainder. Taylor s Series . 89 
II. Singular Points. Maxima and Minima . . . .110 

IV. DEFINITE INTEGRALS ........ 134 

I. Special Methods of Quadrature . . . . .134 
II. Definite Integrals. Allied Geometrical Concepts . . 140 

III. Change of Variable. Integration by Parts . . .166 

IV. Generalizations of the Idea of an Integral. Improper 

Integrals. Line Integrals ...... 175 

V. Functions defined by Definite Integrals .... 192 

VI. Approximate Evaluation of Definite Integrals . .196 

V. INDEFINITE INTEGRALS 208 

I. Integration of Rational Functions ..... 208 

II. Elliptic and Hyperelliptic Integrals .... 226 

III. Integration of Transcendental Functions . . .236 

VI. DOUBLE INTEGRALS ........ 250 

I. Double Integrals. Methods of Evaluation. Green s 

Theorem 250 

II. Change of Variables. Area of a Surface . . . 264 

III. Generalizations of Double Integrals. Improper Integrals. 

Surface Integrals ....... 277 

IV. Analytical and Geometrical Applications . . . 284 



viii CONTENTS 

CHAPTER PAGE 
VII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER 
ENTIALS 296 

I. Multiple Integrals. Change of Variables . . . 296 
II. Integration of Total Differentials . . . . .313 

VIII. INFINITE SERIES . . 327 

I. Series of Real Constant Terms. General Properties. 

Tests for Convergence 327 

II. Series of Complex Terms. Multiple Series . . . 350 
III. Series of Variable Terms. Uniform Convergence . . 360 

IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375 

I. Power Series of a Single Variable . . . . . 375 

II. Power Series in Several Variables ..... S94 

III. Implicit Functions. Analytic Curves and Surfaces . 399 

IV. Trigonometric Series. Miscellaneous Series . . .411 

X. PLANE CURVES 426 

I. Envelopes 426 

II. Curvature 433 

III. Contact of Plane Curves 443 

XI. SKEW CURVES 453 

I. Osculating Plane ........ 453 

II. Envelopes of Surfaces . . . . . . . 459 

III. Curvature and Torsion of Skew Curves .... 468 

IV. Contact between Skew Curves. Contact between Curves 

and Surfaces ........ 486 

XII. SURFACES 497 

I. Curvature of Curves drawn on a Surface . . . 497 

II. Asymptotic Lines. Conjugate Lines .... 506 

III. Lines of Curvature . . . . . . . .514 

IV. Families of Straight Lines 526 

INDEX . 541 




CHAPTER I 

DERIVATIVES AND DIFFERENTIALS 
I. FUNCTIONS OF A SINGLE VARIABLE 

1. Limits. When the successive values of a variable x approach 
nearer and nearer a constant quantity a, in such a way that the 
absolute value of the difference x a finally becomes and remains 
less than any preassigned number, the constant a is called the 
limit of the variable x. This definition furnishes a criterion for 
determining whether a is the limit of the variable x. The neces 
sary and sufficient condition that it should be, is that, given any 
positive number e, no matter how small, the absolute value of x a 
should remain less than e for all values which the variable x can 
assume, after a certain instant. 

Numerous examples of limits are to be found in Geometry 
and Algebra. For example, the limit of the variable quantity 
x = (a 2 m 2 ) / (a m), as m approaches a, is 2 a ; for x 2 a will 
be less than e whenever m a is taken less than e. Likewise, the 
variable x = a 1/n, where n is a positive integer, approaches the 
limit a when n increases indefinitely ; for a x is less than e when 
ever n is greater than 1/e. It is apparent from these examples that 
the successive values of the variable x, as it approaches its limit, may 
form a continuous or a discontinuous sequence. 

It is in general very difficult to determine the limit of a variable 
quantity. The following proposition, which we will assume as self- 
evident, enables us, in many cases, to establish the existence of a limit. 

Any variable quantity which never decreases, and which ahvays 
remains less than a constant quantity L, approaches a limit I, which 
is less than or at most equal to L. 

Similarly, any variable quantity which never increases, and which 
always remains greater than a constant quantity L , approaches a 
limit l } which is greater than or else equal to L . 

1 



2. DERIVATIVES AND DIFFERENTIALS [I, 2 

For example, if each of an infinite series of positive terms is 
less, respectively, than the corresponding term of another infinite 
series of positive terms which is known to converge, then the first 
series converges also ; for the sum 2 n of the first n terms evidently 
increases with n, and this sum is constantly less than the total sum 
5 of the second series. 

2. Functions. When two variable quantities are so related that 
the value of one of them depends upon the value of the other, they 
are said to be functions of each other. If one of them be sup 
posed to vary arbitrarily, it is called the independent variable. Let 
this variable be denoted by x, and let us suppose, for example, 
that it can assume all values between two given numbers a and b 
(a < b). Let y be another variable, such that to each value of x 
between a and b, and also for the values a and b themselves, there 
corresponds one definitely determined value of y. Then y is called 
a function of x, defined in the interval (a, b) ; and this dependence 
is indicated by writing the equation y =/(z). For instance, it may 
happen that y is the result of certain arithmetical operations per 
formed upon x. Such is the case for the very simplest functions 
studied in elementary mathematics, e.g. polynomials, rational func 
tions, radicals, etc. 

A function may also be defined graphically. Let two coordinate 
axes Ox, Oy be taken in a plane ; and let us join any two points A 
and B of this plane by a curvilinear arc .4 CB, of any shape, which 
is not cut in more than one point by any parallel to the axis Oy. 
Then the ordinate of a point of this curve will be a function of the 
abscissa. The arc A CB may be composed of several distinct por 
tions which belong to different curves, such as segments of straight 
lines, arcs of circles, etc. 

In short, any absolutely arbitrary law may be assumed for finding 
the value of y from that of x. The word function, in its most gen 
eral sense, means nothing more nor less than this : to every value of 
x corresponds a value of y. 

3. Continuity. The definition of functions to which the infini 
tesimal calculus applies does not admit of such broad generality. 
Let y =f(x) be a function defined in a certain interval (a, b), and 
let x and x -f h be two values of x in that interval. If the differ 
ence f(x -f A) f(xo) approaches zero as the absolute value of h 
approaches zero, the function f(x} is said to be continuous for the 
value x . From the very definition of a limit we may also say that 



I, 3] FUNCTIONS OF A SINGLE VARIABLE 3 

a function f(x) is continuous for x x if, corresponding to every 
positive number e, no matter how small, we can find a positive num 
ber 77, such that 

|/(*o + A)-/(*o)|< 

for every value of h less than rj in absolute value.* We shall say that 
a function f(x) is continuous in an interval (a, b) if it is continuous 
for every value of x lying in that interval, and if the differences 



each approach zero when h, which is now to be taken only positive, 
approaches zero. 

In elementary text-books it is usually shown that polynomials, 
rational functions, the exponential and the logarithmic function, 
the trigonometric functions, and the inverse trigonometric functions 
are continuous functions, except for certain particular values of 
the variable. It follows directly from the definition of continuity 
that the sum or the product of any number of continuous functions 
is itself a continuous function ; and this holds for the quotient of 
two continuous functions also, except for the values of the variable 
for which the denominator vanishes. 

It seems superfluous to explain here the reasons which lead us to 
assume that functions which are defined by physical conditions are, 
at least in general, continuous. 

Among the properties of continuous functions we shall now state 
only the two following, which one might be tempted to think were 
self-evident, but which really amount to actual theorems, of which 
rigorous demonstrations will be given later, f 

I. If the function yf(x) is continuous in the interval (a, b), and 
if N is a number between f (a) andf(b), then the equation f(x) = N 
has at least one root between a and b. 

II. There exists at least one value of x belonging to the interval 
(a, b ), inclusive of its end points, for which y takes on a value M 
which is greater than, or at least equal to, the value of the function at 
any other point in the interval. Likewise, there exists a value of x 
for which y takes on a value m, than which the function assumes no 
smaller value in the interval. 

The numbers M and m are called the maximum and the minimum 
values of f(x), respectively, in the interval (a, b*). It is clear that 

* The notation | a \ denotes the absolute value of a. 
t See Chapter IV. 



4 DERIVATIVES AND DIFFERENTIALS [I, 4 

the value of x for which /(ce) assumes its maximum value M, or the 
value of x corresponding to the minimum m, may be at one of the 
end points, a or b. It follows at once from the two theorems above, 
that if N is a number between M and m, the equation /() = N has 
at least one root which lies between a and b. 

4. Examples of discontinuities. The functions which we shall study 
will be in general continuous, but they may cease to be so for 
certain exceptional values of the variable. We proceed to give 
several examples of the kinds of discontinuity which occur most 
frequently. 

The function y = 1 / (x a) is continuous for every value x of 
x except a. The operation necessary to determine the value of y 
from that of x ceases to have a meaning when x is assigned the 
value a ; but we note that when x is very near to a the absolute 
value of y is very large, and y is positive or negative with x a. 
As the difference x a diminishes, the absolute value of y increases 
indefinitely, so as eventually to become and remain greater than any 
preassigned number. This phenomenon is described by saying that 
y becomes infinite when x = a. Discontinuity of this kind is of 
great importance in Analysis. 

Let us consider next the function y = sin 1/z. As x approaches 
zero, I/a; increases indefinitely, and y does not approach any limit 
whatever, although it remains between + 1 and 1. The equation 
sin l/a; = ,4, where A \ < 1, has an infinite number of solutions 
which lie between and e, no matter how small e be taken. What 
ever value be assigned to y when x 0, the function under con 
sideration cannot be made continuous for x = 0. 

An example of a still different kind of discontinuity is given by 
the convergent infinite series 






When x approaches zero, S (x~) approaches the limit 1, although 
5 (0) = 0. For, when x = 0, every term of the series is zero, and 
hence 5 (0) = 0. But if x be given a value different from zero, a 
geometric progression is obtained, of which the ratio is 1/(1 + a; 2 ). 
Hence 



~ 



I, 5] FUNCTIONS OF A SINGLE VARIABLE 5 

and the limit of S(x) is seen to be 1. Thus, in this example, the 
function approaches a definite limit as x approaches zero, but that 
limit is different from the value of the function for x = 0. 

5. Derivatives. Let/(x) be a continuous function. Then the two 
terms of the quotient 



k 

approach zero simultaneously, as the absolute value of h approaches 
zero, while x remains fixed. If this quotient approaches a limit, 
this limit is called the derivative of the function /(#), and is denoted 
by y , or by / (x), in the notation due to Lagrange. 

An important geometrical concept is associated with this analytic 
notion of derivative. Let us consider, in a plane XOY, the curve 
A MB, which represents the function y =/(#), which we shall assume 
to be continuous in the interval (a, b). Let M and M be two points 
on this curve, in the interval (a, b), and let their abscissas be x and 
x + A, respectively. The slope of the straight line MM is then 
precisely the quotient above. Now as h approaches zero the point 
M approaches the point M] and, if the function has a derivative, 
the slope of the line MM approaches the limit y . The straight line 
MM , therefore, approaches a limiting position, which is called the 
tangent to the curve. It follows that the equation of the tangent is 

Y-y = y (X-x), 

where X and Y are the running coordinates. 

To generalize, let us consider any curve in space, and let 



be the coordinates of a point on the curve, expressed as functions of 
a variable parameter t. Let M and M be two points of the curve 
corresponding to two values, t and t + h, of the parameter. The 
equations of the chord MM 1 are then 

x-f(t) Y 



f(t + h) - 

If we divide each denominator by h and then let h approach zero, 
the chord MM evidently approaches a limiting position, which is 
given by the equations 

X -f(f) Y- 



f(t) 4, ft) 



6 DERIVATIVES AND DIFFERENTIALS [i, 5 

provided, of course, that each of the three functions f(t), <f> (t), \J/ (t) 
possesses a derivative. The determination of the tangent to a curve 
thus reduces, analytically, to the calculation of derivatives. 

Every function which possesses a derivative is necessarily con 
tinuous, but the converse is not true. It is easy to give examples 
of continuous functions which do not possess derivatives for par 
ticular values of the variable. The function y = xsinl/x, for 
example, is a perfectly continuous function of x, for x = 0,* and y 
approaches zero as x approaches zero. But the ratio y /x = sinl/cc 
does not approach any limit whatever, as we have already seen. 

Let us next consider the function y = x*. Here y is continuous 
for every value of a;; and y = when x = 0. But the ratio y /x = x~* 
increases indefinitely as x approaches zero. For abbreviation the 
derivative is said to be infinite for x = ; the curve which repre 
sents the function is tangent to the axis of y at the origin. 

Finally, the function 



y = 



is continuous at x = 0,* but the ratio y /x approaches two different 
limits according as x is always positive or always negative while 
it is approaching zero. When x is positive and small, e l/x is posi 
tive and very large, and the ratio y /x approaches 1. But if x 
is negative and very small in absolute value, e l/x is very small, and 
the ratio y / x approaches zero. There exist then two values of the 
derivative according to the manner in which x approaches zero : the 
curve which represents this function has a corner at the origin. 

It is clear from these examples that there exist continuous func 
tions which do not possess derivatives for particular values of the 
variable. But the discoverers of the infinitesimal calculus confi 
dently believed that a continuous function had a derivative in gen 
eral. Attempts at proof were even made, but these were, of course, 
fallacious. Finally, Weierstrass succeeded in settling the question 
conclusively by giving examples of continuous functions which do not 
possess derivatives for any values of the variable whatever.! But 
as these functions have not as yet been employed in any applications, 

* After the value zero has been assigned to y for x = 0. TRANSLATOR. 

t Note read at the Academy of Sciences of Berlin, July 18, 1872. Other examples 
are to be found in the memoir by Darboux on discontinuous functions (Annales de 
I Ecole Normale Superieure, Vol. IV, 2d series). One of Weierstrass s examples is 
given later (Chapter IX). 



I, 6] FUNCTIONS OF A SINGLE VARIABLE 7 

we shall not consider them here. In the future, when we say that 
a function f(x) has a derivative in the interval (a, b), we shall mean 
that it has an unique finite derivative for every value of x between 
a and b and also f or x = a (h being positive) and f or x = b (h being 
negative), unless an explicit statement is made to the contrary. 

6. Successive derivatives. The derivative of a function f(x) is in 
general another function of x,f (x). If f (x) in turn has a deriva 
tive, the new function is called the second derivative of /(x), and is 
represented by y" or by f"(x). In the same way the third deriva 
tive y ", or / "(#), is defined to be the derivative of the second, and 
so on. In general, the rath derivative 7/ n) , or f w (x), is the deriva 
tive of the derivative of order (n 1). If, in thus forming the 
successive derivatives, we never obtain a function which has no 
derivative, we may imagine the process carried on indefinitely. In 
this way we obtain an unlimited sequence of derivatives of the func 
tion /(cc) with which we started. Such is the case for all functions 
which have found any considerable application up to the present 
time. 

The above notation is due to Lagrange. The notation D n y, or 
D n f(x), due to Cauchy, is also used occasionally to represent the 
wth derivative. Leibniz notation will be given presently. 

7. Rolle s theorem. The use of derivatives in the study of equa 
tions depends upon the following proposition, which is known as 
Roue s Theorem : 

Let a and b be two roots of the equation f (x) = 0. If the function 
f(x) is continuous and possesses a derivative in the interval (a, b~), 
the equation / (#) = has at least one root which lies between a and b. 

For the function f(x) vanishes, by hypothesis, for x = a and x = b. 
If it vanishes at every point of the interval (a, b), its derivative also 
vanishes at every point of the interval, and the theorem is evidently 
fulfilled. If the function f(x) does not vanish throughout the inter 
val, it will assume either positive or negative values at some points. 
Suppose, for instance, that it has positive values. Then it will have 
a maximum value M for some value of x, say x lf which lies between 
a and b ( 3, Theorem II). The ratio 



8 DERIVATIVES AND DIFFERENTIALS [I, 8 

where h is taken positive, is necessarily negative or else zero. 
Hence the limit of this ratio, i.e. f (x^), cannot be positive ; i.e. 
f ( x i) = 0- But if we consider f (x\) as the limit of the ratio 



> 
h 

where h is positive, it follows in the same manner that f\x\) ^ 0, 
From these two results it is evident that/ ^) = 0. 

8. Law of the mean. It is now easy to deduce from the above 
theorem the important law of the mean : * 

Let f(x) be a continuous function which has a derivative in the 
interval (a, b). Then 

(1) m-f(a) = (b-a)f(c-), 

where c is a number between a and b. 

In order to prove this formula, let < (x) be another function which 
has the same properties as/(x), i.e. it is continuous and possesses a 
derivative in the interval (a, b). Let us determine three constants, 
A, B, C, such that the auxiliary function 



vanishes for x = a and for x = b. The necessary and sufficient 
conditions for this are 



A /(a) +B <()+ C = 0, Af(b) + B<l>(b)+ C = 0; 
and these are satisfied if we set 
A = <l>(a)-4> (b), B =/(&) -/(a), C =/() (*)-/(&) * (a). 

The new function \J/(x) thus defined is continuous and has a derivative 
in the interval (a, b). The derivative if/ (x) = A f (x) + B < (z) there 
fore vanishes for some value c which lies between a and b, whence 
replacing A and B by their values, we find a relation of the form 



It is merely necessary to take < (a;) = x in order to obtain the equality 
which was to be proved. It is to be noticed that this demonstration 
does not presuppose the continuity of the derivative/ ^). 

"Formule des accroissements finis." The French also use " Formule de la 
moyenne" as a synonym. Other English synonyms are "Average value theorem " 
and " Mean value theorem." TRANS. 



I, 8] FUNCTIONS OF A SINGLE VARIABLE 9 

From the theorem just proven it follows that if the derivative 
f (x) is zero at each point of the interval (a, b), the function f(x) 
has the same value at every point of the interval ; for the applica 
tion of the formula to two values Xi, x z , belonging to the interval 
(a, b), gives f(xi)=f(x. 2 ). Hence, if two functions have the same 
derivative, their difference is a constant ; and the converse is evi 
dently true also. If a function F(x) be given whose derivative is 
f(oc), all other functions which have the same derivative are found by 
adding to F(x) an arbitrary constant* 

The geometrical interpretation of the equation (1) is very simple, 
Let us draw the curve A MB which represents the function y = f(x) 
in the interval (a, b). Then the ratio [/(&) /()]/ (b a) is the 
slope of the chord AB, while / () is the slope of the tangent at a 
point C of the curve whose abscissa is c. Hence the equation (1) 
expresses the fact that there exists a point C on the curve A MB, 
between A and B, where the tangent is parallel to the chord AB. 

If the derivative / (a;) is continuous, and if we let a and b approach 
the same limit x according to any law whatever, the number c, 
which lies between a and b, also approaches x 0} and the equation (1) 
shows that the limit of the ratio 



b a 

is f (xo). The geometrical interpretation is as follows. Let us 
consider upon the curve y=f(x) a point M whose abscissa is x , 
and two points A and B whose abscissa are a and b, respectively. 
The ratio [/(&) /()] / (b ) is equal to the slope of the chord 
AB, while / (x ) is the slope of the tangent at M. Hence, when 
the two points A and B approach the point M according to any law 
whatever, the secant AB approaches, as its limiting position, the 
tangent at the point M. 

* This theorem is sometimes applied without due regard to the conditions imposed in 
its statement. Let/(x) and 0(^), f r example, be two continuous functions which have 
derivatives / (a;), </) (x) in an interval (a, 6). If the relation / (z) <t>(x)f(x) 4> (x) = 
is satisfied by these four functions, it is sometimes accepted as proved that the deriva 
tive of the function// <f>, or [/ (a;) (cc) - f(x) (z)] / < 2 , is zero, and that accordingly 
f/<t> is constant in the interval (a, b). But this conclusion is not absolutely rigorous 
unless the function $ (a;) does not vanish in the interval (a, b). Suppose, for instance, 
that (a;) and <j> (x) both vanish for a value c between a and 6. A function/(x) equal 
to Ci<f>(x) between a and c, and to C%<f)(x) between c and b, where Cj and C 2 are dif 
ferent constants, is continuous and has a derivative in the interval (a, b), and we have 
f (x)<t>(x) f(x)<p (x) = for every value of x in the interval. The geometrical 
interpretation is apparent. 



10 



DERIVATIVES AND DIFFERENTIALS 



[I, 9 



This does not hold in general, however, if the derivative is not 
continuous. For instance, if two points be taken on the curve 
y = x*, on opposite sides of the y axis, it is evident from a figure 
that the direction of the secant joining them can be made to approach 
any arbitrarily assigned limiting value by causing the two points to 
approach the origin according to a suitably chosen law. 

The equation (! ) is sometimes called the generalized law of the 
mean. From it de 1 Hospital s theorem on indeterminate forms fol 
lows at once. For, suppose f(a) = and <f> (a) = 0. Replacing b 

by x in (! ). we find 
* \ / \ 



where a^ lies between a and x. This equation shows that if the 
ratio f (x)/(j> (x) approaches a limit as x approaches a, the ratic 
/"(#) / (j> (a;) approaches the same limit, if f(a) = and <f> (a) = 0. 

9. Generalizations of the law of the mean. Various generalizations of the law 
of the mean have been suggested. The following one is due to Stieltjes (Bulletin 
de la Socie te Mathtmatique, Vol. XVI, p. 100). For the sake of defmiteness con 
sider three functions, /(x), g(x), h(x), each of which has derivatives of the first 
and second orders. Let a, 6, c be three particular values of the variable (a < b < c). 
Let A be a number defined by the equation 



and let 



be an auxiliary function. Since this function vanishes when x = b and when 
x = c, its derivative must vanish for some value f between 6 and c. Hence 



/() 


9 (a) 


h(a) 




1 


a 


a 


/(&) 


9(b) 


h(b) 


-A 


1 


b 


6 2 


/(c) 


9(c] 


h(c) 




1 


c 


c 2 


/(a) 


9(0.} 


h(a) 




1 


a 


a 2 


/(&) 


9(b) 


h(b) 


-A 


1 


b 


b 2 


/(*) 


9(x) 


h(x) 




1 


x 


x* 



/(a) g(a) h(a) 
/(&) g(b) h(b) 
/ (f) 



-A 



1 a a 2 
1 b b 2 
1 2f 



If b be replaced by x in the left-hand side of this equation, we obtain a function 
of x which vanishes when x = a and when x = b. Its derivative therefore van 
ishes for some value of x between a and 6, which we shall call . The new 
equation thus obtained is 



/ (a) g (a) h (a) 



/ (f) 



1 2f 



= 0. 



Finally, replacing f by x in the left-hand side of this equation, we obtain a func 
tion of x which vanishes when x = and when x = f . Its derivative vanishes 



I, 10] FUNCTIONS OF SEVERAL VARIABLES 11 

for some value ij, which lies between and f and therefore between a and c. 
Hence A must have the value 



J_ 

1.2 



/ (a) g (a) h (a) 



where lies between a and 6, and 17 lies between a and c. 

This proof does not presuppose the continuity of the second derivatives 
f"(x), g"(x), h"(x). If these derivatives are continuous, and if the values a, 6, c 
approach the same limit XQ, we have, in the limit, 



1 



/ (x ) g (x ) h (x ) 
f (x ) g (xo) h (x ) 
f"(x ) 0"(xo) h"(x Q ) 



Analogous expressions exist for n functions and the proof follows the same 
lines. If only two functions /(x) and g (x) are taken, the formula? reduce to the 
law of the mean if we set g (x) = 1. 

An analogous generalization has been given by Schwarz (Annali di Mathe- 
matica, 2d series, Vol. X). 

II. FUNCTIONS OF SEVERAL VARIABLES 

10. Introduction. A variable quantity w whose value depends on 
the values of several other variables, x, y, z, -, t, which are in 
dependent of each other, is called a function of the independ 
ent variables x, y, z, , t; and this relation is denoted by writing 
w =f(x, y,z,---, t). For definiteness, let us suppose that w = f(x, y) 
is a function of the two independent variables x and y. If we think 
of x and y as the Cartesian coordinates of a point in the plane, 
each pair of values (x, y) determines a point of the plane, and con 
versely. If to each point of a certain region A in the xy plane, 
bounded by one or more contours of any form whatever, there 
corresponds a value of w, the function f(x, y) is said to be defined 
in the region A. 

Let (x , y ) be the coordinates of a point M lying in this region. 
The function f(x, y) is said to be continuous for the pair of values 
( x oi yo) if, corresponding to any preassigned positive number c, another 
positive number 77 exists such that 

|/C*o + h, y + k)-f(x , 2/ ) | < e 

whenever \h < rj and \k\<rj. 

This definition of continuity may be interpreted as follows. Let 
us suppose constructed in the xy plane a square of side 2^ about 
M as center, with its sides parallel to the axes. The point M , 



12 DERIVATIVES AND DIFFERENTIALS [I, 11 

whose coordinates are x + h, y + k, will lie inside this square, if 
| h | < rj and | k \ < rj. To say that the function is continuous for the 
pair of values (x , T/ O ) amounts to saying that by taking this square 
sufficiently small we can make the difference between the value of 
the function at M and its value at any other point of the square less 
than e in absolute value. 

It is evident that we may replace the square by a circle about 
(x , y ) as center. For, if the above condition is satisfied for all 
points inside a square, it will evidently be satisfied for all points 
inside the inscribed circle. And, conversely, if the condition is 
satisfied for all points inside a circle, it will also be satisfied for all 
points inside the square inscribed in that circle. We might then 
define continuity by saying that an rj exists for every c, such that 
whenever V/i 2 + k 2 < 17 we also have 



I /(<> + h > y + k) -f(x , 

The definition of continuity for a function of 3, 4, , n inde 
pendent variables is similar to the above. 

It is clear that any continuous function of the two independent 
variables x and y is a continuous function of each of the variables 
taken separately. However, the converse does not always hold.* 

11. Partial derivatives. If any constant value whatever be substi 
tuted for y, for example, in a continuous function f(x, y), there 
results a continuous function of the single variable x. The deriva 
tive of this function of x, if it exists, is denoted by f x (x, y) or by <a x . 
Likewise the symbol u v , or f y (x, y), is used to denote the derivative 
of the function f(x, y} when x is regarded as constant and y as the 
independent variable. The functions f x (x, y) and f y (x, y) are called 
the partial derivatives of the function f(x, y). They are themselves, 
in general, functions of the two variables x and y. If we form their 
partial derivatives in turn, we get the partial derivatives of the sec 
ond order of the given function f(x, y). Thus there are four partial 
derivatives of the second order, fa (x, y),f x (x, y),f yx (x, y),f+(x, y\ 
The partial derivatives of the third, fourth, and higher orders are 

* Consider, for instance, the f unction /(x, y), which is equal to 2 xy / (x 2 + y 2 ) when 
the two variables x and y are not both zero, and which is zero when x = y = 0. It is 
evident that this is a continuous function of x when y is constant, and vice versa. 
Nevertheless it is not a continuous function of the two independent variables x and y 
for the pair of values x = 0, y = 0. For, if the point (a-, y) approaches the origin upon 
the line x = y. the f unction/ (x, y) approaches the limit 1, and not zero. Such functions 
have been studied by Baire in his thesis. 



I, 11] FUNCTIONS OF SEVERAL VARIABLES 13 

defined similarly. In general, given a function w = /(x, y, z, -, f) 
of any number of independent variables, a partial derivative of the 
nth order is the result of n successive differentiations of the function 
/, in a certain order, with respect to any of the variables which occur 
in /. We will now show that the result does not depend upon the 
order in which the differentiations are carried out. 
Let us first prove the following lemma : 

Let w = f (x, y) be a function of the two variables x and y. Then 
f xij = f tjx , provided that these two derivatives are continuous. 

To prove this let us first write the expression 

U =f(x + Ax, y + Ay) -f(x, y + Ay) -f(x + Ax, y) + /(x, y} 

in two different forms, where we suppose that x, y, Ax, A?/ have 
definite values. Let us introduce the auxiliary function 

< 00 =f( x + Ax, u) -/(x, v), 
where v is an auxiliary variable. Then we may write 



Applying the law of the mean to the function <(w), we 

U = Ay < (y + 0Ay), where < < 1 ; 

or, replacing <j> u by its value, 

U = Ay [/(* + Ax, y + 0Ay) -f y (x, y + 0Ay)]. 

If we now apply the law of the mean to the function f y (u, y + 0Ay), 
regarding u as the independent variable, we find 

U = Ax Ay/^ (x + Ax, y + 0Ay), < < 1. 

From the symmetry of the expression U in x, y, Ax, Ay, we see that 
we would also have, interchanging x and y, 

U = Ay Aaj/q, (x + 0| Ax, y + ^ Ay), 

where 0, and 0[ are again positive constants less than unity. Equat 
ing these two values of U and dividing by Ax Ay, we have 

f xy (x + 0[ Ax, y + ^Ay) =f,, x (x + Ax, y + 0Ay). 

Since the derivatives /,. (x, y) and f vx (x, y) are supposed continuous, 
the two members of the above equation approach f xy (x, y) and 
f yx (x, y), respectively, as Ax and Ay approach zero, and we obtain 
the theorem which we wished to prove. 



14 DERIVATIVES AND DIFFERENTIALS [I, n 

It is to be noticed in the above demonstration that no hypothesis 
whatever is made concerning the other derivatives of the second order, 
f^ and f y t. The proof applies also to the case where the function 
f(x, y) depends upon any number of other independent variables 
besides x and y, since these other variables would merely have to 
be regarded as constants in the preceding developments. 

Let us now consider a function of any number of independent 
variables, 

=/(> y> *>> *)j 

and let n be a partial derivative of order n of this function. Any 
permutation in the order of the differentiations which leads to fi 
can be effected by a series of interchanges between two successive 
differentiations ; and, since these interchanges do not alter the 
result, as we have just seen, the same will be true of the permuta 
tion considered. It follows that in order to have a notation which 
is not ambiguous for the partial derivatives of the nth order, it is 
sufficient to indicate the number of differentiations performed with 
respect to each of the independent variables. For instance, any nth 
derivative of a function of three variables, to =/(x, y, z), will be 
represented by one or the other of the notations 



where p -f- q + r = n* Either of these notations represents the 
result of differentiating / successively p times with respect to x, 
q times with respect to ?/, and r times with respect to 2, these oper 
ations being carried out in any order whatever. There are three 
distinct derivatives of the first order, f x , f , f z \ six of the second 
order, fa, fa fa / 3 . v , fa f xz ; and so on. 

In general, a function of p independent variables has just as many 
distinct derivatives of order n as there are distinct terms in a homo 
geneous polynomial of order n in p independent variables ; that is, 



as is shown in the theory of combinations. 

Practical rules. A certain number of practical rules for the cal 
culation of derivatives are usually derived in elementary books on 

* The notation / a Pyq ..r (x, y, z) is used instead of the notation fxfyn z r (x, y, z) for 
simplicity. Thus the notation f xy (x, y), used in place of f x y (x, y), is simpler and 
equally clear. TRANS. 




I, 11] FUNCTIONS OF SEVERAL VARIABLES 16 

the Calculus. A table of such rules is appended, the function and 
its derivative being placed on the same line : 

y = ax - 1 ; 
y = a x log a, 

where the symbol log denotes the natural logarithm ; 

y = log x, y = -> 

X 

y = sin x, 
y = cos x, 

y = arc sin x, 
y = arc tan x, 

JL -\- X 

y = uv, y = u v 4- uv 1 ; 

_ u f ^ u v uv . 

y =/(), 2/*=/>K; 

The last two rules enable us to find the derivative of a function 
of a function and that of a composite function if f u ,f v ,f w are con 
tinuous. Hence we can find the successive derivatives of the func 
tions studied in elementary mathematics, polynomials, rational 
and irrational functions, exponential and logarithmic functions, 
trigonometric functions and their inverses, and the functions deriv 
able from all of these by combination. 

For functions of several variables there exist certain formulae 
analogous to the law of the mean. Let us consider, for definite- 
ness, a function f(x, y) of the two independent variables x and y. 
The difference f(x + h, y 4- K) f(x, y) may be written in the form 

f(x + h,y + k) -f(x, y) = [/(* + h, y + k) -f(x, y + &)] 



to each part of which we may apply the law of the mean. We 
thus find 



f(x + h,y + k}-f(x, y) = hf x (x + 6h, y + k}+ kf v (x, y + O K), 

where 6 and each lie between zero and unity. 

This formula holds whether the derivatives f x and / are continu 
ous or not. If these derivatives are continuous, another formula, 



16 DERIVATIVES AND DIFFERENTIALS [1,512 

similar to the above, but involving only one undetermined number 
6, may be employed.* In order to derive this second formula, con 
sider the auxiliary function <f>() = f(x + ht, y + kfy, where x, y, h, 
and k have determinate values and t denotes an auxiliary variable. 
Applying the law of the mean to this function, we find 



Now <(>") is a composite function of t, and its derivative 4> (t) is 
equal to hf x (x -f- ht, y + kf) + kf y (x + ht, y -f- kt) ; hence the pre 
ceding formula may be written in the form 



12. Tangent plane to a surface. We have seen that the derivative 
of a function of a single variable gives the tangent to a plane curve. 
Similarly, the partial derivatives of a function of two variables occur 
in the determination of the tangent plane to a surface. Let 

(2) z . F(x, y) 

be the equation of a surface S, and suppose that the function F(x, ?/), 
together with its first partial derivatives, is continuous at a point 
(^o? yo) of the xy plane. Let z be the corresponding value of z, 
and AT (cr , 7/0 > ) the corresponding point on the surface S. If 
the equations 

(3) *=/(*), z/ = <KO> * = ^(9 

represent a curve C on the surface S through the point M , the 
three functions f(f), <j>(t), "A(0> which we shall suppose continuous 
and differentiable, must reduce to x , y , z , respectively, for some 
value t of the parameter t. The tangent to this curve at the point 
M is given by the equations ( 5) 

x x Y z * 



Since the curve C lies on the surface S, the equation \j/(t)=F[f(t~), . 
must hold for all values of t; that is, this relation must be an identity 

* Another formula may be obtained which involves only one undetermined number 0, 
and which holds even when the derivatives/^, and/, are discontinuous. For the applica 
tion of the law of the mean to the auxiliary function <j>(t) =f(x + ht,y + k) +f(x, y + kt) 
gives 

<(!) -0(0) = (0), 0<0<1. 

or 

f(x + h,y + k) -f(x, y) = hf x (x + 6h, y + k) + kf y (x, y + 6k), 0<0<1. 

The operations performed, and hence the final formula, all hold provided the deriva 
tives f x and f y merely exist at the points (x + ht, y + k), (x,y + kt),0^t^\. TRANS. 



I, 13] FUNCTIONS OF SEVERAL VARIABLES 17 

in t. Taking the derivative of the second member by the rule for 
the derivative of a composite function, and setting t = t , we have 

(5) <j, (t )=fi(t )F Xo + <t> (t )F Va . 

We can now eliminate f (t ~), < ( )> i//( ) between the equations (4) 
and (5), and the result of this elimination is 

(6) Z-z = (X- ar ) F Xg + (Y - y ) F^. 

This is the equation of a plane which is the locus of the tangents to 
all curves on the surface through the point M . It is called the tan 
gent plane to the surface. 

13. Passage from increments to derivatives. We have defined the successive 
derivatives in terms of each other, the derivatives of order n being derived from 
those of order (n 1), and so forth. It is natural to inquire whether we may 
not define a derivative of any order as the limit of a certain ratio directly, with 
out the intervention of derivatives of lower order. We have already done some 
thing of this kind for f xy ( 11); for the demonstration given above shows that/rj, 
is the limit of the ratio 

f(x + Ax, y + Ay) -/(x + Ax, y)-f(x, y + Ay) + /(x, y) 
Ax Ay 

as Ax and Ay both approach zero. It can be shown in like manner that the 
second derivative /" of a function f(x) of a single variable is the limit of the 
ratio 

/(x + hi + h*) -f(x + hi) -f(x 

^1^2 

as hi and h 2 both approach zero. 
For, let us set 

/i(x)=/(z + Ai) 

and then write the above ratio in the form 



h\ 

f > (x + 



+ 



hi 

The limit of this ratio is therefore the second derivative /", provided that 
derivative is continuous. 

Passing now to the general case, let us consider, for definiteness, a function of 
three independent variables, w =f(x, y, 2). Let us set 

AW =/(x + h, y, z) -/(x, y, 2), 
AW =/(x, y + k, 2) -/(x, y, 2), 
A^w =/(x, y, 2 -f 1) -/(x, y, z), 

where A* w, A* w, A l z u are the^irsi increments of w. If we consider ^, k, I as given 
constants, then these three first increments are themselves functions of x, y, 2, 
and we may form the relative increments of these functions corresponding to 



18 DERIVATIVES AND DIFFERENTIALS [I, 13 

increments hi, ki, ^ of the variables. This gives us the second increments, 
A* 1 A * w > A* 1 A v w Tnis process can be continued indefinitely ; an increment 
of order n would be defined as a first increment of an increment of order (n 1). 
Since we may invert the order of any two of these operations, it will be suffi 
cient to indicate the successive increments given to each of the variables. An 
increment of order n would be indicated by some such notation as the following : 

A<-> = AX A* p A* AX 1 A^/(z, y, z), 

where p + q + r = n, and where the increments h, k, I may be either equal or 
unequal. This increment may be expressed in terms of a partial derivative of 
order n, being equal to the product 

hihy hpki kgl\ l r 

x f x p*z (x + *i Ai + + d,,h p , y + eiki + + O q k q , z + ffi li + + Kir), 
where every 6 lies between and 1. This formula has already been proved for 
first and for second increments. In order to prove it in general, let us assume 
that it holds for an increment of order (n 1), and let 

(X, y, 2) = A** A h / Ajt AX 1 f. 
Then, by hypothesis, 

$(x,y,z) = h z ---h p ki-- -k q li-- I r f x p-i i f, i r(x + 0sh 2 + ---- \-6 P hp, y-\ ---- ,H ---- ). 
But the nth increment considered is equal to 0(x + hi, y, z) <f>(x, y, z); and if we 
apply the law of the mean to this increment, we finally obtain the formula sought. 
Conversely, the partial derivative f xT ^ z r is the limit of the ratio 

AX - .-AX-.. -AX --- A/-/ 

hi h? h p ki k 2 kg li l r 

as all the increments h, k, I approach zero. 

It is interesting to notice that this definition is sometimes more general than 
the usual definition. Suppose, for example, that w =/(x, y) <f>(x) + ^(y) is a 
function of x and y, where neither <f> nor ^ has a derivative. Then u also has 
no first derivative, and consequently second derivatives are out of the question, 
in the ordinary sense. Nevertheless, if we adopt the new definition, the deriva 
tive fxy is the limit of the fraction 

/(x + h, y + k) -/(x + h, y) -/(x, y + k) +/(x, y) 

hk 
which is equal to 

h) + t( V + k) - <t>(x + h) - 



hk 

But the numerator of this ratio is identically zero. Hence the ratio approaches 
zero as a limit, and we find/ xy = 0.* 

* A similar remark may be made regarding functions of a single variable. For 
example, the f unction /(K) = x s cosl/x has the derivative 

f (x) = 3 x 2 cos - + xsin-i 

and f (x) has no derivative for x 0. But the ratio 

/(2ar)-2/(tt)+/(0) 
o" 

or 8 a cos (I/ 2 a) 2 a cos (I/ or), has the limit zero when a approaches zero. 



l )14 ] THE DIFFERENTIAL NOTATION 19 

III. THE DIFFERENTIAL NOTATION 

The differential notation, which has been in use longer than any 
other,* is due to Leibniz. Although it is by no means indispensable, 
it possesses certain advantages of symmetry and of generality which 
are convenient, especially in the study of functions of several varia 
bles. This notation is founded upon the use of infinitesimals. 

14. Differentials. Any variable quantity which approaches zero as 
a limit is called an infinitely small quantity, or simply an infinitesi 
mal. The condition that the quantity be variable is essential, for 
a constant, however small, is not an infinitesimal unless it is zero. 

Ordinarily several quantities are considered which approach zero 
simultaneously. One of them is chosen as the standard of compari 
son, and is called the principal infinitesimal. Let be the principal 
infinitesimal, and ft another infinitesimal. Then is said to be an 
infinitesimal of higher order with respect to a, if the ratio ft/a 
approaches zero with a. On the other hand, ft is called an infini 
tesimal of the first order with respect to a, if the ratio ft/a 
approaches a limit K different from zero as a approaches zero. In 
this case 

^ = K + e, 


where c is another infinitesimal with respect to a. Hence 
ft=a(K + c)= Ka + at, 

and Ka is called the principal part of ft. The complementary term 
at is an infinitesimal of higher order with respect to a. In general, 
if we can find a positive power of a, say a", such that ft /a" 
approaches a finite limit K different from zero as a approaches 
zero, ft is called an infinitesimal of order n with respect to a. Then 
we have 

4 ; = K + e, 

a 

or 

ft = a n (K -f e) = Ka* + ". 

The term Ka" is again called the principal part of ft. 

Having given these definitions, let us consider a continuous func 
tion y=f(x), which possesses a derivative f (x). Let Aa; be an 



* With the possible exception of Newton s notation. TRANS. 



20 DERIVATIVES AND DIFFERENTIALS [I, 14 

increment of x, and let A?/ denote the corresponding increment of y. 
From the very definition of a derivative, we have 



where c approaches zero with Ace. If Ax be taken as the principal 
infinitesimal, AT/ is itself an infinitesimal whose principal part is 
f (x) Ax.* This principal part is called the differential of y and is 
denoted by dy. 

dy=f(x)&x. 

When /(x) reduces to x itself, the above formula becomes dx = Ax ; 
and hence we shall write, for symmetry, 



where the increment dx of the independent variable x is to be given 
the same fixed value, which is otherwise arbitrary and of course 
variable, for all of the several dependent 
functions of x which may be under consid 
eration at the same time. 

Let us take a curve C whose equation is 
y = f(x), and consider two points on it, M 
and M , whose abscissae are x and x -f dx, 
respectively. In the triangle MTN we have 

NT = MN tan Z TMN = dxf (x). 

Hence NT represents the differential dy, 

while Ay is equal to NM . It is evident from the figure that M T 
is an infinitesimal of higher order, in general, with respect to NT, 
as M approaches M, unless MT is parallel to the x axis. 

Successive differentials may be defined, as were successive deriv 
atives, each in terms of the preceding. Thus we call the differ 
ential of the differential of the first order the differential of the 
second order, where dx is given the same value in both cases, as 
above. It is denoted by d 2 y: 

d*y = d (dy) = [/"(x) dx] dx = f"(x) (dx}*. 
Similarly, the third differential is 

d*y = d(d*y) = [_f(x)dx*]dx =f"(x)(dx)*, 




* Strictly speaking, we should here exclude the case where f (x) = 0. It is, how 
ever, convenient to retain the same definition of dy =f (x)&x in this case also. 
even though it is not the principal part of Ay. TRANS. 



I, 14] THE DIFFERENTIAL NOTATION 21 

and so on. In general, the differential of the differential of order 
(n 1) is 

The derivatives / (or), /"(a), -, f (n \x), ... can be expressed, on 
the other hand, in terms of differentials, and we have a new nota 
tion for the derivatives : 

t dy ,, _ <Py M d n y 

y ~ dx ~ dx 2 ~dtf t> 

To each of the rules for the calculation of a derivative corresponds 
a rule for the calculation of a differential. For example, we have 

d x m = mx m - l dx, da x = a x log a dx ; 

, , dx 

d log x = j d sin x = cos x dx ; ; 

SC 

, . dx dx 
aarcsmcc = > darctanx = - - 

Vl - a; 2 1 + x 2 

Let us consider for a moment the case of a function of a function. 
y =/(), where u is a function of the independent variable x. 



whence, multiplying both sides by dx, we get 

y x dx =/(M) X u x dx; 
that is, 

dy =f(u)du. 

The formula for dy is therefore the same as if u were the inde 
pendent variable. This is one of the advantages of the differential 
notation. In the derivative notation there are two distinct formulae, 

&=/(*)> yx=f(u)u xy 

to represent the derivative of y with respect to cc, according as y is 
given directly as a function of x or is given as a function of x by 
means of an auxiliary function u. In the differential notation the 
same formula applies in each case.* 

If y = f(u, v, w) is a composite function, we have 

Vx = U xfu + V x f v + W x f n , 

at least if f u ,f v ,f w are continuous, or, multiplying by dx, 
y x dx = u x dxf u + v x dxf v + w x dxf w ; 

* This particular advantage is slight, however ; for the last formula ahove is equally 
well a general one and covers both the cases mentioned. TRANS. 



22 DERIVATIVES AND DIFFERENTIALS [I, 15 

that is, 

dll = f u du + f v dv +f w dw. 

Thus we have, for example, 

V du, 



V 

The same rules enable us to calculate the successive differentials. 
Let us seek to calculate the successive differentials of a function 
y = /(u), for instance. We have already 

dy=f (u}du. 

In order to calculate d?y, it must be noted that du cannot be regarded 
as fixed, since u is not the independent variable. We must then 
calculate the differential of the composite function f (u) du, where u 
and du are the auxiliary functions. We thus find 



To calculate d*y, we must consider d*y as a composite function, with 
u, du, d 2 u as auxiliary functions, which leads to the expression 

d*y =f "(u)du 8 + 3f"(u)dud*u +f (u)d*u ; 

and so on. It should be noticed that these formulae for d*y, d*y, 
etc., are not the same as if u were the independent variable, on 
account of the terms d*u, d z u, etc.* 

A similar notation is used for the partial derivatives of a function 
of several variables. Thus the partial derivative of order n of 
f(x, y, s), which is represented by f xf>flzr in our previous notation, 
is represented by 



in the differential notation.f This notation is purely symbolic, and 
in no sense represents a quotient, as it does in the case of functions 
of a single variable. _ _ . __ 



15. Total differentials. Let w =f(x, y, z) be a function of the 
three independent variables x, y, z. The expression 

o / o / Q / 

du = ^- dx + ^ dy + -^ dz 

ex dy - dz 

* This disadvantage would seem completely to offset the advantage mentioned 
above. Strictly speaking, we should distinguish between d^y and d? u y, etc. TRANS. 

t This use of the letter d to denote the partial derivatives of a function of several 
variables is due to Jacob! . Before his time the same letter d was used as is used for 
the derivatives of a function of a single variable. 



I, 15] THE DIFFERENTIAL NOTATION 23 

is called the total differential of o>, where dx, dy, dz are three fixed 
increments, which are otherwise arbitrary, assigned to the three 
independent variables x, y, z. The three products 

8 f 7 d f j d f j 

TT- dx. - dy, ~ dz 

ex dy cz 

are called partial differentials. 

The total differential of the second order d*<a is the total differ 
ential of the" total differential of the first order, the increments 
dx, dy, dz remaining the same as we pass from one differential to 
the next higher. Hence 

_ , 7 . ddia ddia cdw 

d 2 u = d(dta) = dx -f -^ dy + -= dz ; 
Ox oy cz 

or, expanding, 



ex* dx oy ex cz 



! 

Oy Oz 



+ 2 - dxdy + 2 dxdz + 2 = dy dz. 

Ox oy ox cz Oy Oz 

If cPf be replaced by df 2 , the right-hand side of this equation 
becomes the square of 



We may then write, symbolically, 



0x cy oz 

it being agreed that df* is to be replaced by 8 2 f after expansion. 

In general, if we call the total differential of the total differential 
of order (n 1) the total differential of order n, and denote it by 
d n (a, we may write, in the same symbolism, 

*.-(*** +*)", 

\0x Oy Oz / 

where df n is to be replaced by d n f after expansion ; that is, in our 
ordinary notation, 



DERIVATIVES AND DIFFERENTIALS [I, 15 



where 

A n 

pqr p\q\r\ 

is the coefficient of the term a p & c r in the development of (a. + b + c) n . 
For, suppose this formula holds for d n w. We will show that it then 
holds for d n+l <o; and this will prove it in general, since we have 
already proved it for n = 2. From the definition, we find 

d n+l w=d(d n (o) 

r zn+if d n+l f 

- + 



whence, replacing e n + 1 /by cf n + l , the right-hand side becomes 



f ( 7T- dx -f 7f- dy + rf I , 

1 C7 ^V <7 



or 



cy cz I \ox cy 

Hence, using the same symbolism, we may write 



- - 

cy cz 

Note. Let us suppose that the expression for dw, obtained in any 
way whatever, is 

(7) dw = P dx -f- Q dy + R dz, 

where P, Q, R are any functions x, y, z. Since by definition 

d<a 8<a d<a 

rfw = ^- <c + ay + ^- dz, 

dx cy cz 

we must have 



where dx, dy, dz are any constants. Hence 

/\ S<a - P go) - n 8<a - P 

(o) "5~ - .r, ^~ y, "5~~ -ft. 

^X ^ KB 

The single equation (7) is therefore equivalent to the three separate 
equations (8) ; and it determines all three partial derivatives at once. 



I, 16] THE DIFFERENTIAL NOTATION 25 

In general, if the nth total differential be obtained in any way 

whatever, 

d" w = 2 C pqr dx" dy" dz r ; 

then the coefficients C yqr are respectively equal to the corresponding 
nth derivatives multiplied by certain numerical factors. Thus all 
these derivatives are determined at once. We shall have occasion 
to use these facts presently. 

16. Successive differentials of composite functions. Let w = F(u, v, w~) 
be a composite function, u, v, w being themselves functions of the 
independent variables x, y, z, t. The partial derivatives may then be 
written down as follows : 

dia_dFdii dFdv dFdw 
dx du dx dv dx dw dx 



d> _d_F_d_u d_F_d_v_ ___ 
dy du dy dv dy dw dy 



d_F_d_v 

dz du dz do dz dw dz 

dw _ dF du dF dv dF dw 

dt du dt dv dt dw dt 

If these four equations be multiplied by dx, dy, dz, dt, respectively, 
and added, the left-hand side becomes 

d( , <? W 7 , ^< 7 ,^ W J. 

3- dx + -r- d y + -^- dz + -^ dt, 
dx dy dz d 

that is, do* ; and the coefficients of 

d_F d]F 0F 

du do dw 

on the right-hand side are du, dv, dw, respectively. Hence 

dF dF dF 

(9) do) = ^ du + -r dv + ^ dw, 

cu dv cw 

and we see that the expression of the total differential of the first 
order of a composite function is the same as if the auxiliary functions 
were the independent variables. This is one of the main advantages 
of the differential notation. The equation (9) does not depend, in 
form, either upon the number or upon the choice of the independent 
variables ; and it is equivalent to as many separate equations as 
there are independent variables. 

To calculate d 2 w, let us apply the rule just found for dta, noting 
that the second member of (9) involves the six auxiliary functions 
u, v, w, du, dv, dw. We thus find 



26 DERIVATIVES AND DIFFERENTIALS [I, lu 



d 2 F dF 

= -i^- du 2 4- -z du dv + - du dw + -^- d z u 
Ctr cucu cucw en 



4- - dudv 4- ^ dv 2 + ff dvdw + ^ 

du dv cv 2 cu cw cv 



d 2 F d 2 F d 2 F dF 

+ du dw 4- o Q dv dw -f TT-^ t?w + ^ 

^gw ^y^M> Cw 1 cw 

or, simplifying and using the same symbolism as above, 



7 , ^ , ^ ^ , 

d 2 w = [7^- du+ ^- dv + dw\ + TT- * + c? 2 w 4- ^ . 
Vc/w ^y CM; / cu Co cw 

This formula is somewhat complicated on account of the terms in 
d 2 u, d z v, d z w, which drop out when u, v, w are the independent 
variables. This limitation of the differential notation should be 
borne in mind, and the distinction between d 2 w in the two cases 
carefully noted. To determine d s w, we would apply the same rule 
to <2 2 o>, noting that d 2 w depends upon the nine auxiliary functions 
u, v, w, du, dv, dw, d 2 u, d 2 v,d 2 w; and so forth. The general expres 
sions for these differentials become more and more complicated ; 
d n w is an integral function of du, dv, dw, d 2 u, , d n u, d n v, d n w, and 
the terms containing d n u, d n v, d n w are 

dF 7 dF , dF 7 
d n u 4- d n v 4- d n w. 
cu cv cw 

If, in the expression for d" w, u, v, w, du, dv, dw, be replaced by 
their values in terms of the independent variables, d n t becomes an 
integral polynomial in dx, dy, dz, whose coefficients are equal 
(cf. Note, 15) to the partial derivatives of w of order n, multiplied 
by certain numerical factors. We thus obtain all these derivatives 
at once. 

Suppose, for example, that we wished to calculate the first and 
second derivatives of a composite function <a=f(ii), where w is a 
function of two independent variables u = <f> (x, y). If we calculate 
these derivatives separately, we find for the two partial derivatives 
of the first order 

1ft 8 w _ 8 w d u 8u) _ du> du 

dx du dx dy du dy 

Again, taking the derivatives of these two equations with respect 
to x, and then with respect to y, we find only the three following 
distinct equations, which give the second derivatives : 



THE DIFFERENTIAL NOTATION 



27 



(11) 



dx* 



dx dy 



du\* 

ex 



du 



d* 



<i> U C U C <a 

du* Cx dy du dx dy 



d- 22 
da dy 



&u t 

,2* 



The second of these equations is obtained by differentiating the 
first of equations (10) with respect to y, or the second of them with 
respect to x. In the differential notation these five relations (10) 
and (11) may be written in the form 



(12) 



en 



cu 



If du and d*u in these formulae be replaced by 
du 



TT- dy and -^ a 



dx dy 



respectively, the coefficients of dx and dy in the first give the first 
partial derivatives of o, while the coefficients of dx z , 2 dx dy, and 
dy 2 in the second give the second partial derivatives of w. 

17. Differentials of a product. The formula for the total differential 
of order n of a composite function becomes considerably simpler 
in certain special cases which often arise in practical applications. 
Thus, let us seek the differential of order n of the product of two 
functions o> = uv. For the first values of n we have 

dw = v dti + u dv, d* a) = v d* u + 2 du dv -f ud* v, ; 
and, in general, it is evident from the law of formation that 
d" w = v d" u 4- r, dr d n ~^u + Cd*v d n ~ 2 n -f + 



where C lt C 2 , are positive integers. It might be shown by alge 
braic induction that these coefficients are equal to those of the 
expansion of (a + &)" ; but the same end may be reached by the 
following method, which is much more elegant, and which applies 
to many similar problems. Observing that C l , C 2 , do not depend 
upon the particular functions n and v employed, let us take the 



28 DERIVATIVES AND DIFFERENTIALS [I, 17 

special functions u = e*, v = &, where x and y are the two inde 
pendent variables, and determine the coefficients for this case. We 
thus find 

w = e x+y , dw = e x+y (dx + dy), -, d n <* = e x + y (dx + dy) n , 
du = e*dx, d z u = e x dx*, , 
dv e v dy, d*v = e y dy 2 , ; 

and the general formula, after division by e x+ J , becomes 

(dx + di/} n = dx* + C^dydx*- 1 + C t dy 2 dx n - 2 -\ [-dp. 

Since dx and dy are arbitrary, it follows that 

r _n n(n-l) n(n -1) - (n - p + 1) 

Cl ~l 1.2 " p ~ 1.2-..p 

and consequently the general formula may be written 

(13) d n (uv) = vd n u+^dud n - l u + 7 ^ ^ d 2 vd n ~ 2 u -\ \-ud*v. 

1 1 . 4 

This formula applies for any number of independent variables. 
In particular, if u and v are functions of a single variable x, we 
have, after division by dx n , the expression for the nth derivative of 
the product of two functions of a single variable. 

It is easy to prove in a similar manner formulae analogous to 
(13) for a product of any number of functions. 

Another special case in which the general formula reduces to a 
simpler form is that in which u, v, w are integral linear functions 
of the independent variables x, y, z. 

u= ax -f by + cz+f, 
v = a x + b y + c z +/ , 
w = a"x + b"y + c"z +/", 

where the coefficients a, a , a", b, b , are constants. For then we 

have 

du = a dx + b dy + c dz, 

dv = a dx -f- b dy + c dz, 
dw = a"dx + b"dy + c"dz, 

and all the differentials of higher order d n u, d n v, d n iv, where n>l, 
vanish. Hence the formula for d n <j> is the same as if u, v, w were 
the independent variables ; that is, 



I, 18] THE DIFFERENTIAL NOTATION 29 

(dF . 8F . 8F , V"> 
d n w = -5- du + -T- dv 4- 5 dw I . 



We proceed to apply this remark. 

18. Homogeneous functions. A function <f>(x, y, z) is said to be 
homogeneous of degree m, if the equation 

<(w, v, w)= t m $(x, y, z) 
is identically satisfied when we set 

u = tx, v = ty, w = tz. 

Let xis equate the differentials of order n of the two sides of this 
equation with respect to t, noting that u, v, w are linear in t, and that 

du = x dt, dv = y dt, dw = z dt. 
The remark just made shovvs that 

ihi + y d fo + *^) ( " >== m(m ~ 1} " (m " n +1 ) <m ""*( a; y*)- 

If we now set # = 1, w, v, w reduce to #, ?/, 2, and any term of 
the development of the first member, 



becomes 

d"<> 



whence we may write, symbolically, 



which reduces, for n = 1, to the well-known formula 



Various notations. We have then, altogether, three systems of nota 
tion for the partial derivatives of a function of several variables, 
that of Leibniz, that of Lagrange, and that of Cauchy. Each of 
these is somewhat inconveniently long, especially in a complicated 
calculation. For this reason various shorter notations have been 
devised. Among these one first used by Monge for the first and 



30 



DERIVATIVES AND DIFFERENTIALS 



[I, 19 



second derivatives of a function of two variables is now in common 
use. If z be the function of the two variables x and y, we set 



P 



t = 



dy ex 2 ex 8y o if 

and the total differentials dz and d 2 z are given by the formulae 

dz = p dx + q dy, 
d 2 z = r dx 2 -f- 2 s dx dy + t dy~. 

Another notation which is now coming into general use is the 
following. Let z be a function of any number of independent vari 
ables x 1} x z , x 3) , x n ; then the notation 



ex l ex. 2 ox 
is used, where some of the indices a lt a. 2) , a n may be zeros. 

19. Applications. Let y f(x) be the equation of a plane curve C with 
respect to a set of rectangular axes. The equation of the tangent at a point 
M(x, y) is 

Y-y = y (X-x). 

The slope of the normal, which is perpendicular to the tangent at the point of 
tangency, is l/y ; and the equation of the normal is, therefore, 



Let P be the foot of the ordinate of the point Jlf, and let T and N be the 
points of intersection of the x axis with the tangent and the normal, respectively. 

The distance PN is called the subnormal ; 
FT, the subtangent; MN, the normal; and 
M T, the tangent. 

From the equation of the normal the ab 
scissa of the point N is x + yy , whence the 
subnormal is yy . If we agree to call the 
length PN the subnormal, and to attach the 
sign + or the sign according as the direc 
tion PN is positive or negative, the subnormal 
will always be yy for any position of the curve 
C. Likewise the subtangent is y /y . 
The lengths MN and M T are given by the triangles MPN and MPT: 





Various problems may be given regarding these lines. Let us find, for 
instance, all the curves for which the subnormal is constant and equal to a given 
number a. This amounts to finding all the functions y=f(x) which satisfy 
the equation yy = a. The left-hand side is the derivative of 2/ 2 /2, while the 



I, EXS.] EXERCISES 31 

right-hand side is the derivative of ax. These functions can therefore differ 

only by a constant ; whence 

y 2 = 2ax + C, 

which is the equation of a parabola along the x axis. Again, if we seek the 
curves for which the subtangent is constant, we are led to write down the equa 
tion y /y = l/; whence 

log2/ = - + logC, or y = Ce?, 



a 



which is the equation of a transcendental curve to which the x axis is an asymp 
tote. To find the curves for which the normal is constant, we have the equation 



or 




/a 2 - y* 
The first member is the derivative of - Vo^- y 2 ; hence 



(x + C) 2 + y 2 = a 2 , 

which is the equation of a circle of radius a, whose center lies on the x axis. 

The curves for which the tangent is constant are transcendental curves, which 
we shall study later. 

Let y = f(x) and Y F(x) be the equations of two curves C and C", and let 
M, M be the two points which correspond to the same value of x. In order that 
the two subnormals should have equal lengths it is necessary and sufficient that 

YY =yy ; 

that is, that Y 2 y 2 + C, where the double sign admits of the normals being 
directed in like or in opposite senses. This relation is satisfied by the cirfves 



and also by the curves 



which gives an easy construction for the normal to the ellipse and to the hyperbola. 

EXERCISES 

1. Let p = f(6) be the equation of a plane curve in polar coordinates. Through 
the pole O draw a line perpendicular to the radius 

vector OM, and let T and N be the points where this 
line cuts the tangent and the normal. Find expres 
sions for the distances OT, ON, MN, and MT in 
terms of /(0) and / (<?). 

Find the curves for which each of these distances, 
in turn, is constant. 

2. Let y = f(x), z<t>(x) be the equations of a 

skew curve T, i.e. of a general space curve. Let N FIG. 3 




32 DERIVATIVES AND DIFFERENTIALS [I, Exs. 

be the point where the normal plane at a point Af, that is, the plane perpendicu 
lar to the tangent at .M", meets the z axis ; and let P be the foot of the perpen 
dicular from M to the z axis. Find the curves for which each of the distances 
PN and JOT, in turn, is constant. 

[Note. These curves lie on paraboloids of revolution or on spheres.] 

3. Determine an integral polynomial /(z) of the seventh degree in x, given 
that f(x) + 1 is divisible by (x - I) 4 and f(x) - 1 by (x+1)*. Generalize the 
problem. 

4. Show that if the two integral polynomials P and Q satisfy the relation 

Vl -p-t = Q Vl - x 2 , 
then 

dP ndx 



Vl - p* Vl - x 2 
where n is a positive integer. 
[Note. From the relation 

(a) l-P2 = Q2(l-x) 
it follows that 

(b) - 2 PP = Q [2 Q (l - x*) - 2 Qx]. 

The equation (a) shows that Q is prime to P ; and (b) shows that P is divisible 

by Q-] 

5*. Let E (x) be a polynomial of the fourth degree whose roots are all dif 
ferent, and let x = U / V be a rational function of t, such that 



where R\ (t) is a polynomial of the fourth degree and P / Q is a rational function. 
Show that the function U/ V satisfies a relation of the form 

dx kdt _ 

VR(X) Vfl!() 

where A; is a constant. [JACOBI.] 

[Note. Each root of the equation R(U/ V) = 0, since it cannot cause R (x) 
to vanish, must cause UV VU , and hence also dx/dt, to vanish.] 

6*. Show that the nth derivative of a function y = $ (u), where u is a func 
tion of the independent variable x, may be written in the form 



where 



1.2 

~ (*=1, 2, 

ctx 



[First notice that the nth derivative may be written in the form (a), where the 
coefficients AI, A*, -, A n are independent of the form of the function <j>(u). 



I, EXS.] EXERCISES 33 

To find their values, set (M) equal to w, 2 , , u n successively, and solve the 
resulting equations for A it A^, , A n . The result is the form (b).] 

7*. Show that the nth derivative of <f> (x 2 ) is 
" 2 (>(x 2 ) + n(n - 



dx n 

+ n ( n -V--( n -*P+V (2 X )n- 2P 0("-P)(x 2 ) + , 

1 . ju p 

where p varies from zero to the last positive integer not greater than n/2, and 
where 0(0 (x 2 ) denotes the ith derivative with respect to x. 
Apply this result to the functions er 3 ?, arc sin x, arc tan x. 

8*. If x = cos u, show that 

d-i(l -x 2 )">-* , 1.3.5- -(2m- 1) . 

S = ( l) m ~i sin mu. 

dx m ~ l m 

[OLINDE RODRIGUES.] 

9. Show that Legendre s polynomial, 

2 . 4 . 6 2 n dx" 
satisfies the differential equation 



ax- 1 ax 

Hence deduce the coefficients of the polynomial. 

10. Show that the four functions 

y t = sin (n arc sin x), 2/3 = sin (n arc cos x), 

y 2 = cos (n arc sin x), 2/4 = cos (n arc cos x), 

satisfy the differential equation 

(1 - x 2 ) y" - xy + ri*y = Q. 

Hence deduce the developments of these functions when they reduce to poly 
nomials. 

11*. Prove the formula 

i 

d n G* 

_(x-iei) = (-!)" - 

dx V x+! 

[HALPHEN.] 

12. Every function of the form z = x$(y/x) + $ (y/x) satisfies the equation 

rx 2 + 2 sxy + ty* = 0, 
whatever be the functions <f> and ^. 

13. The function z = x0(x + y) + y^(x + y) satisfies the equation 

r - 2 s + t = 0, 
whatever be the functions and \f/. 



34 DERIVATIVES AND DIFFERENTIALS [I, Exs 



14. The function z =f[x + </>(y)] satisfies the equation ps = qr, whatever 
be the functions / and 0. 

15. The function z = x<j>(y/x) + y~ n ^(y/x) satisfies the equation 

rx 2 + 2 sxy + ty 2 + px + qy = n 2 z, 
whatever be the functions <j> and \f/. 

16. Show that the function 

y - x - ai | 0! (x) + x - a z \ fa (x) + + | x - a n \ n (x), 

where fa (x), fa (x), , n (x), together with their derivatives, 0i (x), (x), , 
0n(x), are continuous functions of x, has a derivative which is discontinuous 
for x = a\ , Oz , , a n 

17. Find a relation between the first and second derivatives of the function 
=/(&! M), where M = 0(x 2 , x 3 ); x t , x 2 , x 3 being three independent variables, 
and /and two arbitrary functions. 

18. Let/"(x) be the derivative of an arbitrary f unction /(x). Show that 

1 d*u, _ 1 #2 
u dx 2 v dx 2 
where u = [/ (x)]-i and =/(x) [/ (x)]-*. 

19*. The nth derivative of a function of a function u-<p(y), where y = ^ (x), 
may be written in the form 



^1.2, 

where the sign of summation extends over all the positive integral solutions of 

the equation i + 2 j + 3 h -\ + Ik = n, and where p = i + j + . + k. 

[FA A DE BRUNO, Quarterly Journal of Mathematics, Vol. I, p. 359.] 



CHAPTER II 

IMPLICIT FUNCTIONS FUNCTIONAL DETERMINANTS 
CHANGE OF VARIABLE 

I. IMPLICIT FUNCTIONS 

20. A particular case. We frequently have to study functions for 
which no explicit expressions are known, but which are given by 
means of unsolved equations. Let us consider, for instance, an 
equation between the three variables x, y, z, 

(1) F(x, y, z) = 0. 

This equation defines, under certain conditions which we are about 
to investigate, a function of the two independent variables x and y. 
We shall prove the following theorem : 

Let x = x , y =. y , z = z b & a set of values which satisfy the equa 
tion (1), and let us suppose that the function F, together with its first 
derivatives, is continuous in the neighborhood of this set of values* 
If the derivative F z does not vanish for x = x , y = y , z = z , there 
exists one and only one continuous function of the independent variables 
x and y which satisfies the equation (1), and which assumes the value z 
when x and y assume the values x and y , respectively. 

The derivative F z not being zero for x = x , y = y , z = z , let us 
suppose, for defmiteness, that it is positive. Since F, F x , F v , F z are 
supposed continuous in the neighborhood, let us choose a positive 
number I so small that these four functions are continuous for all 
sets of values x, y, z which satisfy the relations 

(2) \x-x \<l, \y-y \<l, \*-z <l, 
and that, for these sets of values of x, y, z, 

F z (x,y,z} > P, 

*Iu a recent article (Bulletin de la Societe Mathematique de France, Vol. XXXI, 
190. ?, pp. 184-192) Goursat has shown, by a method of successive approximations, that 
it is not necessary to make any assumption whatever regarding F x and F t/ , even as to 
their existence. His proof makes no use of the existence of F x and F y . His general 
theorem and a sketch of his proof are given in a footnote to 25. TRANS. 

35 



36 FUNCTIONAL RELATIONS [II, 20 

where P is some positive number. Let Q be another positive num 
ber greater than the absolute values of the other two derivatives 
F x , F y in the same region. 

Giving x, y, z values which satisfy the relations (2), we may then 
write down the following identity : 

F (*> V, *) - F(*o, 7/0, ) = F(x, y, z} - F(x , y, z) + F(x , y, z) 

-F(x , T/o, z) +F(x , 7/ 0j z) F(x , 7/0, ) 5 

or, applying the law of the mean to each of these differences, and 
observing that F(x , y , ) = 0, 

F(x>y>*) = (* *o)-F T a r[o + 8(x o), y, *] 
+ - yo) F v [*., 2/o + ff(y - y ), ] 

+ (z - ) F 2 [>o, ?/o, *o + 0"0 - )]. 
Hence -F(cc, T/, 2) is of the form 

(3) S F ^ y ^ = A ( x > y > ^ ( x ~ x d 

I +B(x, y, z) (y - T/ O ) + C (x, y, z) (z - * ), 

where the absolute values of the functions A(x, y, z), B(x, y, z), 
C(x, y, z) satisfy the inequalities 

M|<Q, \B\<Q, \C\>P 

for all sets of values of x, y, z which satisfy (2). Now let c be a 
positive number less than Z, and rj the smaller of the two numbers 
I and Pe/2Q. Suppose that x and y in the equation (1) are given 
definite values which satisfy the conditions 



and that we seek the number of roots of that equation, z being 
regarded as the unknown, which lie between z e and z + c. In 
the expression (3), for F(x, y, z} the sum of the first two terms is 
always less than 2Qrj in absolute value, while the absolute value of 
the third term is greater than Pe when z is replaced by z e. From 
the manner in which 77 was chosen it is evident that this last term 
determines the sign of F. It follows, therefore, that F(x, y, z e ) < 
and F(x, y, z + e ) > ; hence the equation (1) has at least one root 
which lies between z e and z + e . Moreover this root is unique, 
since the derivative F z is positive for all values of z between z e 
and z + e. It is therefore clear that the equation (1) has one and 
only one root, and that this root approaches Z Q as x and y approach 
X Q and ?/ , respectively. 



II, 20] 



IMPLICIT FUNCTIONS 



37 



Let us investigate for just what values of the variables x and y 
the root whose existence we have just proved is denned. Let h be 
the smaller of the two numbers I and PI/2Q; the foregoing reason 
ing shows that if the values of the variables x and y satisfy the 



inequalities \x x^\ 



< h, the equation (1) will have one 

and only one root which lies between z I and z -f I- Let R be a 
square of side 2 h, about the point M (x , y ), with its sides parallel 
to the axes. As long as the point (x, y) lies inside this square, 
the equation (1) uniquely determines a function of x and y, which 
remains between z I and z + I. This function is continuous, by 
the above, at the point M , and this is likewise true for any other 
point M l of R; for, by the hypotheses made regarding the func 
tion F and its derivatives, the derivative F t (x lf y l} i) will be posi 
tive at the point M lt since \x l x <l, \y\ ya\<l, \ z i~ z o\<l- 
The condition of things at M l is then exactly the same as at M , 
and hence the root under consideration will be continuous for 

Since the root considered is defined only in the interior of the 
region R, we have thus far only an element of an implicit function. 
In order to define this function out 
side of R, we proceed by successive 
steps, as follows. Let L be a con 
tinuous path starting at the point 
(x , y ~) and ending at a point (X, F) 
outside of R. Let us suppose that 
the variables x and y vary simul 
taneously in such a way that the ~ 
point (x, y) describes the path L. 
If we start at (x , y ) with the value 

z of z, we have a definite value of this root as long as we remain 
inside the region R. Let M 1 (x l , y^ be a point of the path inside R, 
and z t the corresponding value of z. The conditions of the theorem 
being satisfied for x = x lt y = y l} z = z v , there exists another region 
R l} about the point MI, inside which the root which reduces to z l for 
x = Xi, y = yi is uniquely determined. This new region #! will 
have, in general, points outside of R. Taking then such a point M t 
on the path L, inside R but outside R, we may repeat the same con 
struction and determine a new region R 2 , inside of which the solu 
tion of the equation (1) is defined; and this process could be 
repeated indefinitely, as long as we did not find a set of values of 
x, y, z for which F z = 0. We shall content ourselves for the present 




Fm 4 



38 FUNCTIONAL RELATIONS [II, 21 

with these statements; we shall find occasion in later chapters to 
treat certain analogous problems in detail. 

21. Derivatives of implicit functions. Let us return to the region 
R, and to the solution z = <f>(x, y) of the equation (1), which is a 
continuous function of the two variables x and y in this region. 
This function possesses derivatives of the first order. For, keeping 
y fixed, let us give x an increment Ax. Then z will have an incre 
ment Az, and we find, by the formula derived in 20, 

F(x + As, y,z + A) - F(x, ij, z) 
= Az F x (x + 0Az, y,e + Az) -f Aa F t (x, y, z + Az) = 0. 

Hence 



and when A# approaches zero, As does also, since z is a continuous 
function of a;. The right-hand side therefore approaches a limit, 
and z has a derivative with respect to x : 



In a similar manner we find 



If the equation F = is of degree m in z, it defines m 
functions of the variables x and y, and the partial derivatives cz/cx, 
3z/dy also have m values for each set of values of the variables 
x and y. The preceding formulas give these derivatives without 
ambiguity, if the variable z in the second member be replaced by 
the value of that function whose derivative is sought. 
For example, the equation 

defines the two continuous functions 

+ Vl x 2 y* and Vl x 2 y 2 

for values of x and y which satisfy the inequality x- + y 2 < 1. 
The first partial derivatives of the first are 

- y 



II, 2] IMPLICIT FUNCTIONS 39 

and the partial derivatives of the second are found by merely chang 
ing the signs. The same results would be obtained by using the 

formulae 

dz _ x Cz _ y 

dx z cy z 

replacing z by its two values, successively. 

22. Applications to surfaces. If we interpret x, y, z as the Cartesian 
coordinates of a point in space, any equation of the form 

(4) F(x,y, z)=0 

represents a surface S. Let (cc , y , z^) be the coordinates of a point 
A of this surface. If the function F, together with its first deriva 
tives, is continuous in the neighborhood of the set of values x , y w z , 
and if all three of these derivatives do not vanish simultaneously 
at the point A, the surface S has a tangent plane at A. Suppose, 
for instance, that F z is not zero for x = x , y = y , z = . Accord 
ing to the general theorem we may think of the equation solved 
for z near the point A, and we may write the equation of the surface 

in the form 

z = 4(x, y}, 

where <f> (x, y) is a continuous function ; and the equation of the 
tangent plane at A is 



Replacing dz /dx and dz /dy by the values found above, the equation 
of the tangent plane becomes 



If F z = 0, but F x = 0, at A , we would consider y and z as inde 
pendent variables and a; as a function of them. We would then 
find the same equation (5) for the tangent plane, which is also evi 
dent a priori from the symmetry of the left-hand side. Likewise 
the tangent to a plane curve F(x, y) = 0, at a point (x , y ~), is 



If the three first derivatives vanish simultaneously at the point A. 

dF 



40 FUNCTIONAL RELATIONS [II, 23 

the preceding reasoning is no longer applicable. We shall see later 
(Chapter III) that the tangents to the various curves which lie on 
the surface and which pass through A form, in general, a cone and 
not a plane. 

In the demonstration of the general theorem on implicit functions 
we assumed that the derivative F^ did not vanish. Our geometrical 
intuition explains the necessity of this condition in general. For, 
if F^ = but F^ 3= 0, the tangent plane is parallel to the % axis, 
and a line parallel to the z axis and near the line x = x w y = y 
meets the surface, in general, in two points near the point of 
tangency. Hence, in general, the equation (4) would have two 
roots which both approach z when x and y approach x and y , 
respectively. 

If the sphere a; 2 + y 2 -+- 2 1 = 0, for instance, be cut by the line 
y = 0, x = 1 + c, we find two values of z, which both approach zero 
with e ; they are real if c is negative, and imaginary if c is positive. 

23. Successive derivatives. In the formulae for the first derivatives, 

3z = _Fx dz_ = _F JL 

dx~ F, cy~ F, 

we may consider the second members as composite functions, z being 
an auxiliary function. We might then calculate the successive deriv 
atives, one after another, by the rules for composite functions. The 
existence of these partial derivatives depends, of course, upon the 
existence of the successive partial derivatives of F(x, y, K). 

The following proposition leads to a simpler method of determin 
ing these derivatives. 

If several functions of an independent variable satisfy a relation 
F = 0, their derivatives satisfy the equation obtained by equating to 
zero the derivative of the left-hand side formed by the rule for differ 
entiating composite functions. For it is clear that if F vanishes 
identically when the variables which occur are replaced by func 
tions of the independent variable, then the derivative will also van 
ish identically. The same theorem holds even when the functions 
which satisfy the relation F = depend upon several independent 
variables. 

Now suppose that we wished to calculate the successive derivatives 
of an implicit function y of a single independent variable x defined 
by the relation 



II, 23] IMPLICIT FUNCTIONS 41 

We find successively 

dF 

T~ + 
ox cy 



dF SF , 

~ ~ 



d 2 F dF 

+ 2 v + v 2 + v" =0 
* ^ dxdy y -dy ^ dy y 

2 

3 



dx* ox 2 dy y ox dy* * dxdy dy s 

32 7,1 



from which we could calculate successively y , y", y 



Example. Given a function y =/(x), we may, inversely, consider y as the 
independent variable and x as an implicit function of y defined by the equation 
y=f(x). If the derivative / (x) does not vanish for the value XQ, where 
2/o =/(zo)i there exists, by the general theorem proved above, one and only one 
function of y which satisfies the relation y = f(x) and which takes on the value 
XQ for y = 2/0- This function is called the inverse of the f unction /(x). To cal 
culate the successive derivatives x y , x y t, ay, of this function, we need merely 
differentiate, regarding y as the independent variable, and we get 

1 = / (x) x y , 

= /"(x) (X,) 2 + / (x) ay, 

=/" (x) (x y )* + 3f"(x)x y x? +/ (x)x 2/ 3, 



whence 

1 f"(z) _8[/"(x)]- 

~7^) ~[7w [/ ( 

It should be noticed that these formulae are not altered if we exchange x v and 
/ (x), Xy2 and /"(x), Xj,s and /" (x), , for it is evident that the relation between 
the two functions y = /(x) and x = (y) is a reciprocal one. 

As an application of these formulae, let us determine all those functions 
y=f(x) which satisfy the equation 

y y " - 3y"* = 0. 

Taking y as the independent variable and x as the function, this equation 
becomes 

Xj/> = 0. 

But the only functions whose third derivatives are zero are polynomials of at 
most the second deree. Hence x must be of the form 



where Ci, C 2 , C 3 are three arbitrary constants. Solving this equation for y, 
we see that the only functions y = /(x) which satisfy the given equation are 
of the form _ 

y = a V bx + c, 



42 FUNCTIONAL RELATIONS [II, 24 

where a, 6, c are three arbitrary constants. This equation represents a parabola 
whose axis is parallel to the x axis. 

24. Partial derivatives. Let us now consider an implicit function 
of two variables, denned by the equation 

(6) F(x,y,z) = 0. 

The partial derivatives of the first order are given, as we have seen, 
by the equations 

9F.9F9* ?l dFdz_ 

(7) o p 7T- u, -^ h fl ;p u. 
0x 9* 0z <?y </* 0# 

To determine the partial derivatives of the second order we need 
only differentiate the two equations (7) again with respect to x and 
with respect to y. This gives, however, only three new equations, 
for the derivative of the first of the equations (7) with respect to y 
is identical with the derivative of the second with respect to x. 
The new equations are the following: 



.OJL.? + /!?)%- =o 

dx 2 dxdzdx dz 2 \dx] dz dx 2 
d 2 F , d*F dz ^ d 2 F d_z_ <P_F dz dz d_F d*z _ 

"" r\ n r\ o I O O O ^J 

dx ftr dx dy c~ 



d 2 F d 2 F dz d 2 F (dz\ 2 dF d 2 z 
~ + " fe dy + dz 2 (dy) + ^ dy 2 : 



The third and higher derivatives may be found in a similar manner. 
By the use of total differentials we can find all the partial deriva 
tives of a given order at the same time. This depends upon the 
following theorem : 

If several functions u, v, w, of any number of independent vari 
ables x, y, z, satisfy a relation F = 0, the total differentials satisfy 
the relation dF= 0, which is obtained by forming the total differential 
of F as if all the variables which occur in F ivere independent variables. 

In order to prove this let F(u, v, w) = be the given relation between 
the three functions u, v, iv of the independent variables x, y, z, t. The 
first partial derivatives of M, v, w satisfy the four equations 

dFdu 



__ __ 

du dx dv dx dw dx 



d_Fd_u , 

du dy dv dy dw dy 



II,24J IMPLICIT FUNCTIONS 43 

dFdu dFdv d_F_d-w_ 

o ~ ~o ~ I o ~a ^) 
CM tfS tf OW CZ 

dFdu d_Fd_ ^^! = 

aw & a? a< a^ st = 

Multiplying these equations by dx, dy, dz, dt, respectively, and 
adding, we find 

-^du + ^-dv + d -/-dw = dF=0. 

du dv OW 

This shows again the advantage of the differential notation, for the 
preceding equation is independent of the choice and of the number 
of independent variables. To find a relation between the second 
total differentials, we need merely apply the general theorem to the 
equation dF = 0, considered as an equation between u, v, w, du, 
dv, dw, and so forth. The differentials of higher order than the 
first of those variables which are chosen for independent variables 
must, of course, be replaced by zeros. 

Let us apply this theorem to calculate the successive total differ 
entials of the implicit function defined by the equation (6), where 
x and y are regarded as the independent variables. We find 

* F i ^ F j ^ 3F j 

dx + 7 a;/ + 7 dz = 0, 

ox cy cz 

dF dF 8F V 2 > , dF n 

T- dx + -z- dy + -r- dz ) + <P* = 0, 

dx dy dz / cz 



and the first two of these equations may be used instead of the five 
equations (7) and (8) ; from the expression for dz we may find the 
two first derivatives, from that for d^z the three of the second order, 
etc. Consider for example, the equation 

Ax 2 + A y* + A"z* = l, 
which gives, after two differentiations, 

Ax dx + A ydy + A "z dz = 0, 
A dx 2 + A dy 2 + A "dz 2 + A " zd*z = 0, 

whence 

Axdx + A ydy, 

dg ~ -- TTi - 
A"z 

and, introducing this value of dz in the second equation, we find 
A (A x* + A "z 2 } dx* + 2AA xy dx dy + A (A y* + A "z 2 } dy* 



44 FUNCTIONAL RELATIONS [II, 24 

Using Monge s notation, we have then 

Ax A y 

p= ~IV q ~ ~IV 

A(Ax* + A"z*) _ AA xy 

~ "** 



This method is evidently general, whatever be the number of the 
independent variables or the order of the partial derivatives which 
it is desired to calculate. 

Example. Let z = /(x, y) be a function of x and y. Let us try to calculate 
the differentials of the first and second orders dx and d 2 x, regarding y and z as 
the independent variables, and x as an implicit function of them. First of all, 
we have 

dz = dx + dy. 
dx dy 

Since y and z are now the independent variables, we must set 

d*y = d 2 z = 0, 
and consequently a second differentiation gives 

= ^dx + 2 ^- dxdy + ?^-dy* + ^d?x. 
x 2 dxdy dy* dx 

In Monge s notation, using p, q, r, s, t for the derivatives of /(x, y), these 
equations may be written in the form 

dz p dx + q dy, 
= r dx 2 + 2 s dx dy + tdy* + p d 2 x. 



From the first we find 

, dz q dy 
dx= - -, 
P 

and, substituting this value of dx in the second equation, 
rdz* + 2(ps-qr)dydz + (q*r -2pqs 



<Px= - 



The first and second partial derivatives of x, regarded as a function of y and 
z, therefore, have the following values : 



dx _ 1 8x _ q 

dz p dy p 

d*x _ r d z x _qr ps d 2 x _ 2pqs pH q*r 

dz 2 p 8 dy dz p 3 dy 2 p s 

As an application of these formulae, let us find all those functions /(x, y) 
which satisfy the equation 

= 2pqs. 



If, in the equation z =/(x, y), x be considered as a function of the two inde 
pendent variables y and z, the given equation reduces to Xyt = 0. This means 



II, 25] 



IMPLICIT FUNCTIONS 



45 



that x v is independent of y ; and hence x v = 0(z), where <f>(z) is an arbitrary 
function of z. This, in turn, may be written in the form 



which shows that x - y <f>(z) is independent of y. Hence we may write 



where ^ (z) is another arbitrary function of z. It is clear, therefore, that all the 
functions z =/(x, y) which satisfy the given equation, except those for which f x 
vanishes, are found by solving this last equation for z. This equation represents 
a surface generated by a straight line which is always parallel to the xy plane. 

25. The general theorem. Let us consider a system of n equations 



(E) 



} x pi u l> **ll " i 



* (x x x u u w) = 

between the n-\-p variables u i} u 3 , , u n ; x l} x y , , x p . Suppose 
that these equations are satisfied for the values x v x\, , x p = x p , 
u = wj, j u n = u n ; that the functions F i are continuous and possess 
first partial derivatives which are continuous, in the neighborhood of 
this system of values; and, finally, that the determinant 



du 



does not vanish for 



x,- 



u k = 



Under these conditions there exists one and only one system of con 
tinuous functions u^ = <f>i(xi, x 2 , , x^), , u n <}> n (x 1 , x 2 , , x p ~) 
which satisfy the equations (E) and which reduce to u\, u\, , u n , 
for x, = x\, , x p = x* p * 

*In his paper quoted above (ftn., p. 35) Goursat proves that the same conclusion 
may be reached without making any hypotheses whatever regarding the derivatives 
cFi/dXj of the functions F { with regard to the x s. Otherwise the hypotheses remain 
exactly as stated above. It is to be noticed that the later theorems regarding the 
existence of the derivatives of the functions 4> would not follow, however, without 
some assumptions regarding dF f /dXj. The proof given is based on the following 



46 FUNCTIONAL RELATIONS [II, 26 

The determinant A is called the Jacobian,* or the Functional Deter 
minant, of the n functions F u F 2 , -, F n with respect to the n vari 
ables u l} it?, , u n . It is represented by the notation 

D(F lf F 2 , ...,F,,) 



We will begin by proving the theorem in the special case of a 
system of two equations in three independent variables x, y, z and 
two unknowns u and v. 

(9) Fi(x, y, z, u, v) = 0, 

(10) Fi(x,y,z,u,v) = Q. 

These equations are satisfied, by hypothesis, for x = x ,y = y ,z = z , 
u = u , v = v ; and the determinant 

dF\ dFj _ dF\ dFt 

du cv dv cu 

does not vanish for this set of values. It follows that at least one 
of the derivatives dF^/dv, dF 2 /dv does not vanish for these same 
values. Suppose, for definiteness, that oF l /8u does not vanish. 
According to the theorem proved above for a single equation, the 
relation (9) defines a function v of the variables x, y, z, u, 

v =f( x , y, *> )> 

which reduces to v for x = x , y = y , z = z w u = u . Replacing v 
in the equation (10) by this function, we obtain an equation between 
x, y, z, and u, 

$(, y, z, u} = F t [x, y, z, u, f(x, y, z, )] = 0, 

lemma: Let f\(x\,3kt,---,v p ; MI, u 2 , ,u n ), ,/(!, x?, ,x p ; MI, u 2 , , u n ) be n 
functions of the n + p variables X{ and u^, which, together with the n 2 partial deriva 
tives cfi/GUfr, are continuous near Xi 0, x z = 0, , x p = 0, HI = 0, , u n = 0. If 
the n functions f{ and the n 2 derivatives dfi/^Uf. all vanish for this system of values, 
then the n equations 

i=/i. 2 =/2, " =/ 
admit one and only one system of solutions of the form 



where 1( 2 > > n a continuous functions of the p variables Xi, x 2 , , x p which 
all approach zero as the variables all approach zero. The lemma is proved by means of 
a suite of functions u^ =f i (x 1 ,x z , ,x p \ u[ m ~ l \ w^" -1) , , u^ ^) (i = l, 2, , n), 
where M^ O) = 0. It is shown that the suite of functions u\ m) thus denned approaches a 
limiting function U { , which 1) satisfies the given equations, and 2) constitutes the only 
solution. The passage from the lemma to the theorem consists in an easy transforma 
tion of the equations (E) into a form similar to that of the lemma. TRANS. 
*JACOBI, Crelle s Journal, Vol. XXII. 



II, 25] IMPLICIT FUNCTIONS 47 

which is satisfied for x = x , y y , z = z w u = u . Now 



^ t , . 

du 8u dv du 



and from equation (9), 



du ov ou 
whence, replacing df/du by this value in the expression for 

WP nhfaun 



we obtain 



d D(u, v) 

~du ~ dF^ 

dv 

It is evident that this derivative does not vanish for the values x , 
y< z o> u o- Hence the equation <I> = is satisfied when u is replaced 
by a certain continuous function u = < (x, y, ), which is equal to 
MO when x = x , y = y , z = z ; and, replacing u by < (x, y, z) in 
f(x, y, z, ?/), we obtain for v also a certain continuous function. 
The proposition is then proved for a system of two equations. 

We can show, as in 21, that these functions possess partial 
derivatives of the first order. Keeping y and z constant, let us 
give x an increment Ax, and let AM and Ay be the corresponding 
increments of the functions u and v. The equations (9) and (10) 
then give us the equations 

- + . + A. + . + A,, + ." = 



+ + A. + , + A, + ,- = 0, 





where e, e , e", rj, rj , rj" approach zero with Aa-, A, Av. It follows 
that 



, ^ + c ^ + 77" - ^ + V- 2 + 77 
A;/. \ da; / \ go 7 / V g / \ Ox 

8Fi , A/^ , ,\ /^i , "V aF2 4-V 

\- f. II p t] 

y / \ ou 

When Ax approaches zero, AM and Av also approach zero ; and hence 
e, e , e", 77, 77 , 77" do so at the same time. The ratio Aw /Ax therefore 
approaches a limit ; that is, u possesses a derivative with respect to x : 



48 FUNCTIONAL RELATIONS [II, 26 

dF l cF 2 dF l dF 2 

du dx dv dv dx 



dx dF l dF 2 dFi dF z 

du dv dv du 

It follows in like manner that the ratio Av/Aa; approaches a finite 
limit dv /dx, which is given by an analogous formula. Practically, 
these derivatives may be calculated by means of the two equations 

8 Ft dFj du dF l dv _ 

dx du dx dv dx 

dF 2 dF^du dF^dv 

o "T" <~\ ~r\~ T ~ ~^~~ == " ! 
OX CU OX CV OX 

and the partial derivatives with respect to y and z may be found in 
a similar manner. 

In order to prove the general theorem it will be sufficient to show 
that if the proposition holds for a system of (n 1) equations, it 
will hold also for a system of n equations. Since, by hypothesis, 
the functional determinant A does not vanish for the initial values 
of the variables, at least one of the first minors corresponding to the 
elements of the last row is different from zero for these same values. 
Suppose, for definiteness, that it is the minor which corresponds to 
dF n /du n which is not zero. This minor is precisely 

D(F l} F 2 , -,F n _ 1 ). 
D(UI, 2 , ..., _,) 

and, since the theorem is assumed to hold for a system of (n 1) 
equations, it is clear that we may obtain solutions of the first (n 1) 
of the equations (E) in the form 

M 1 = ^ 1 (a? 1 , a;,, ,*; M n ), -, u n _ l = fa^fa, x 2 , -, x p ; u n ), 

where the functions <,. are continuous. Then, replacing u^ , w n _, 
by the functions ^ 1? ,<_! in the last of equations (E), we obtain 
a new equation for the determination of u n , 



ai >*,; u n ) = F n (x lt ,,-.., x p ; ^ t , <^ 2 , ., B _,, MII ) = 0. 

It only remains for us to show that the derivative d<b/du n does 
not vanish for the given set of values x\, x 2 , -, x p , <; for, if so, we 
can solve this last equation in the form 

= ^0*i, *a, , *p), 

where $ is continuous. Then, substituting this value of u n ir 
<^i5 > <f> n -i> we would obtain certain continuous functions foi 



II, 25] IMPLICIT FUNCTIONS 49 

HI, u 2 , -, u n _ l also. In order to show that the derivative in ques 
tion does not vanish, let us consider the equation 



- . . . 

du n dui du n d u n-i ^ u n du n 

The derivatives 8<j> } /8u n , d<jj 2 /du n , , d* n _i/d n are given by the 
(n 1) equations 



(12) 



7J ~7, r * " T o ~o i o V} 



n-1 tj *Pl i i ^-^n-1 ^yn-1 g -^n-l r\ . 



and we may consider the equations (11) and (12) as n linear equa 
tions for d<f>!/du n , , d<f> n -i/du n , d/du n) from which we find 



cu n D (MU t<2, ,- i) D (M!, ?< 2 , , w n ) 

It follows that the derivative d/du n does not vanish for the initial 
values, and hence the general theorem is proved. 

The successive derivatives of implicit functions defined by several 
equations may be calculated in a manner analogous to that used in 
the case of a single equation. When there are several independent 
variables it is advantageous to form the total differentials, from 
which the partial derivatives of the same order may be found. 
Consider the case of two functions u and v of the three variables 
x, y, z defined by the two equations 

F(x, y, z, u, v)=0, 



The total differentials of the first order du and dv are given by the 
two equations 

3F . , 0F _ . SF . . 8F , .dF 

-%- dx + ^- dy + ^- dz + -5- du + -r- dv = 0, 
dx dy dz du cv 

d d$ .. d& , . d , , d -, A 

-z- dx + -^- du + -^- dz + -r du + -5 c?v = 0. 
Sec cy ^s du cv 

Likewise, the second total differentials d z u and d*v are given by the 
equations 



50 FUNCTIONAL RELATIONS [II, 26 

dF V 2) dF dF 

- dx + -.. + dv) + G /-d*u+ d*v = 0, 
cv du co 

2 > d& d& 

+ ?-*d*u + ~d*v = 0, 

CU CV 

and so forth. In the equations which give d n u and d n v the deter 
minant of the coefficients of those differentials is equal for all vahies 
of n to the Jacobian D(F, <)/Z>(w, v), which, by hypothesis, does not 
vanish. 

26. Inversion. Let MI, 2 , , u n be n functions of the n independent vari 
ables xi, x 2 , , n, such that the Jacobian D(UI, 2, , u,,)/D(xi, x 2 , , x n ) 
does not vanish identically. The n equations 



/i g\ 

( U n = n (X 1( X 2 , -, X n ) 

define, inversely, Xi, x 2 , , x n as functions of u\, M 2 , ,. For, taking any 
system of values x?, x, , x, for which the Jacobian does not vanish, and 
denoting the corresponding values of MI, w 2 , , Un by uj, w!j, , M, there 
exists, according to the general theorem, a system of functions 

which satisfy (13), and which take on the values x", x", -, x, respectively, 
when MI = wj, , u n = u n n . These functions are called the inverses of the func 
tions 0i, 2 , -, n , and the process of actually determining them is called 
an inversion. 

In order to compute the derivatives of these inverse functions we need merely 
apply the general rule. Thus, in the case of two functions 

u=f(x, y), v = <f>(x,y) t 

if we consider u and v as the independent variables and x and y as inverse 
functions, we have the two equations 



whence 



, 8f, , Bf , , d<f> , 30 , 

du = dx + dy, dv = - dx + - dy. 

dx cy dx cy 



^0 j %f -i c0 . df 

ait av du H - dv 

, dy dy , dx dx 

dx = , dy = 

a/^0_?/a0 a/ a0 _ ^/ ^0 

dx dy dy dx dx dy dy dx 



We have then, finally, the formulae 



50 _ d_f 

dx dy dx dy 



du 8/a0_a/c0 dv d_fdj>_d_fd_$ 

dx dy dy dx dx dy dy dx 



II, 27] IMPLICIT FUNCTIONS 51 



8f 

dx dy ex 



eu ~" df _ cf d<t> eB^ 

ex e^ ey ex ex ey ey ex 

27. Tangents to skew curves. Let us consider a curve C repre 
sented by the two equations 

l*i(*,y,)-0, 

(14) < 

JF,(a5,y,) = 5 

and let x , T/ O , 2 be the coordinates of a point M of this curve, such 
that at least one of the three Jacobians 

dF\dF^ _d_F\d_F\ dF 1 dFt_dF\ 8F* 8F l gF 2 dF_i dF 

dy dz dz dy vz dx dx dz ex dy dij dx 

does not vanish when x, y, z are replaced by x , %, z ot respectively. 
Suppose, for defmiteness, that D(F l} Fj/D(y, z) is one which does 
not vanish at the point M n . Then the equations (14) may be solved 
in the form 

y = ^(x) ) z = t( x )> 

where $ and \j/ are continuous functions of x which reduce to y and 
z , respectively, when x = x . The tangent to the curve C at the 
point 3/o is therefore represented by the two equations 

X-x = F-7/Q = Z -g 
1 ^ (x ) " f (x ) 

where the derivatives <#> (cc) and i//(ce) may be found from the two 
equations 



In these two equations let us set x = x 0) y = y ,z = 2^, and replace 
* (*) and ^o) by ( F - T/ O ) / (X - *) and (Z - )/(X - x ), 
respectively. The equations of the tangent then become 




62 FUNCTIONAL RELATIONS [II, 28 

or 

X x a Y y Z z 



^(y, *) o ^>(, ) o fl(,y 

The geometrical interpretation of this result is very easy. The 
two equations (14) represent, respectively, two surfaces Sj and S 2 , of 
which C is the line of intersection. The equations (15) represent 
the two tangent planes to these two surfaces at the point 1/ ; and 
the tangent to C is the intersection of these two planes. 

The formulae become illusory when the three Jacobians above all 
vanish at the point M . In this case the two equations (15) reduce 
to a single equation, and the surfaces Si and S 2 are tangent at the 
point A/ . The intersection of the two surfaces will then consist, in 
general, as we shall see, of several distinct branches through the 
point M . 

II. FUNCTIONAL DETERMINANTS 

28. Fundamental property. We have just seen what an important 
role functional determinants play in the theory of implicit functions. 
All the above demonstrations expressly presuppose that a certain 
Jacobian does not vanish for the assumed set of initial values. 
Omitting the case in which the Jacobian vanishes only for certain 
particular values of the variables, we shall proceed to examine the 
very important case in which the Jacobian vanishes identically. 
The following theorem is fundamental. 

Let HI, u 2 , , u n be n functions of the n independent variables 
x \i x ii ") x n" I n order that there exist between these n functions 
a relation II (M I} w 2 > > u n) == 0, which does not involve explicitly any 
of the variables x ly x z , , x n , it is necessary and sufficient that the 
functional determinant 

D(UI, M 2 , , ?y) 



should vanish identically. 

In th.3 first place this condition is necessary. For, if such a rela 
tion TL(UI, w 2 , , w n ) = exists between the n functions HI, u%, , u n , 
the following n equations, deduced by differentiating with respect to 
each of the z s in order, must hold : 



II, 28.1 FUNCTIONAL DETERMINANTS 53 

end Ul 8udu 2 an 8u n __ 

7; ^ ~T~ "^ "o ~r T Q /-, " , 



jfi , jf2 , = Q. 

dui 8x n du 2 dx n du n dx n 

and, since we cannot have, at the same time, 

^5 = = = ^5 = 

! <7U 2 CU H 

since the relation considered would in that case reduce to a trivial 
identity, it is clear that the determinant of the coefficients, which is 
precisely the Jacobian of the theorem, must vanish.* 

The condition is also sufficient. To prove this, we shall make 
use of certain facts which follow immediately from the general 
theorems. 

1) Let u, v, w be three functions of the three independent variables 
x, y, z, such that the functional determinant D(u, v, w)/D(x, y, z) 
is not zero. Then no relation of the form 

A du + /u, dv + v dw = 

can exist between the total differentials du, dv, dw, except for 
X = p, = v = 0. For, equating the coefficients of dx, dy, dz in the 
foregoing equation to zero, there result three equations for X, p., v 
which have no other solutions than X = /u, = v = 0. 

2) Let w, u, v, w be four functions of the three independent 
variables x, y, z, such that the determinant D (u, v, w} / D (x, y, s) 
is not zero. We can then express x, y, z inversely as functions of 
u, v, w t and substituting these values for x, y, z in o>, we obtain 

a function 

a, = $ (u, v, w) 

of the three variables u, v, ^v. If by any process ivhatever we can 
obtain a relation of the form 

(16) du = P du + Q dv + R dw 

*As Professor Osgood has pointed out, the reasoning here supposes that the 
partial derivatives an / Si/i , dU /, , ^H / dUn do not all vanish simultaneously 
for any system of values which cause U (u lf u 2 , -,) to vanish. This supposition 
is certainly justified when the relation II = is solved for one of the variables u t . 



54 FUNCTIONAL RELATIONS [II, 28 

between the total differentials dw, du, dv, dw, taken with respect to the 
independent variables x, y, z, then the coefficients P, Q, R are equal, 
respectively, to the three first partial derivatives of < (u, v, w) : 

d& d$ 8<b 

P = o Q = ~o ** = o 

Cu cv ow 

For, by the rule for the total differential of a composite function 
( 16), we have 

d& d<b d& 

<D = - du + -^ dv -|- dw : 
du cv cw 

and there cannot exist any other relation of the form (16) between 
d<a, du, dv, dw, for that would lead to a relation of the form 

A. du + p. do + v dw = 0, 

where X, /t, v do not all vanish. We have just seen that this is 
impossible. 

It is clear that these remarks apply to the general case of any 
number of independent variables. 

Let us then consider, for definiteness, a system of four functions 
of four independent variables 



(17) 



X = F l (x,y,z, *), 
Y=Fi (x, y, z, t), 
Z = F 3 (x, y, z, t), 
T=F t (x,y,z, t), 



where the Jacobian D(F l} F 2 , F 3 , F i )/D(x, y, z, t) is identically 
zero by hypothesis ; and let us suppose, first, that one of the first 
minors, say D(F^ F 2 , F s )/D(x, y, z), is not zero. We may then 
think of the first three of equations (17) as solved for x, y, z as 
functions of X, Y, Z, t ; and, substituting these values for x, y, z in 
the last of equations (17), we obtain T as a function of A , Y, Z, t: 

(18) T=*(X,Y,Z,t). 

We proceed to show that this function $ does not contain the vari 
able t, that is, that 8$ /dt vanishes identically. For this purpose 
let us consider the determinant 



II, 



FUNCTIONAL DETERMINANTS 



55 



A = 



QFi 


dj\ 


df\ 


dX 


dx 


dt/ 


dz 




dx 


dF, 


dF z 

dz 


dY 


~dx~ 


dF s 

ty 


dF s 

dz 


dZ 


dx 


dF\ 


dz 


dT 



If, in this determinant, dX, dY, dZ, dT be replaced by their values 



ox 



tiy 



Ct 



and if the determinant be developed in terms of dx, dy, dz, dt, it turns 
out that the coefficients of these four differentials are each zero ; the 
first three being determinants with two identical columns, while the 
last is precisely the functional determinant. Hence A = 0. But if 
we develop this determinant with respect to the elements of the last 
column, the coefficient of dTis not zero, and we obtain a relation of 
the form 

dT = P dX + Q dY + R dZ. 

By the remark made above, the coefficient of dt in the right-hand 
side is equal to d<i?/dt. But this right-hand side does not contain 
dt, hence d&/dt = 0. It follows that the relation (18) is of the form 



which proves the theorem stated. 

It can be shown that there exists no other relation, distinct from 
that just found, between the four functions X, Y, Z, T, independent 
of x, y, z, t. For, if one existed, and if we replaced T by $>(X, Y, Z) 
in it, we would obtain a relation between X, Y, Z of the form 
U(X, Y, Z)=0, which is a contradiction of the hypothesis that 
D(X, Y, Z)/D(x, y, z) does not vanish. 

Let us now pass to the case in which all the first minors of the 
Jacobian vanish identically, but where at least one of the second 
minors, say D(F lt F^)/D(x, y}, is not zero. Then the first two of 
equations (17) may be solved for x and y as functions of X. Y, z, t, 
and the last two become 

Z = *! (X, Y, z, t), T = . 2 (A , Y, z, t). 



56 FUNCTIONAL RELATIONS [n, 

On the other hand we can show, as before, that the determinant 

dX 
dY 
dZ 



ex 



ex 



fy 





ex 



dy 



vanishes identically ; and, developing it with respect to the elements 
of the last column, we find a relation of the form 



dZ = FdX + QdY, 



whence it follows that 



In like manner it can be shown that 

!r= 



dt 



= 0; 



and there exist in this case two distinct relations between the four 
functions X, Y, Z, T, of the form 



There exists, however, no third relation distinct from these two; 
for, if there were, we could find a relation between X and Y, which 
would be in contradiction with the hypothesis that D(X, Y} / D(x, y) 
is not zero. 

Finally, if all the second minors of the Jacobian are zeros, but 
not all four functions X, Z, Y, T are constants, three of them are 
functions of the fourth. The above reasoning is evidently general. 
If the Jacobian of the n functions F 1} F 2 , , F H of the n independ 
ent variables x ly x 2 , , x n , together with all its (n r + 1) -rowed 
minors, vanishes identically, but at least one of the (n r)- rowed 
minors is not zero, there exist precisely r distinct relations between 
the n functions ; and certain r of them can be expressed in terms 
of the remaining (n r), between which there exists no relation. 

The proof of the following proposition, which is similar to the 
above demonstration, will be left to the reader. The necessary and 
sufficient condition that n functions of n + p independent variables be 
connected by a relation which does not involve these variables is that 
every one of the Jacobians of these n functions, with respect to any n 



II, 28] FUNCTIONAL DETERMINANTS 57 

of the independent variables, should vanish identically. In par 
ticular, the necessary and sufficient condition that two functions 
F i(#i , x z , , CC B ) and F 2 (x l , x z ,---, #) should be functions of each 
other is that the corresponding partial derivatives dF 1 /dx i and 
dF 2 /dXf should be proportional. 

Note. The functions F 19 F 2 , , F n in the foregoing theorems may 
involve certain other variables y 1} y 2) -, y m , besides x l , x 2 , , x n . 
If the Jacobian D(F l} F z , , F n )/D(x l , x 2 , -, oj n ) is zero, the 
functions JF\, F 2 , , F n are connected by one or more relations 
which do not involve explicitly the variables x 1} x 2 , , x n , but 
which may involve the other variables y 1} y 2 , , y m . 

Applications. The preceding theorem is of great importance. The funda 
mental property of the logarithm, for instance, can be demonstrated by means 
of it, without using the arithmetic definition of the logarithm. For it is proved 
at the beginning of the Integral Calculus that there exists a function which is 
defined for all positive values of the variable, which is zero when x 1, and 
whose derivative is l/x. Let/(x) be this function, and let 

u=f(x)+f(y), v = xy. 
Then 

D (u, v) _ 



D (x, y) 



x y =0. 

y x 

Hence there exists a relation of the form 
f(x)+f( V ) = 

and to determine we need only set y = 1, which gives f(x) = <j> (x). Hence, 
since x is arbitrary, 

f(z)+f(y)=f(xy). 

It is clear that the preceding definition might have led to the discovery of 
the fundamental properties of the logarithm had they not been known before the 
Integral Calculus. 

As another application let us consider a system of n equations in n unknowns 



(MI, 2 , 



(19) 



. Fn(Ul, W 2 , 



where J/i, JT 2 ) ) H n are constants or functions of certain other variables 
*i *2 Xmi which may also occur in the functions .F,-. If the Jacobian 
Z)(Fi, F 2 , , F n )/D(u\, 2i > u n ) vanishes identically, there exist between 
the n functions F, a certain number, say n fc, of distinct relations of the form 

, F t ) t , F lt = U n - k (F!, , F k ). 



58 



FUNCTIONAL RELATIONS 



[II, 29 



In order that the equations (19) be compatible, it is evidently necessary that 
H t + l = Hi (Hn .,H k ),..-,H n = U H - t (Hi, , H k ), 

and, if this be true, the n equations (19) reduce to k distinct equations. We 
have then the same cases as in the discussion of a system of linear equations. 

29. Another property of the Jacobian. The Jacobian of a system of n 
functions of n variables possesses properties analogous to those of 
the derivative of a function of a single variable. Thus the preceding 
theorem may be regarded as a generalization of the theorem of 8. 

The formula for the derivative of a function of a function may be 
extended to Jacobians. Let F lf F 2 , , F n be a system of n func 
tions of the variables M I} u 2 , , u n , and let us suppose that u^ w 2 > 
-, u a themselves are functions of the n independent variables x lf 
x x. Then the formula 



D(F 



l , 



-, F n ) D( UI , 



D(x lt 



D(x 1} 



follows at once from the rule for the multiplication of determinants 
and the formula for the derivative of a composite function. For, 
let us write down the two functional determinants 



cj\ 

ou 



dF 



du 



dx n dx u 



cx 



where the rows and the columns in the second have been inter 
changed. The first element of the product is equal to 



, 



dF l 



i 



du,, 



that is, to 



?!, and similarly for the other elements. 



30. Hessians. Let/(x, ?/, z) be a function of the three variables x, y, z. Then 
the functional determinant of the three first partial derivatives cf/dx, Sf/cy, 
df/dz, 

a 2 / a 2 / a 2 / 

ax 2 ex 5y dx az 

a 2 / a 2 / a 2 / 
ax cy a?/ 2 
a 2 / a 2 / 



ex cz cy oz 



dydz 

a 2 / 

cz- 



II, 30] 



FUNCTIONAL DETERMINA NTS 



59 



is called the Hessian of f(x, y, z). The Hessian of a function of n variables is 
defined in like manner, and plays a role analogous to that of the second deriva 
tive of a function of a single variable. We proceed to prove a remarkable 
invariant property of this determinant. Let us suppose the independent vari 
ables transformed by the linear substitution 



(X= aX+ [3Y+ yZ, 
y= a X + p Y+ y Z, 



(19 ) 



where X, F, Z are the transformed variables, and or, 0, 7, , 7" are constants 
such that the determinant of the substitution, 

a J8 7 
A = a /3 7 
a" /3" 7" 

is not zero. This substitution carries the function /(x, y, z) over into a new 
function F(X, Y, Z) of the three variables X, Y, Z. Let II (X, F, Z) be the 
Hessian of this new function. We shall show that we have identically 

II (X, F, Z) = A 2 /t(x, ?/, z), 

where x, ?/, z are supposed replaced in /i(x, y, z) by their expressions from (19 ). 
For we have 

fZF dF dF^ 



H = 



7 ^ ?I\ 

BY cZ ) ~\dX aT aZ/ D(x, y, 



D(X, Y, Z) D(x, y, z) D(X, Y, Z) 

and if we consider cf/cx, cf/dy, df/dz, for a moment, as auxiliary variables, 
we may write 



By cz D(x, y, 



^, ^, Kl 

cx dy dz / 



D(x,y,z) U(X, Y, Z) 



But from the relation F(X, Y, Z) =f(x, y, z), we find 

dF cf ,cf ,,Bf 
- = a + a + a" , 
dX ex cy dz 



dY 



dy 



whence 



3F a/ , a/ , c/ 

-^ = 7 + 7 + 7 
c : Z cx cy cz 



d_F dF 

er ez 



and hence, finally, 



dx dy dz 



a a a" 



7 7 



= A; 



H= 



D(x, y, z) 



D(X, Y, Z) 
It is clear that this theorem is general. 



- = A 2 /i- 



60 FUNCTIONAL RELATIONS [II, 30 

Let us now consider an application of this property of the Hessian. Let 
/(x, y) = ox 3 + 3 bx*y + 3 cxy 2 + dy* 

be a given binary cubic form whose coefficients a, b, c, d are any constants. 
Then, neglecting a numerical factor, 



h = 



ax + by bx 4 cy 
bx + cy ex + dy 



= (ac - & 2 )x 2 + (ad - bc)xy + (bd - c 2 )y 2 , 



and the Hessian is seen to be a binary quadratic form. First, discarding the 
case in which the Hessian is a perfect square, we may write it as the product of 

two linear factors : 

h = (mx + ny) (px + qy). 

If, now, we perform the linear substitution 

mx + ny = X, px + qy = Y, 
the form/(x, y) goes over into a new form, 

F(X, Y) = AX* + 3 BX 2 Y+3 CXY 2 + DY 8 , 
whose Hessian is 

H(X, Y) = (AC - B 2 ) X 2 + (AD - EC] XY + (BD - C 2 ) F 2 , 

and this must reduce, by the invariant property proved above, to a product of 
the form KXY. Hence the coefficients A, B, C, D must satisfy the relations 



If one of the two coefficients B, C be different from zero, the other must be so, 
and we shall have 

-? -f- 

F(X, Y) = (B*X* + 3 B 2 CX* Y + 3 BC* XY 2 + C* Y 3 ) = ( B ^+^ Y )\ 

whence F(X, Y), and hence /(x, y), will be a perfect cube. Discarding this 
particular case, it is evident that we shall have B = C = ; and the polynomial 
F(X, Y) will be of the canonical form 

AX* + DY 3 . 

Hence the reduction of the form /(x, y) to its canonical form only involves the 
solution of an equation of the second degree, obtained by equating the Hessian 
of the given form to zero. The canonical variables X, Y are precisely the two 
factors of the Hessian. 

It is easy to see, in like manner, that the form/(x, y) is reducible to the form 
AX 3 + BX 2 Y when the Hessian is a perfect square. When the Hessian van 
ishes identically /(x, y) is a perfect cube : 

/(x, y) = (ax 



II, 31] TRANSFORMATIONS 61 



III. TRANSFORMATIONS 

It often happens, in many problems which arise in Mathematical 
Analysis, that we are led to change the independent variables. It 
therefore becomes necessary to be able to express the derivatives 
with respect to the old variables in terms of the derivatives with 
respect to the new variables. We have already considered a problem 
of this kind in the case of inversion. Let us now consider the 
question from a general point of view, and treat those problems 
which occur most frequently. 

31. Problem I. Let y be a function of the independent variable x, 
and let t be a new independent variable connected luith x by the relation 
x = <(). It is required to express the successive derivatives of y with 
respect to x in terms of t and the successive derivatives of y with 
respect to t. 

Let y=f(x) be the given function, and F(t) =/[<()] the func 
tion obtained by replacing x by <j>(t) in the given function. By the 
rule for the derivative of a function of a function, we find 

dy dy ,. . 
37 = ~r~ X 9 m, 
at ax 

whence 

dy 

dt y t 



This result may be stated as follows : To find the derivative of y 
with respect to x, take the derivative of that function with respect to t 
and divide it by the derivative of x with respect to t. 

The second derivative d 2 y/dx* may be found by applying this 
rule to the expression just found for the first derivative. We find : 

-I 

d Ll = _ -y^ (0-y^"(0. 
dx* w) [> (0] 

and another application of the same rule gives the third derivative 



62 FUNCTIONAL RELATIONS [H, 32 

or, performing the operations indicated, 



_ 

<* [> (OJ 6 

The remaining derivatives may be calculated in succession by 
repeated applications of the same rule. In general, the nth deriva 
tive of y with respect to x may be expressed in terms of <}> (), <j>"(t), 
, < (n) (), and the first n successive derivatives of y with respect to 
t. These formulae may be arranged in more symmetrical form. 
Denoting the successive differentials of x and y with respect to t by 
dx, dy, d z x, d*y, -, d"x, d n y, and the successive derivatives of y 
with respect to x by y , y", , y (n \ we may write the preceding 
formulae in the form 



(20) 



y 7 

9 . dx 



f _ dx d 2 y dy 6? 



dx 3 
x 2 - 3 d?y dx d z x + 3dy (d*x)* - dy d*x dx 

y ~~ 5 



The independent variable t, with respect to which the differentials 
on the right-hand sides of these formulae are formed, is entirely 
arbitrary ; and we pass from one derivative to the next by the 
recurrent formula 



, , 

<>= 

the second member being regarded as the quotient of two differen 
tials. 

32. Applications. These formulas are used in the study of plane 
curves, when the coordinates of a point of the curve are expressed in 
terms of an auxiliary variable t. 

=/(*) y = * co 

in order to study this curve in the neighborhood of one of its points 
it is necessary to calculate the successive derivatives y , ?/", of y 
with respect to x at the given point. But the preceding formulas 
give us precisely these derivatives, expressed in terms of the succes 
sive derivatives of the functions f(t) and <j> (#), without the necessity 



II, 32] TRANSFORMATIONS 63 

of having recourse to the explicit expression of y as a function of x, 
which it might be very difficult, practically, to obtain. Thus the 
first formula 

y> = dx = f (t) 

gives the slope of the tangent. The value of y" occurs in an impor 
tant geometrical concept, the radius of curvature, which is given by 
the formula 






which we shall derive later. In order to find the value of R, when 
the coordinates x and y are given as functions of a parameter t, we 
need only replace y and y" by the preceding expressions, and we 
find 

(dx 2 4- dy^Y 
R = . , ^r~ " 



where the second member contains only the first and second deriva 
tives of x and y with respect to t. 

The following interesting remark is taken from M. Bertrand s Traitt de 
Calcul differentiel et integral (Vol. I, p. 170). Suppose that, in calculating some 
geometrical concept allied to a given plane curve whose coordinates x and y are 
supposed given in terms of a parameter , we had obtained the expression 

F(x, y, dx, dy, d 2 x, d 2 y, -, d n x, dy), 

where all the differentials are taken with respect to t. Since, by hypothesis, 
this concept has a geometrical significance, its value cannot depend upon the 
choice of the independent variable t. But, if we take x = t, we shall have 
dx dt, d z x = d 3 x = = d a x = 0, and the preceding expression becomes 

f(x, y, y , y", > 2/ ( ) ; 

which is the same as the expression we would have obtained by supposing at the 
start that the equation of the given curve was solved with respect to y in the 
form y = *(). To return from this particular case to the case where the inde 
pendent variable is arbitrary, we need only replace y , y", by their values 
from the formulae (20). Performing this substitution in 



we should get back to the expression F(x, y, dx, dy, d 2 x, d 2 y, ) with which 
we started. If we do not, we can assert that the result obtained is incorrect. 
For example, the expression 

dxd 2 y + dyd 2 x 



64 FUNCTIONAL RELATIONS [II, 33 

cannot have any geometrical significance for a plane curve which is independent 
of the choice of the independent variable. For, if we set x = t, this expression 
reduces to y" /(I + y 2 )$ ; and, replacing y and y" by their values from (20), we 
do not get back to the preceding expression. 

33. The formulae (20) are also used frequently in the study of 
differential equations. Suppose, for example, that we wished to 
determine all the functions y of the independent variable x, which 
satisfy the equation 

(21) (1 _^*_ e eg + . = , 

where n is a constant. Let us introduce a new independent variable 
t, where x = cos t. Then we have 

dy 
dy dt 

dx sin t 

d*y dy 

smt-jfi. cost 
d?y at* dt < 

dx 2 sin 8 1 

and the equation (21) becomes, after the substitution, 

(22) 

It is easy to find all the functions of t which satisfy this equation, 
for it may be written, after multiplication by 2 dy /dt, 



whence 



where a is an arbitrary constant. Consequently 



or 




71 = 0. 



II, 34] TRANSFORMATIONS 65 

The left-hand side is the derivative of arc sin (y/a) nt. It follows 
that this difference must be another arbitrary constant b, whence 

y = a sm(nt + &), 
which may also be written in the form 

y = A sin nt + B cos nt. 

Returning to the original variable x, we see that all the functions of 
x which satisfy the given equation (21) are given by the formula 

y = A sin (n arc cos a) + B cos (n arc cos a), 
where A and B are two arbitrary constants. 

34. Problem II. To every relation between x and y there corresponds, 
by means of the transformation x = f(t, u), y = <f>(t, u*), a relation 
between t and u. It is required to express the derivatives of y with 
respect to x in terms of t, u. and the derivatives of u with respect to t. 

This problem is seen to depend upon the preceding when it is 
noticed that the formulae of transformation, 



give us the expressions for the original variables x and y as func 
tions of the* variable t , if we imagine that u has been replaced in 
these formulas by its value as a function of t. We need merely 
apply the general method, therefore, always regarding x and y as 
composite functions of t, and u as an auxiliary function of t. We 
find then, first, 

8<jt 8<ft du 

dy _dy dx dt du dt 
dx dt dt df df du 
dt du dt 
and then 

d?y __ d (dy\ dx 
dx" 2 dt \dxj dt 

or, performing the operations indicated, 



, _ _ , , 

SuBtdt du*\dt) gu dfr* \ dt + du dt/\dt* 



t du dt 



66 FUNCTIONAL RELATIONS [II, 33 

In general, the nth derivative y (n) is expressible in terms of t, u, and 
the derivatives du/dt, d?vi/dt 2 , , d n u/dt n . 

Suppose, for instance, that the equation of a curve be given in 
polar coordinates p = /(o>). The formulae for the rectangular coor 
dinates of a point are then the following : 

x = p cos <D, y p sin w. 

Let p , p", be the successive derivatives of p with respect to w, 
considered as the independent variable. From the preceding formulae 

we find 

dx = cos (a dp p sin w e?w, 

dy = sin o> dp + p cos w d<a, 
d 2 x = cosu) d z p 2 sin w dai dp p cos w da?, 
d 2 y sinw d 2 p + 2 cosw rfw ffy p sin w 7(o 2 , 
whence 

<&e 2 + dif 1 = dp 2 + p 2 rfw 2 , 
dij d^x = 2 du dp 2 p d<a d 2 p + p 2 c?w 3 . 



The expression found above for the radius of curvature becomes 



p + *p m -pp 

35. Transformations of plane curves. Let us suppose that to every 
point m of a plane we make another point M of the same plane cor 
respond by some known construction. If we denote the coordinates 
of the point m by (x, y) and those of M by (X, F), there will exist, 
in general, two relations between these coordinates of the form 

(23) X=f(x,y), Y=4>(x, y}. 

These formulae define a point transformation of which numerous 
examples arise in Geometry, such as projective transformations, the 
transformation of reciprocal radii, etc. When the point m describes 
a curve c, the corresponding point M describes another curve C, whose 
properties may be deduced from those of the curve c and from the 
nature of the transformation employed. Let y , ?/", be the suc 
cessive derivatives of y with respect to x, and F , F", the succes 
sive derivatives of F with respect to X. To study the curve C it 
is necessary to be able to express F , F", in terms of x, y, y , 
y", -. This is precisely the problem which we have just discussed ; 
and we find 



II, 36] 



TRANSFORMATIONS 



67 



dY 



Y 


dx 


dx dy 


Y" 


dX 
dx 
dY 
dx 


dx dy 




dX 
dx 


/df df ,V 

1 o i ~ y } 

\dx dy I 



and so forth. It is seen that Y depends only on x, y, y . Hence, 
if the transformation (23) be applied to two curves c, c , which are 
tangent at the point (x, ?/), the transformed curves C, C will also 
be tangent at the corresponding point (A , F). This remark enables 
us to replace the curve c by any other curve which is tangent to it 
in questions which involve only the tangent to the transformed 
curve C. 

Let us consider, for example, the transformation defined by the 
formulae 



Y = 




<- A 



x 2 + y 2 

which is the transformation of reciprocal radii, or inversion, with 
the origin as pole. Let m be a point of a curve c and M the cor 
responding point of the curve C. In 
order to find the tangent to this curve 
C we need only apply the result of 
ordinary Geometry, that an inversion 
carries a straight line into a circle 
through the pole. 

Let us replace the curve c by its 
tangent mt. The inverse of mt is a 
circle through the two points Mand O, 
whose center lies on the perpendicular 

Ot let fall from the origin upon mt. The tangent MT to this circle 
is perpendicular to AM, and the angles Mmt and mMT are equal, 
since each is the complement of the angle mOt. The tangents mt 
and MT are therefore antiparallel with respect to the radius vector. 

36. Contact transformations. The preceding transformations are 
not the most general transformations which carry two tangent 
curves into two other tangent curves. Let us suppose that a point 
M is determined from each point m of a curve c by a construction 



FIG. 5 



68 FUNCTIONAL RELATIONS [II, 36 

which depends not only upon the point m, but also upon the tangent 
to the curve c at this point. The formulae which define the trans 
formation are then of the form 

(24) X = /(*, y, y-), Y=<j>(x, y, y 1 ) ; 

and the slope Y of the tangent to the transformed curve is given 
by the formula 



dx dy dy> y 

In general, F depends on the four variables x, y, y , y" ; and if we 
apply the transformation (24) to two carves c, c which are tangent 
at a point (x, y~), the transformed curves C, C will have a point 
(X, Y) in common, but they will not be tangent, in general, unless 
y" happens to have the same value for each of the curves c and c . 
In order that the two curves C and C should always be tangent, it 
is necessary and sufficient that Y should not depend on y"; that is, 
that the two functions f(x, y, y ) and < (x, y, y ) should satisfy the 
condition 



In case this condition is satisfied, the transformation is called a 
contact transformation. It is clear that a point transformation is a 
particular case of a contact transformation.* 

Let us consider, for example, Legendre s transformation, in which 
the point M, which corresponds to a point (x, y) of a curve c, is given 

by the equations 

X = y , Y=xy -y; 

from which we find 

Y , _dY _xjf _ 
~ dX - y" 

which shows that the transformation is a contact transformation. 
In like manner we find 

dY dx 1 






V 11 =r 



dX y"dx y" 

y 



dX 



*Legendre and Ampere gave many examples of contact transformations. Sophus 
Lie developed the general theory in various works ; see in particular his Geometric 
der Beruhrungstransformationen. See also JACOBI, Vorlesungen iiber Dynamik. 



II, 37] TRANSFORMATIONS 69 

and so forth. From the preceding formulae it follows that 
x = Y , y = XY -Y, y = X, 

which shows that the transformation is involutory.* All these prop 
erties are explained by the remark that the point whose coordinates 
are X = y , Y = xy y is the pole of the tangent to the curve c at 
the point (x, y) with respect to the parabola x 2 2 y = 0. But, in 
general, if M denote the pole of the tangent at m to a curve c with 
respect to a directing conic 2, then the locus of the point M is a, 
curve C whose tangent at M is precisely the polar of the point m 
with respect to 2. The relation between the two curves c and C is 
therefore a reciprocal one ; and, further, if we replace the curve c by 
another curve c , tangent to c at the point m, the reciprocal curve C 
will be tangent to the curve C at the point M. 

Pedal curves. If, from a fixed point O in the plane of a curve c, a perpen 
dicular OM be let fall upon the tangent to the curve at the point m, the locus of 
the foot M of this perpendicular is a curve (7, which is called the pedal of the 
given curve. It would be easy to obtain, by a direct calculation, the coordinates 
of the point Jlf, and to show that the trans 
formation thus defined is a contact transfor 
mation, but it is simpler to proceed as follows. 
Let us consider a circle 7 of radius E, de 
scribed about the point as center; and let ?MI 
be a point on OM such that Om\ x OM= E 2 . 
The point mi is the pole of the tangent mt 
with respect to the circle ; and hence the 
transformation which carries c into C is the 
result of a transformation of reciprocal po- 
lars, followed by an inversion. When the 
point m describes the curve c, the point mi, 
the pole of mt, describes a curve Ci tangent 
to the polar of the point m with respect to 

the circle 7, that is, tangent to the straight line miti, a perpendicular let fall 
from mi upon Om. The tangent M Tto the curve C and the tangent m\ti to the 
curve Ci make equal angles with the radius vector OmiM. Hence, if we draw 
the normal MA, the angles AMO and AOM are equal, since they are the comple 
ments of equal angles, and the point A is the middle point of the line Om. It 
follows that the normal to the pedal is found by joining the point Mto the center 
of the line Om. 

37. Projective transformations. Every function y which satisfies the equation 
y" = is a linear function of x, and conversely. But, if we subject x and y to 
the projective transformation 

* That is, two successive applications of the transformation lead us back to the 
original coordinates. TRANS. 




70 FUNCTIONAL RELATIONS [II, 38 

_ aX + bY+c _ a X+ b Y + c 

a" X + b" Y + c" a" X -f b" Y + c" 

a straight line goes over into a straight line. Hence the equation y" = should 
become d ^Y/dX" 2 0. In order to verify this we will first remark that the 
general projective transformation may be resolved into a sequence of particular 
transformations of simple form. If the two coefficients a" and b" are not both 
zero, we will set X\ = a" X -\- b" Y + c" ; and since we cannot have at the same 
time ab" ba" = and a b" b a" = 0, we will also set YI a X + b Y + c , 
on the supposition that a b" b a" is not zero. The preceding formulae may 
then be written, replacing X and Y by their values in terms of Xi and F l5 in 
the form 

YI a Xi + /3 FI + 7 YI 7 

A! Xi AI Xi 

It follows that the general projective transformation can be reduced to a 
succession of integral transformations of the form 

x aX + bY + c, y - a X + b Y + c , 
combined with the particular transformation 

1 Y 

x = , y = . 

X X 

Performing this latter transformation, we find 



and 



dy _-.-_ 

~ dx ~ ~~ 



y" = ~y~ = - XY"(- X*) = X s Y". 
dx 



Likewise, performing an integral projective transformation, we have 
d y a + b Y 



y - 



a + bY 



_ dt/ _ (ab -ba )Y" 
~ dx (a + bY ) 3 

In each case the equation y" goes over into Y" = 0. 

We shall now consider functions of several independent variables, and, for 
definiteness, we shall give the argument for a function of two variables. 

38. Problem III. Let w = f(x, y) be a function of the two independ 
ent variables x and y, and let u and v be two new variables connected 
with the old ones by the relations 



It is required to express the partial derivatives of u with respect to the 
variables x and y in terms of u, v, and the partial derivatives of u> with 
respect to u and v. 



II, 



TRANSFORMATIONS 71 



Let w = F(u, v) be the function which results from/(x, y) by the 
substitution. Then the rule for the differentiation of composite 

functions gives 

c oj 8 a) 8 <J> d o d\ff 

cu dx cu dy du 

Cd) C w C (jt d ta C\}/ 

dv dx cv dy dv 

whence we may find d<a/dx and du/dy; for, if the determinant 
D(<f>, \ji)/D(u, v) vanished, the change of variables performed 
would have no meaning. Hence we obtain the equations 



(25) 



dw d(a d d> 

o~ = A a r B ~z~ 
ex cu cv 

Cu) __ Cd) Coi 

c -^ + u ~^~~> 

cy cu cv 



where A, B, C, D are determinate functions of u and v ; and these 
formulae solve the problem for derivatives of the first order. They 
show that the derivative of a function with respect to x is the sum of 
the two products formed by multiplying the two derivatives with respect 
to u and v by A and B, respectively. The derivative with respect to 
y is obtained in like manner, using C and D instead of A and B, 
respectively. In order to calculate the second derivatives we need 
only apply to the first derivatives the rule expressed by the preced 
ing formulae ; doing so, we find 

2 <o d /8u>\ d I t d& i 
575 == 2~\ "-~/ == 2\p r 

C/X Q<Kf \ t/JC / GOT/ \ C/it 

\ / \ 

(31 C o> v to \ .- w . _ _ _ 

= A (A - + B - + B y- (A r- + B -5- 
cu\ Cu cv / dv\ cu 

or, performing the operations indicated, 

^= A[A-z+Br-%:+-^-- + -z 

^-^ 2 v ^"^ CU CV CU CU CU CV 

-B t-^ + 




My P cy </ 0v cv 

and we could find 8 2 (a/dxdy, 3 2 <a/di/ 2 and the following derivatives 
in like manner. In all differentiations which are to be carried out 
we need only replace the operations d /dx and d /dy by the operations 

d 8 d d 

A-Z-+B-Z-* C + D^-, 

du Cv cu cv 



72 



FUNCTIONAL RELATIONS 



[II, 38 



respectively. Hence everything depends upon the calculation of the 
coefficients A, B, C, D. 

Example I. Let us consider the equation 



(26) 



C CO , (i3 dJ 

a +26 -- + c = 0, 



ex ci/ cy* 

where the coefficients a, 6, c are constants ; and let us try to reduce this equa 
tion to as simple a form as possible. We observe first that if a = c = 0, it would 
be superfluous to try to simplify the equation. We may then suppose that c, 
for example, does not vanish. Let us take two new independent variables u 
and v, defined by the equations 

u = x + ay, v = x + py, 
where a and /3 are constants. Then we have 

c u < <jj cu 



cx 

d<a 

- 

8y 



du I w 

a + P , 

du cv 



and hence, in this case, A = B = 1, C = a, D = p. The general formulae then 
give 



dx? 



~dii? cucv "ai? 2 " 



ducv 



eu 2 



and the given equation becomes 

(a + 26a + ca 2 )^ + 2 [a + b(a + ft) + ca/3]-^- + (a -(- 

au-^ au a CB 

It remains to distinguish several cases. 

First case. Let 6 2 ac> 0. Taking for a and the two roots of the equation 
a .)- 2 6r + cr 2 = 0, the given equation takes the simple form 



cudv 



= 0. 



Since this may be written 



we see that dta/Su must be a function of the single variable, w, say/(w). Let 
F(u) denote a function of u such that F (u) =f(u). Then, since the derivative 
of w F(u) with respect to u is zero, this difference must be independent of w, 
and, accordingly, u = F(u) + *(). The converse is apparent. Returning to 
the variables x and y, it follows that all the functions w which satisfy the equation 
(26) are of the form 



II, 38] TRANSFORMATIONS 73 

where F and $ are arbitrary functions. For example, the general integral of 

the equation 

c 2 w O c 2 w 

= a 2 , 
cy 2 dx 2 

which occurs in the theory of the stretched string, is 
w =f(x + ay) + <f> (x - ay). 

Second case. Let b 1 ac = 0. Taking a equal to the double root of the equa 
tion a -f 26r -f cr 2 = 0, and |3 some other number, the coefficient of d^w/dudv 
becomes zero, for it is equal to a + ba + p(b + car). Hence the given equation 
reduces to 5 2 w/cz> 2 = 0. It is evident that w must be a linear function of t>, 
w = vf(u) + <f> (u), where f(u) and <f> (u) are arbitrary functions. Returning to 
the variables x and y, the expression for w becomes 

w = (x + Py)f(x + ay) + <f>(x + ay), 
which may be written 

w = [x + ay + (p - a)y]f(x + ay) + <f>(x + ay), 
or. finally, 

w = yF(x + ay) + <l>(x + ay). 

Third case. If 6 2 ac < 0, the preceding transformation cannot be applied 
without the introduction of imaginary variables. The quantities a and /3 may 
then be determined by the equations 

a + 26a + c a 2 = 
a + b(a + p) 

which give 

2b 2& 2 -ac 

a + /3= , a/3= 

c c 2 

The equation of the second degree, 

26 26 2 -ac 

r 2 H r H = 0, 

c c 2 

whose roots are a and , has, in fact, real roots. The given equation then 

becomes 

a 2 w c 2 w 

Aw = H = 0. 

du 2 c 2 

This equation Aw = 0, which is known as Laplace s Equation, is of fundamental 
importance in many branches of mathematics and mathematical physics. 

Example II. Let us see what form the preceding equation assumes when we 
set x = p cos <, y = p sin 0. For the first derivatives we find 

8 u du Su 

= cos^ H 

8p dx dy 

i u ("ui d u 

= p sin <b -\ p cos <z>, 

p * 



74 FUNCTIONAL RELATIONS [H,a9 

or, solving for dw/dx and du/cy, 

du du sin rf> du 

- COS <p i 

dx dp p dip 

du du cosrf) du 

- =sm0 + 



dy dp p dip 

Hence 

a I du sin0 du\ sind> 8 t 
( costf> --- - ) -- ( 
dp\ dp p 8<f>/ p d<t>\ 



du simp du 

--- - 
dp p 

d 2 u sin 2 0a 2 w 2 sin </> cos S 2 w 2sin0cos<dw sin 2 0cw 

= COS 2 <t> -- 1 ------- 1 ---- 1 --- i 

a/? 2 p 2 dtf> 2 p dp dip p 2 d<f> p dp 

and the expression for d 2 u/dy 2 is analogous to this. Adding the two, we find 



39. Another method. The preceding method is the most practical 
when the function whose partial derivatives are sought is unknown. 
But in certain cases it is more advantageous to use the following 
method. 

Let z =f(x, y) be a function of the two independent variables x 
and y. If x, y, and z are supposed expressed in terms of two aux 
iliary variables u and v, the total differentials dx, dy, dz satisfy the 
relation 

O /> o / 

dz = -^- dx + -- dy. 
ex cy 

which is equivalent to the two distinct equations 



_ 
du dx du dy du 



dz _d_fdx 

dv dx dv dy dv 

whence df/dx and df/dy may be found as functions of u, v, dz/du, 
dz/dv, as in the preceding method. But to find the succeeding 
derivatives we will continue to apply the same rule. Thus, to find 
d 2 f/dx 2 and d 2 f/dxdy, we start with the identity 



> 

dx 2 dxcy 

which is equivalent to the two equations 

d (dx) = d 2 fdx | ay dy } 

du dx 2 du dx dy du 



II, 39] TRANSFORMATIONS 75 




: | } 

dec 2 ov dx dy dv 

where it is supposed that df/dx has been replaced by its value cal 
culated above. Likewise, we should find the values of d 2 f/dx dy and 
2 by starting with the identity 



df\ a 2 / 8 2 f 

a I Q ?T *** ~cT~a *% 
dy/ dxdy dy 2 

The work may be checked by the fact that the two values of 
c 2 f/8x dy found must agree. Derivatives of higher order may be 
calculated in like manner. 

Application to surfaces. The preceding method is used in the study 
of surfaces. Suppose that the coordinates of a point of a surface S 
are given as functions of two variable parameters u and v by means 
of the formulae 

(27) x=f(u,v), y = $(u,v), z = f(u,v). 

The equation of the surface may be found by eliminating the vari 
ables u and v between the three equations (27); but we may also 
study the properties of the surface S directly from these equations 
themselves, without carrying out the elimination, which might be 
practically impossible. It should be noticed that the three Jacobians 



D(, -y) D(u, v) 

cannot all vanish identically, for then the elimination of w and v 
would lead to two distinct relations between x, y, z, and the point 
whose coordinates are (x, y, %) would map out a curve, and not a sur 
face. Let us suppose, for definiteness, that the first of these does not 
vanish : D(f, <j>)/D(u, v) = 0. Then the first two of equations (27) 
may be solved for u and v, and the substitution of these values in the 
third would give the equation of the surface in the form z = F(x, y). 
In order to study this surface in the neighborhood of a point we need 
to know the partial derivatives p, q, r,s,t, of this function F(x, y) 
in terms of the parameters u and v. The first derivatives p and q 
are given by the equation 

dz = p dx -f- q dy, 
which is equivalent to the two equations 



76 FUNCTIONAL RELATIONS [II, 40 

^ - n ^4-n^ 

TT = p T; -- r q r~ 

du f du du 

Q a 

cv cv cv 

from which p and q may be found. The equation of the tangent 
plane is found by substituting these values of p and q in the equation 

Z - z = p(X - *) + q(Y - y), 
and doing so we find the equation 



The equations (28) have a geometrical meaning which is easily 
remembered. They express the fact that the tangent plane to the 
surface contains the tangents to those two curves on the surface which 
are obtained by keeping v constant while u varies, and vice versa* 

Having found p and q, p =/i(w, v), q = f 2 (u, v), we may proceed 
to find r, s, t by means of the equations 

dp = r dx + s dy, 
\ 

dq = s dx + t dy, 

each of which is equivalent to two equations ; and so forth. 

40. Problem IV. To every relation between x, y, z there corresponds 
by means of the equations 

(30) x =/(w, v, w), y = < (M, v, w), z = \j/(u, v, w), 

a new relation between u, v, w. It is required to express the partial 
derivatives of z with respect to the variables x and y in terms of u, v, w, 
and the partial derivatives of iv with respect to the variables u and v. 

This problem can be made to depend upon the preceding. For, 
if we suppose that w has been replaced in the formulae (30) by a 
function of u and v, we have x, y, z expressed as functions of the 

* The equation of the tangent plane may also be found directly. Every curve on 
the surface is defined by a relation between u and w, say v = U (u) ; and the equations 
of the tangent to this curve are 

X-x Y-y Z-z 

df df ~ dd> d4> ~ d4> d& 

f- + -f n () ^ + ~ n () -^- + - H (M) 

du dv du dv du dv 

The elimination of IT(a) leads to the equation (29) of the tangent plane. 



II, 41] TRANSFORMATIONS 77 

two parameters u and v; and we need only follow the preceding 
method, considering /, <, ^ as composite functions of u and v, and 
w as an auxiliary function of u and v. In order to calculate the 
first derivatives p and y, for instance, we have the two equations 



_ , , 

P a Q T 5 I cT~ i T~ ~o 1+7 "^ -- P ^ -- o 

* cu ow 8u du dw cu 

_ a^ a_w 

dv dw dv \d 8w dv \8v + 8w 

The succeeding derivatives may be calculated in a similar manner. 

In geometrical language the above problem may be stated as fol 
lows : To every point m of space, whose coordinates are (x, y, z), 
there corresponds, by a given construction, another point M, whose 
coordinates are X, Y, Z. When the point m maps out a surface S, 
the point M maps out another surface 2, whose properties it is pro 
posed to deduce from those of the given surface S. 

The formulae which define the transformation are of the form 



x =f( x > y> ), y = <t> (*, y, *), 

Let 



Y) 



be the equations of the two surfaces S and 2, respectively. The 
problem is to express the partial derivatives P, Q, R, S,T, of the 
function $(A", Y) in terms of x, y, z and the partial derivatives 
p, q, r, s, t, of the function F(x, y). But this is precisely the 
above problem, except for the notation. 

The first derivatives P and Q depend only on x, y, z, p, q ; and 
hence the transformation carries tangent surfaces into tangent sur 
faces. But this is not the most general transformation which enjoys 
this property, as we shall see in the following example. 

41. Legendre s transformation. Let z =f(x, y) be the equation of 
a surface S, and let any point m (x, ?/, z) of this surface be carried 
into a point M, whose coordinates are X, Y, Z, by the transformation 

X=p, Y = q, Z px + qy z. 

Let Z = $ (X, Y) be the equation of the surface 2 described by the 
point M. If we imagine z, p, q replaced by /, df/dx, df/dy, respec 
tively, we have the three coordinates of the point M expressed as 
functions of the two independent variables x and y. 



78 FUNCTIONAL RELATIONS [II, 41 

Let P, Q, R, S, T denote the partial derivatives of the function 
$>(X, Y). Then the relation 

dZ = PdX+ QdY 
becomes 

p dx + q dy + x dp + y dq dz = P dp + Q, dq, 
or 

x dp + y dq = P dp -f Q dq. 

Let us suppose that p and q, for the surface S, are not functions of each 
other, in which case there exists no identity of the form \dp + p.dq=. 0, 
unless X = fi 0. Then, from the preceding equation, it follows that 



In order to find R, S, T we may start with the analogous relations 

dP = RdX + SdY, 
dQ = SdX+ TdY, 

which, when X, Y, P, Q are replaced by their values, become 

dx = R (r dx + s dy) + S (s dx + t dy) , 
dy = S (r dx + s dy~) + T(sdx + t dy} ; 
whence 



and consequently 

t s r 

rt s 2 rt s 2 rt s 2 

From the preceding formulae we find, conversely, 

x = P, y=Q, z = PX+QY-Z, p = X, q = Y, 

T - S R 

t 



RT- ^ RT- .S 2 RT - S 2 

which proves that the transformation is involutory. Moreover, it 
is a contact transformation, since X, Y, Z, P, Q depend only on x, 
y, z, p, q. These properties become self-explanatory, if we notice 
that the formulae define a transformation of reciprocal polars with 
respect to the paraboloid 

x 2 + y 2 - 2 z = 0. 

Note. The expressions for R, S, T become infinite, if the relation 
rt s 2 = holds at every point of the surface S. In this case the 
point M describes a curve, and not a surface, for we have 



II, 42] TRANSFORMATIONS 79 

= ,, = 

*,y) D*,y 

and likewise 

D(X, Z) = Jfo ;>* + gy - *) = _ ^ = 
D(*,y) (*, ?/) 

This is precisely the case which we had not considered. 

42. Ampfere s transformation. Retaining the notation of the preceding article, 
let us consider the transformation 

X x, Y = q, Z qy z. 
The relation 

dZ = PdX + QdY 
becomes 

qdy + ydq - dz = Pdx + Qdq, 
or 

y dq p dx = Pdx + Qdq. 
Hence 

P=-P, Q = y ; 

and conversely we find 

x = X, y=Q, z=QY-Z, p = - P, q = Y. 

It follows that this transformation also is an involutory contact transformation. 

The relation 

dP = EdX+ SdY 
next becomes 

r dx s dy = R dx + S (s dx + t dy) ; 
that is, 

R -f Ss = - r, St = - s, 

whence 



Starting with the relation dQ = SdX + TdY, we find, in like manner, 

r=l. 
t 

As an application of these formulae, let vis try to find all the functions /(x, y) 
which satisfy the equation rt s 2 = 0. Let S be the surface represented by the 
equation z =f(x, y), S the transformed surface, and Z = 4>(X, Y) the equation 
of S. From the formulae for R it is clear that we must have 

R - ^ - 
~ S~ ~ 

and * must be a linear function of X : 



where and ^ are arbitrary functions of F. It follows that 



80 FUNCTIONAL RELATIONS [II, 43 

and, conversely, the coordinates (a;, y, z) of a point of the surface S are given 
as functions of the two variables X and Y by the formulae 

x = X, y = Xt (Y) + y(Y), z = Y[X<t> (Y) + f (Y)] - X<f,(Y) - t(Y). 

The equation of the surface may be obtained by eliminating X and Y ; or, what 
amounts to the same thing, by eliminating a between the equations 

z = ay -x<t>(a)-t (a), 
= y x (a) ^ (a). 

The first of these equations represents a moving plane which depends upon the 
parameter a, while the second is found by differentiating the first with respect 
to this parameter. The surfaces defined by the two equations are the so-called 
developable surfaces, which we shall study later. 

43. The potential equation in curvilinear coordinates. The calculation to which 
a change of variable leads may be simplified in very many cases by various 
devices. We shall take as an example the potential equation in orthogonal 
curvilinear coordinates.* Let 

F (x, y, z) - p, 

FI(X, y, Z)=PI, 

F*(X, y, )=P2i 

be the equations of three families of surfaces which form a triply orthogonal 
system, such that any two surfaces belonging to two different families intersect 
at right angles. Solving these equations for x, y, z as functions of the parame 
ters p, pi, PQ, ^6 obtain equations of the form 

fx-<t>(p, PI, pa), 

( 31 ) j y = <Pi(p, Pi, Pa), 

l*= 02 (p, Pi, pa); 

and we may take p, p t , p^ as a system of orthogonal curvilinear coordinates. 

Since the three given surfaces are orthogonal, the taagents to their curves of 
intersection must form a trirectangular trihedron. It follows that the equations 



must be satisfied where the symbol x> indicates that we are to replace by <i, 
then by 2 , and add. These conditions for orthogonalism may be written in the 
following form, which is equivalent to the above : 





dp dpi 8p dpi dp dpi _ 


0, 


(33) 


x dx By dy dz cz 

dp dpz , n dpi Spy 
-H \j. -+- . . 


V 7 

= 0. 


. Bx dx dx dx 





* Lame", TraiU des coordonnees curvilignes. See also Bertrand, Traitt de Calcul 
differ entiel, Vol. I, p. 181. 



H, 43] TRANSFORMATIONS 81 

Let us then see what form the potential equation 



ax 2 dy* az 2 
assumes in the variables p, p 1? p 2 . First of all, we find 

dv _ dv dp 8V dpi aF apa 

ax dp dx dpi dx dp z dx 
and then 

a 2 F_ a^F/aA 2 a 2 F ap a^. aF av 

ax 2 ~ a/ 2 \ax/ apapi ax "aaT "a7 ax 2 



| 2 a 2 F a Pl a P2 | 



p^ \dx/ dpidpz ax ax api ax 2 



, 



-- 
\ax/ a/>ap 2 ax ax ap a ax 2 



Adding the three analogous equations, the terms containing derivatives of the 
second order like a 2 V / dp dpi fall out, by reason of the relations (33), and we have 



a 2 F . . , v a 2 F 

(34) 



A 2 (p) ^ + A 2 ( P1 ) ~ + A 2 (p 2 ) pT, 
op cpi apa 



where AI and A 2 denote Lam&s differential parameters : 



The differential parameters of the first order Ai(p), Ai(pi), Aj(p 2 ) are easily 
calculated. From the equations (31) we have 

a^ ap a^ api aj^ apg _ 

ap ax api ax ap 2 ax 

a 01 ap a 01 api a 01 ap 2 _ 

ap ax api ax apa ax 

a0 2 a_p a0 2 api a0 2 ap 2 _ . _ 

ap ax api ax ap 2 ax 

whence, multiplying by , -, -, respectively, and adding, we find 
dp dp dp 

a0 

ap _ 

ax ~ 



/a0\ 2 /a0A 2 /a0 2 
\dp) + \dp/ + \dpj 



Then, calculating dp/dy and dp/dz in like manner, it is easy to see that 
i 2 /ap\ 2 1 



*/ \dy/ \dz/ (d<f>\ , ... + 



82 FUNCTIONAL RELATIONS [II, 43 



Let us now set 

a = 



dp 

where the symbol $ indicates, as before, that we are to replace <f> by 0i, then 
by 02) and add. Then the preceding equation and the two analogous equations 
may be written 

A!(P) = , Ai(pi) = , A!(p 2 ) = 
H HI HZ 

Lame* obtained the expressions for A 2 (p), A 2 (pi), A 2 (p 2 ) as functions of p, pi, 
pa by a rather long calculation, which we may condense in the following form. 
In the identity (34) 

1 a 2 F 1 &V , 1 a 2 F , .8V , .8V , . , ,aF 

A 2 F = -- + T + -^ + A 2 (p) + A 2 (pi) + A 2 (p 2 ) i 
H dp z HI dpi H 2 epjj dp dpi dpz 

let us set successively V x, V = y, V = z. This gives the three equations 

i a 2 i 52^ i c> 2 a0 



2 -- =0, 

dp* HI cpj HZ cpz dp 



. , -TT + W VF + A2 ( p ) iF + **( pl ) -^ + ^(pz) 1 - = 0, 
H Cfr HI Cpj .a 2 cp 2 op cpi cpz 



which we need only solve for A a (p), A 2 (pi), A 2 (p 2 ). For instance, multiplying 
by d<p/dp, d<j>i/dp, dfa/dp, respectively, and adding, we find 



Moreover, we have 

oe>4> 6 2 _ 1 ag 
*-> ~d~p Up* ~ 2 lp~ 

and differentiating the first of equations (32) with respect to pi, we find 



~dp ~dp\ ~~ V dpi dp dpi ~ 2 
In like manner we have 

o0 ev _ _ i a g 2 

^ ap" "ap| ~ 2 ap 

and consequently 



A 2 (p)= - - + -- 



2HHi dp 2HH* dp 2Hp_ \H 1 H 2 



H. EXS.] EXERCISES 83 

Setting 

*=i, zr,= , *,= , 

this formula becomes 

A 2 (p) = ^ 

and in like manner we find 

A 2 (p!) = h\ A (log A.) , A 2 (p 2 ) = A" A (log A 
api \ hhzj cpz \ fifii 

Hence the formula (34) finally becomes 



(35) 



^ ^ * ll*z 

ax 2 a 2 az 2 



ra 2 F , a /. A \aF~l 

-- h I log - I I 

L V <>P \ *!*/ ^/ J 



* 

a 



or, in condensed form, 

. Fa / h dV\ , a / ftj aF\ a / & 2 dF\~| 
2 * 2 1 \ / \ nr / I ^T^ s / I 

Let us apply this formula to polar coordinates. The formulae of transforma 
tion are 

x = p sin0cos</>, y 



where and <f replace pi and p 2 , and the coefficients ft, fti, hq have the following 
values : 

& = 1 hi = - h% = : - 
p p sin 

Hence the general formula becomes 



i Fa / aF\ a / . aF\ a / i aF\~| 

A 2 F= _ - _ Ip 2 sin(? - J + (sin^ ) + (- -J h 

P 2 sine \_dp \ dp] de\ c0J a0 \sin^ a^ / J 



or, expanding, 

a 2 F i a 2 F i a 2 F 2 aF cot0aF 

Ao F = -- 1 ---- h - -- h --- 1 --- 

ap 2 p 2 a*? 2 p 2 sin 2 a^> 2 p ap p 2 a<? 

which is susceptible of direct verification. 



EXERCISES 

1. Setting u = x 2 + y 2 + z 2 , v = x + y + z, w = xy + yz + zx, the functional 
determinant D (u, t>, w>) /D (, y, 2) vanishes identically. Find the relation which 
exists between M, v, w. 

Generalize the problem. 



84 FUNCTIONAL RELATIONS pi, EXS. 



2. Let 

i = ==. 1 = , u n - 



Vi _ -r 2 a- 2 

1 x l *n 

Derive the equation 

i, W2, , M,,) . 1 



3. Using the notation 

Xi = COS 0i, 

X2 = sin 0i cos 02 , 

x 3 = sin 0i sin 02 cos0 3 , 



Xn = sin 0i sin 02 sin0 n _ 1 cos0 n , 
show that 

(Xl, Xg, , X n ) _ ^_ 1 )n s i nn ^ lS J n n-l^ 2Sm n-2^ g . . . 8 in 2 n _ ! sin n . 

4. Prove directly that the function z = F(x, y} defined by the two equations 

z = ax+ yf(a) + 0(or), 
= x + yf (a) + (a), 

where a. is an auxiliary variable, satisfies the equation rt s 2 = 0, where /(a) 
and (a) are arbitrary functions. 

5. Show in like manner that any implicit function z = F(x, y) defined by 
an equation of the form 



where (z) and ^ (z) are arbitrary functions, satisfies the equation 
rg 2 - 2pqs + tp 2 = 0. 

6. Prove that the function z = F(x, y) defined by the two equations 

z (a) = [y - (a)] 2, ( x + a) (a) = y - (a), 

where a is an auxiliary variable and (a) an arbitrary function, satisfies the 
equation pq = z. 

7. Prove that the function z F(x, y) defined by the two equations 

[Z - ()] 2 = Z 2 (y2 _ a 2), [z _ ((r)] (Q ,) = aa .2 

satisfies in like manner the equation pq = xy. 

8*. Lagrange s formulae. Let y be an implicit function of the two variables 
x and a, defined by the relation y = a + x<j>(y); and let u =f(y) be any func 
tion of y whatever. Show that, in general, 



[LAPLACE.] 



II, EM.] EXERCISES 85 

Note. The proof is based upon the two formulas 

d F, x du~\ d F 7 \dM~l d u i \ cu 

F(u)~ = F(u) . <(,(y) , 

da |_ dx J dx ]_ da_\ dx da 

where u is any function of y whatever, and F(u) is an arbitrary function of u. 
It is shown that if the formula holds for any value of n, it must hold for the 
value n + I. 

Setting x = 0, y reduces to a and M to /(a); and the nth derivative of u with 
respect to x becomes 

ff/*\ 
^dx n 

9. If x =f(u, v), y = (J>(u, v) are two functions which satisfy the equations 

dj d dj d <j> 

du dv dv du 

show that the following equation is satisfied identically : 



\ -! i r~ ~i 

M \ S" ,., I 

:/o da- l \_ J 



, 

10. If the function F(x, y, 2) satisfies the equation 



show that the function 



satisfies the same equation, where A; is a constant and r 2 = x 2 + y z + z 2 . 

[LORD KELVIN.] 



11. If V(x, y, z) and Vi(x, y, z) are two solutions of the equation A^V = 0, 
show that the function 

U = F(z, y, z) + (x 2 + y 2 + z 2 ) FI (x, y, z) 
satisfies the equation 



12. What form does the equation 

(x - x 8 )7/"+ (1 - Sx 2 )?/ - xy = 
assume when we make the transformation x = Vl t- ? 

13. What form does the equation 

2 

= 



Sx 2 dx dy 

assume when we make the transformation x = u, y = l/v? 

14*. Let 0(xi, x 2 , , x n ; MI, MS, , u n ) be a function of the 2 n independent 
variables Xi, x 2 , , x n , MI, u 2 , , M n , homogeneous and of the second degree 
with respect to the variables MI, M 2 , ,. If we set 



86 FUNCTIONAL RELATIONS [II, Exs. 

S C(t> cd> 

-=pi, -=pz, , ~^-=p n , 

CUi CUz CU n 

and then take p\ , pz , , p n as independent variables in the place of Ui , u z , , u n , 
the function <f> goes over into a function of the form 



d<f> 



i, X 2 , 

Derive the formulae : 



15. Let N be the point of intersection of a fixed plane P with the normal MN 
erected at any point M of a given surface S. Lay off on the perpendicular to the 
plane P at the point N a length Nm = NM. Find the tangent plane to the 
surface described by the point m, as M describes the surface S. 

The preceding transformation is a contact transformation. Study the inverse 
transformation. 

16. Starting from each point of a given surface S, lay off on the normal to 
the surface a constant length I. Find the tangent plane to the surface 2 (the 
parallel surface) which is the locus of the end points. 

Solve the analogous problem for a plane curve. 

17*. Given a surface S and a fixed point O ; join the point to any point M of 
the surface S, and pass a plane OM N through OM and the normal MN to the 
surface S at the point M. In this plane OMN draw through the point O a per 
pendicular to the line OM, and lay off on it a length OP = OM. The point P 
describes a surface 2, which is called the apsidal surface to the given surface S. 
Find the tangent plane to this surface. 

The transformation is a contact transformation, and the relation between the 
surfaces <S and 2 is a reciprocal one. When the given surface S is an ellipsoid 
and the point is its center, the surface 2 is Fresnel s wave surface. 

18*. Halphen s differential invariants. Show that the differential equation 



dx 2 / da* dx 2 dx dx* \dx* 

remains unchanged when the variables x, y undergo any projective transfor 
mation ( 37). 

19. If in the expression Pdx + Qdy + fidz, where P, Q, R are any functions 
of x, y, z, we set 

x=/(u, B, 10), y = <f> (u, v, w) , Z = ^(M, v, >), 

where , v, w are new variables, it goes over into an expression of the form 
Pidu + Q\dv + Ridw, 

where PI, Qi, RI are functions of , v, w. Show that the following equation is 
satisfied identically: 

gi = -P(Si ^ z) g> 
I) (M, v, w) 



II, Exs.] EXERCISES 87 

where 



/* _ 

\dv du 



20*. Bilinear covariants. Let 0</ be a linear differential form : 



where JTx, X 2 , , -^ n are functions of the n variables x lt x 2 , , x n . Let us 
consider the expression 



where 



and where there are two systems of differentials, d and 5. If we make any 
transformation 

Xi = 4>i(yi, y 2 , , 2/ n ), (i = 1, 2, ., n), 

the expression Qj goes over into an expression of the same form 

Q d = Y l dy l + + Y n dy n , 
where FI, F 2 , , Y n are functions of yi, y 2 , , y n . Let us also set 



and 



Show that H = H , identically, provided that we replace dx, and dxt, respec 
tively, by the expressions 



Syi -\ 5j/ 2 + H Sy n . 

The expression // is called a bilinear covariant of Qj. 

21*. Beltrami s differential parameters. If in a given expression of the form 
Edx* + 2Fdxdy + Gdy*, 

where E, F, G are functions of the variables x and y, we make a transformation 
z =/(M, v), y = <f>(u, ), we obtain an expression of the same form: 

EI du 2 + 2Fidudv+ Gj dv 2 , 



88 



FUNCTIONAL RELATIONS 



[II, Exs. 



where EI, FI, G\ are functions of u and v. Let 0(x, y) be any function of the 
variables x and y, and 0i(u, v) the transformed function. Then we have, iden 
tically, 



ax 



- 2 F + * 

ex dy \dy 



du dv 



dv 



EG- F 2 


E l G l - F? 


IG _ F \ 


(E F 


i 


8 


dx 


gyl | 


1 




dy 


ax 


VF.G - F 2 


dx \ 


^EG- 


"F 2 / 


^EG - F 2 




^EG- 


F" 2 


i 


-( 


J1 au 


? i-~\ 

4- 


1 


d 


^~&o~ 


1 du 



ffj G! - Ff 



- F 



I 



22. Schwarzian. Setting y (ax + b) / (ex +<8), where x is a function of t and 
a, 6, c, d are arbitrary constants, show that the relation 



y 2 yy 

is identically satisfied, where x , x", x ", y , T/", y " denote the derivatives with 
respect to the variable t. 

23*. Let u and v be any two functions of the two independent variables x and y, 
and let us set 



U = 



au + bv + c 



F = 



a w + 6 + c 



a" + 6" u + c" a" w + 6" u + c" 

where a, &, c, , c" are constants. Prove the formulae : 

c*udv_G*vdu d*U dV _ cPV SU 
dx 2 dx gx 2 dx ax 2 dx dx* dx 



(u, v) 

d*u dv d*v du /dv 

dx 2 dy ex 2 dy \dx dxdy dx dxdy/ 

(u, v) 

dV a 2 V d U 
ax 2 ~dy ~ ~d& Hy 



dx dxdy ex dxdy/ 



(ff.F) 

and the analogous formulae obtained by interchanging x and y, where 

du dv du dv r dU dV dV dU 

dx dy dy dx dx dy dx dy 

[GOURSAT and PAINLEVE, Comptes rendus, 1887.] 



CHAPTER III 

TAYLOR S SERIES ELEMENTARY APPLICATIONS 
MAXIMA AND MINIMA 

I. TAYLOR S SERIES WITH A REMAINDER 
TAYLOR S SERIES 

44. Taylor s series with a remainder. In elementary texts on the 
Calculus it is shown that, if f(x) is an integral polynomial of 
degree n, the following formula holds for all values of a and h : 



This development stops of itself, since all the derivatives past the 
(n + l)th vanish. If we try to apply this formula to a function 
/(x) which is not a polynomial, the second member contains an 
infinite number of terms. In order to find the proper value to 
assign to this development, we will first try to find an expression 
for the difference 

f> li 2 J) n 

f(a + h) -f(a) - 2 f (a ) - f^ /"(a) ___ /W(a) , 

with the hypotheses that the function /(#), together with its first n 
derivatives / (a:), f"(x), , f^ n) (x), is continuous when x lies in the 
interval (a, a -f A), and that f (n \x) itself possesses a derivative 
/ ( " + J) (x) in the same interval. The numbers a and a -f h being 
given, let us set 



(2) 



where p is any positive integer, and where P is a number which is 
defined by this equation itself. Let us then consider the auxiliary 
function 

89 



90 TAYLOR S SERIES [in, 44 



=f(a + A) -/() - = f , (x) _ ~ 

_ O + h ~ *)" /-A _ (a + h-x 
" 



J 



1.2- "ii 1.2..- 

It is clear from equation (2), which defines the number P, that 



and it results from the hypotheses regarding f(x) that the func 
tion <(x) possesses a derivative throughout the interval (a, a -f A). 
Hence, by Rolle s theorem, the equation </> (#) = must have a root 
a + Oh which lies in that interval, where is a positive number 
which lies between zero and unity. The value of <t> (x), after some 
easy reductions, turns out to be 



The first factor (a -f h x~) p ~ l cannot vanish for any value of x 
other than a + h. Hence we must have 

P= h n - p + l (l - 0)"-* + !/( + J > (a + Ofy, where 0<^<1; 
whence, substituting this value for P in equation (2), we find 



(3) /I 
where 

JL = 




. 2 n .p 



We shall call this formula Taylor s scries with a remainder, and 
the last term or R n the remainder. This remainder depends upon the 
positive integer p, which we have left undetermined. In practice, 
about the only values which are ever given to p are p = n + 1 and 
p = 1. Setting p = n + 1, we find the following expression for the 
remainder, which is due to Lagrange : 



setting p 1, we find 



Ill, 44] TAYLOR S SERIES WITH A REMAINDER 91 

an expression for the remainder which is due to Cauchy. It is 
clear, moreover, that the number will not be the same, in general, 
in these two special formulae. If we assume further that / (n + 1) (a:) 
is continuous when x = a, the remainder may be written in the form 



where e approaches zero with h. 

Let us consider, for definiteness, Lagrange s form. If, in the gen 
eral formula (3), n be taken equal to 2, 3, 4, , successively, we 
get a succession of distinct formulae which give closer and closer 
approximations for f(a -f- A) for small values of h. Thus for n = 1 
we find 

1 1.2 

which shows that the difference 

/(* + *) -/(a) -*/*() 

is an infinitesimal of at least the second order with respect to h, 
provided that /" is finite near x = a. Likewise, the difference 

_// , 7 \ _// \ *" _/l / \ *^ //// \ 

f(a + A) f(a) - f (a) - - f (a) 

/\ / / \ / 1 V \ / 1 O */ \ / 

J. JL * 

is an infinitesimal of the third order ; and, in general, the expression 

A a + h) f(a) - f (a) : f (n) (a) 
J v \ / ~1 v \ / -w! \ f 

is an infinitesimal of order n + 1. But, in order to have an exact 
idea of the approximation obtained by neglecting R, we need to 
know an upper limit of this remainder. Let us denote by M* an 
upper limit of the absolute value of y ( " + 1) (#) in the neighborhood 
of x = a, say in the interval (a 17, a -f rj). Then we evidently have 

Kt i< i^r 1 M) 

provided that | h | < 77. 

* That is, 3/>|/( + J )(z) I when |z a\<ij. The expression " the upper limit," 
defined in 68, must be carefully distinguished from the expression " an upper limit," 
which is used here to denote a number greater than or equal to the absolute value of 
the function at any point in a certain interval. In this paragraph and in the next 
/( + i)(a;) is supposed to have an upper limit near x a. TRANS. 



92 TAYLOR S SERIES [III, 45 

45. Application to curves. This result may be interpreted geomet 
rically. Suppose that we wished to study a curve C, whose equa 
tion is y =f(x), in the neighborhood of a point A, whose abscissa 
is a. Let us consider at the same time an auxiliary curve C", whose 
equation is 



A line x = a -f- h, parallel to the axis of y, meets these two curves 
in two points M and M , which are near A. The difference of their 
ordinates, by the general formula, is equal to 



This difference is an infinitesimal of order not less than n -+- 1 ; and 
consequently, restricting ourselves to a small interval (a 77, a + rj), 
the curve C sensibly coincides with the curve C . By taking larger 
and larger values of n we may obtain in this way curves which 
differ less and less from the given curve C; and this gives us a 
more and more exact idea of the appearance of the curve near the 
point A. 

Let us first set n = \. Then the curve C" is the tangent to the 
curve C at the point A : 



and the difference between the ordinates of the points M and M 
of the curve and its tangent, respectively, which have the same 
abscissa a -f h, is 



Let us suppose that /"() = 0, which is the case in general. The 
preceding formula may be written in the form 



where c approaches zero with h. Since f"(a) = 0, a positive num 
ber rj can be found such that e | < | /"(a) | , when h lies between rj 
and + 77. For such values of h the quantity /"() + e will have 
the same sign as /"( a )> an( i hence y Y will also have the same 
sign as /"(a). If /"(a) is positive, the ordinate y of the curve is 



Ill, 46] TAYLOR S SERIES WITH A REMAINDER 93 

greater than the ordinate F of the tangent, whatever the sign of h ; 
and the curve C lies wholly above the tangent, near the point A. 
On the other hand, if /"( a ) is negative, y is less than Y, and the 
curve lies entirely below the tangent, near the point of tangency. 
If f"(a) = 0, let / (p) () be the first succeeding derivative which 
does not vanish for x = a. Then we have, as before, if f (p) (x) is 
continuous when x = a, 



and it can be shown, as above, that in a sufficiently small interval 
( a ^ a -f- 17) the difference y Y has the same sign as the product 
A p / (p) (a). When p is even, this difference does not change sign 
with h, and the curve lies entirely on the same side of the tangent, 
near the point of tangency. But if p be odd, the difference y Y 
changes sign with h, and the curve C crosses its tangent at the 
point of tangency. In the latter case the point A is called a point 
of inflection ; it occurs, for example, if f "(a) = 0. 

Let us now take n = 2. The curve C is in this case a parabola : 



Y =f(a) + (x 

whose axis is parallel to the axis of y\ and the difference of the 
ordinates is 



If / "(a) does not vanish, y Y has the same sign as A 8 / "() for 
sufficiently small values of h, and the curve C crosses the parabola 
C at the point A. This parabola is called the osculatory parabola 
to the curve C ; for, of the parabolas of the family 

Y = mx z + nx + p, 

this one comes nearest to coincidence with the curve C near the 
point A (see 213). 

46. General method of development. The formula (3) affords a 
method for the development of the infinitesimal f(a + K) ~f( a } 
according to ascending powers of h. But, still more generally, let 
x be a principal infinitesimal, which, to avoid any ambiguity, we 



94 TAYLOR S SERIES [III, 46 

will suppose positive ; and let y be another infinitesimal of the 
form 

(4) y = A lX i + A 2 x* + ...+x P (A p + c ), 

where n l} n z , , n p are ascending positive numbers, not necessarily 
integers, A l} A t , -, A p are constants different from zero, andc is 
another infinitesimal. The numbers HI, A l , n 2 , A 2 , may be cal 
culated successively by the following process. First of all, it is 
clear that HI is equal to the order of the infinitesimal y with 
respect to x, and that A v is equal to the limit of the ratio y/x n i when 
x approaches zero. Next we have 



y A : x n ^ = ui = A z x"* -| ---- + (A p + c) x"p, 

which shows that n z is equal to the order of the infinitesimal M I? 
and A 2 to the limit of the ratio u^/x n -i. A continuation of this 
process gives the succeeding terms. It is then clear that an infini 
tesimal y does not admit of two essentially different developments of 
the form (4). If the developments have the same number of terms, 
they coincide ; while if one of them has p terms and the other 
p + q terms, the terms of the first occur also in the second. This 
method applies, in particular, to the development of f(a + A) f(a) 
according to powers of h ; and it is not necessary to have obtained 
the general expression for the successive derivatives of the func 
tion f(x) in advance. On the contrary, this method furnishes 
us a practical means of calculating the values of the derivatives 



Examples. Let us consider the equation 

(5) F(x, y) = Ax n + By + xy<S>(x, y) + Cx n + l + + Dy 2 + - - - = 0, 

where 4> (x, y) is an integral polynomial in x and y, and where the 
terms not written down consist of two polynomials P(x) and Q(y), 
which are divisible, respectively, by x n + 1 and y 2 . The coefficients A 
and B are each supposed to be different from zero. As x approaches 
zero there is one and only one root of the equation (5) which ap 
proaches zero ( 20). In order to apply Taylor s series with a 
remainder to this root, we should have to know the successive deriv 
atives, which could be calculated by means of the general rules. 
But we may proceed more directly by employing the preceding 
method. For this purpose we first observe that the principal part 



Ill, 46] TAYLOR S SERIES WITH A REMAINDER 95 

of the infinitesimal root is equal to (.4 /E)x n . For if in the equa 
tion (5) we make the substitution 



y = 

and then divide by x n , we obtain an equation of the same form : 

(cc, yi) 



which has only one term in y lt namely By^. As x approaches zero 
the equation (6) possesses an infinitesimal root in y lt and conse 
quently the infinitesimal root of the equation (5) has the principal 
part (A/B)x n , as stated above. Likewise, the principal part of 
?/! is (Al B)x**] and we may set 



where y z is another infinitesimal whose principal part may be found 
by making the substitution 



in the equation (6). 

Continuing in this way, we may obtain for this root y an expres 
sion of the form 



(a p + e)x 



.n + MJ H 1- n 



n 



which we may carry out as far as we wish. All the numbers 
H!, n z , , n p are indeed positive integers, as they should be, since 
we are working under conditions where the general formula (3) is 
applicable. In fact the development thus obtained is precisely the 
same as that which we should find by applying Taylor s series with 
a remainder, where a = and h = x. 

Let us consider a second example where the exponents are not 
necessarily positive integers. Let us set 



y 



96 TAYLOR S SERIES 



[III, 46 



where a, ft, y, and fa, y u are two ascending series of positive 
numbers, and the coefficient A is not zero. It is clear that the prin 
cipal part of y is Ax a , and that we have 



tt _ + Cx* + - A x a (B l x^ + Cix* + ) 

y J]_ 00 - - y 

1 + B^ + dx* H ---- 

which is an expression of the same form as the original, and whose 
principal part is simply the term of least degree in the numerator. 
It is evident that we might go on to find by the same process as 
many terms of the development as we wished. 

Let / (x) be a function which possesses n + 1 successive derivatives. Then 
replacing a by x in the formula (3), we find 

f(x + h) -/(x) + -f(x) + ~f"(x) + ..-+ *" [/00(x) + e] , 
l . fi 1 . 2 n 

where e approaches zero with h. Let us suppose, on the other hand, that we 
had obtained by any process whatever another expression of the same form for 



f(x + h) = /(x) + hfa (x) + A 2 02 (X ) + . . . + h n faty + e /] . 

These two developments must coincide term by term, and hence the coefficients 
0i> 02> , <t>n are equal, save for certain numerical factors, to the successive 
derivatives of /(x) : 



, -_. 

1.2 1. 2- -n 

This remark is sometimes useful in the calculation of the derivatives of certain 
functions. Suppose, for instance, that we wished to calculate the nth derivative 
of a function of a function : 

y=f(u), where u = 0(x). 
Neglecting the terms of order higher than n with respect to h, we have 

* = 0(X + h) - 0(X) = J (X) + -^ 0"(X) + + , ** 0W( Z ); 

1 . 2i 1 . 2 n 

and likewise neglecting terms of order higher than n with respect to k, 

" 



f(u + k) ~f(u) = f(u) 



i . * 1 . 2 n 

If in the right-hand side k be replaced by the expression 

^ + *"( x ) + + 



T 

i . & n 



and the resulting expression arranged according to ascending powers of A, it is 
evident that the terms omitted will not affect the terms in h, h 2 , , h n . The 



HI, 47] TAYLOR S SERIES WITH A REMAINDER 97 

coefficient of /i", for instance, will be equal to the nth derivative of /[</>()] 
divided by 1 . 2 n ; and hence we may write 



where Ai denotes the coefficient of h n in the development of 



For greater detail concerning this method, the reader is referred to Hermite s 
Cours d Analyse (p. 59). 

47. Indeterminate forms.* Let f(x) and </> (x) be two functions 
which vanish for the same value of the variable x = a. Let us try 
to find the limit approached by the ratio 

f(a + K) 
<f>(a + h) 

as h approaches zero. This is merely a special case of the problem 
of finding the limit approached by the ratio of two infinitesimals 
The limit in question may be determined immediately if the prin 
cipal part of each of the infinitesimals is known, which is the case 
whenever the formula (3) is applicable to each of the functions 
/(cc) and <f> (x) in the neighborhood of the point a. Let us suppose 
that the first derivative of f(x) which does not vanish for x = a is 
that of order p, f (p \a) ; and that likewise the first derivative of 
< (cc) which does not vanish for x = a is that of order q, < (s) (a). 
Applying the formula (3) to each of the functions f(x) and < (x) 
and dividing, we find 



where c and e are two infinitesimals. It is clear from this result 
that the given ratio increases indefinitely when h approaches zero, if 
q is greater than p ; and that it approaches zero if q is less than p. 
If q = p, however, the given ratio approaches / (p) ( a )/^ (7) ( a ) as ^ s 
limit, and this limit is different from zero. 

Indeterminate forms of this sort are sometimes encountered in finding the 
tangent to a curve. Let 

x =/(<), y = *(t), z = *(l) 
* See also 7. 



98 TAYLOR S SERIES [m, J 4 8 

be the equations of a curve C in terms of a parameter t. The equations of the 
tangent to this curve at a point M, which corresponds to a value t of the param 
eter, are, as we saw in 5, 

Z - 



f (tv) * (*o) f (o) 

These equations reduce to identities if the three derivatives / (), <f> (t), $ (t) all 
vanish for t = t . In order to avoid this difficulty, let us review the reasoning 
by which we found the equations of the tangent. Let M be a point of the 
curve C near to M, and let to + h be the corresponding value of the parameter. 
Then the equations of the secant MM are 



For the sake of generality let us suppose that all the derivatives of order less 
than p (p> 1) of the functions /(), <t> (t), \f/ (t) vanish for t = t , but that at least 
one of the derivatives of order p, say /<*>> ( ), is not zero. Dividing each of the 
denominators in the preceding equations by hp and applying the general for 
mula (3), we may then write these equations in the form 



to) + e &> J) (to) + e f ( "> (to) + e" 

where e, e , e" are three infinitesimals. If we now let h approach zero, these 
equations become in the limit 



in which form all indetermination has disappeared. 

The points of a curve C where this happens are, in general, singular points 
where the curve has some peculiarity of form. Thus the plane curve whose 
equations are 

X = , y = <3 

passes through the origin, and dx/dt = dy / dt = at that point. The tangent 
is the axis of x, and the origin is a cusp of the first kind. 

48. Taylor s series. If the sequence of derivatives of the function 
f(x) is unlimited in the interval (a, a -f h), the number n in the 
formula (3) may be taken as large as we please. If the remainder 
R n approaches zero when n increases indefinitely, we are led to write 
down the following formula : 



which expresses that the series 

/() + \ / () + + iT^r^ ^"W + 



Ill, 48] TAYLOR S SERIES WITH A REMAINDER 99 

is convergent, and that its " sum " * is the quantity f(a + h). This 
formula (7) is Taylor s series, properly speaking. But it is not justi 
fiable unless we can show that the remainder R n approaches zero when 
n is infinite, whereas the general formula (3) assumes only the exist 
ence of the first n + 1 derivatives. Replacing a by x, the equation 
(7) may be written in the form 



Or, again, replacing h by x and setting a, = 0, we find the formula 
(8) /(*) =/(0) + / (O) + - + -- / 



This latter form is often called Maclaurin s series; but it should 
be noticed that all these different forms are essentially equivalent. 
The equation (8) gives the development of a function of x accord 
ing to powers of x ; the formula (7) gives the development of a func 
tion of h according to powers of h : a simple change of notation is 
all that is necessary in order to pass from one to the other of these 
forms. 

It is only in rather specialized cases that we are able to show 
that the remainder R n approaches zero when n increases indefinitely. 
If, for instance, the absolute value of any derivative whatever is less 
than a fixed number M when x lies between a and a + h, it follows, 
from Lagrange s form for the remainder, that 

I h\" + 1 
I* ^l. 2- ..( + !) 

an inequality whose right-hand member is the general term of a 
convergent series. f Such is the case, for instance, for the functions 
e x , sin x, cos x. All the derivatives of e x are themselves equal to 
e x , and have, therefore, the same maximum in the interval con 
sidered. In the case of sin x and cos x the absolute values never 
exceed unity. Hence the formula (7) is applicable to these three 
functions for all values of a and h. Let us restrict ourselves to 
the form (8) and apply it first to the function f(x) = e x . We find 



* That is to say, the limit of the sum of the first n terms as n becomes infinite. 
For a definition of the meaning of the technical phrase " the sum of a series," see 
157. TRANS. 

t The order of choice is a, h, M, n, not a, h, n, M. This is essential to the con 
vergence of the series in question. TRANS. 



100 TAYLOR S SERIES [III, 49 

and consequently we have the formula 



which applies to all values, positive or negative, of x. If a is any 
positive number, we have a x = e rl Ka , and the preceding formula 
becomes 



Let us now take f(x) = sin x. The successive derivatives form a 
recurrent sequence of four terms cos x, sin x, cos a;, sin x ; and 
their values for x = form another recurrent sequence 1, 0, 1, 0. 
Hence for any positive or negative value of x we have 



(11) 



and, similarly, 

(12) .o.,-l- + J!L_ 



Let us return to the general case. The discussion of the remain 
der R n is seldom so easy as in the preceding examples; but the 
problem is somewhat simplified by the remark that if the remain 
der approaches zero the series 



necessarily converges. In general it is better, before examining 
R n , to see whether this series converges. If for the given values of 
a and h the series diverges, it is useless to carry the discussion 
further ; we can say at once that R n does not approach zero when n 
increases indefinitely. 

49. Development of log(l 4- x). The function log(l + x), together 
with all its derivatives, is continuous provided that x is greater 
than 1. The successive derivatives are as follows : 



HI, 49] TAYLOR S SERIES WITH A REMAINDER 101 

_ i 



Let us see for what values of a? Maclaurin s formula (8) may be 
applied to this fu action. Writing first the series with a remainder, 
we have, under any circumstances, 



*) = - + + + (- !)- + *, 
The remainder R n does not approach zero unless the series 



converges, which it does only for the values of x between 1 and 
+ 1, including the upper limit -f 1. When x lies in this interval 
the remainder may be written in the Cauchy form as follows : 



_ iyl.2-.-n 

~ 



1.2 
or 



( 



1 (9 



Let us consider first the case where | x < 1. The first factor x 
approaches zero with x, and the second factor (1 6)/(l + Ox) is 
less than unity, whether x be positive or negative, for the numer 
ator is always less than the denominator. The last factor remains 
finite, for it is always less than 1/(1 |x|). Hence the remainder 
R n actually approaches zero when n increases indefinitely. This 
form of the remainder gives us no information as to what happens 
when x = 1 ; but if we write the remainder in Lagrange s form, 



it is evident that R n approaches zero when n increases indefinitely. 
An examination of the remainder for x = 1 would be useless, 



1:02 TAYLOR S SERIES [III, 49 

sin f . e the series diverges for that value of x. We have then, when 
x lies between 1 and -f 1, the formula 

(13) log(l+*)^-f + | 3 - . + (-l)-i + .... 

This formula still holds when x = 1, which gives the curious 
relation 

(14) Iog2l-|+|-j+. . + (-1)-^ + .... 

The formula (13), not holding except when x is less than or equal 
to unity, cannot be used for the calculation of logarithms of whole 
numbers. Let us replace x by x. The new formula obtained, 



still holds for values of x between 1 and + 1 ; and, subtracting 
the corresponding sides, we find the formula 



2 + + f--- + T-T 

1 x] \1 3 o 2 n + 1 

When x varies from to 1 the rational fraction (1 +#)/(! a:) 
steadily increases from 1 to + <x>, and hence we may now easily cal 
culate the logarithms of all integers. A still more rapidly con 
verging series may be obtained, however, by forming the difference 
of the logarithms of two consecutive integers. For this purpose 
let us set 

1+ x N + 1 1 

or x = 



1-x N 2N+1 

Then the preceding formula becomes 



an equation whose right-hand member is a series which converges 
very rapidly, especially for large values of N. 

Note. Let us apply the general formula (3) to the function log (1 + x), setting 
a = 0, h = x, n = 1, and taking Lagrange s form for the remainder. We find in 
this way 

x 2 
log(l + x) = x ---- 



Ill, 49] TAYLOR S SERIES WITH A REMAINDER 103 

If we now replace x by the reciprocal of an integer n, this may be written 



n 2n 2 

where n is a positive number less than unity. Some interesting consequences 
may be deduced from this equation. 

1) The harmonic series being divergent, the sum v-r - \ \\/ % . 

31 r \ 

1 1 1 



+ n 



, \ 
increases indefinitely with n. But the difference 

vw 

2 n log n 
approaches a finite limit. For, let us write this difference in the form 



I n + 1\ n+ 1 

-log ) + log 



n n / n 



Now 1 / p log (1 + 1 / p) is the general term of a convergent series, for by the 
equation above 



which shows that this term is smaller than the general term of the convergent 
series 2(1 /p 2 ). When n increases indefinitely the expression 



n+1 / n 1\ 

log = log ( 1 + - 

n V */ 



approaches zero. Hence the difference under consideration approaches a finite 
limit, which is called Euler s constant. Its exact value, to twenty places of 
decimals, is C = 0.57721566490153286060. 

2) Consider the expression 



n+1 n + 2 n + p 

where n and p are two positive integers which are to increase indefinitely. Then 
7?e may write 



2 n + p/ 

1 1 

2 + + = log (n 

1 1 

2 + n 



104 TAYLOR S SERIES [III, 50 

where p n + P and p n approach the same value C when n and p increase indefi 
nitely. Hence we have also 



Now the difference p n +p p n approaches zero. Hence the sum 2 approaches 
no limit unless the ratio p/n approaches a limit. If this ratio does approach a 
limit a, the sum S approaches the limit log (1 + a). 
Setting p n, for instance, we see that the sum 

n+ 1 n+2 2n 

approaches the limit log 2. 

50. Development of (1 + x) m . The function (1 + x) m is denned and 
continuous, and its derivatives all exist and are continuous func 
tions of x, when 1 + x is positive, for any value of m ; for the 
derivatives are of the same form as the given function : 



fM(x) = m(m - 1) - - - (m - n + 1) (1 + 
/( + 1 )(x) = m(m - 1) (m - n) (1 + a)" 

Applying the general formula (3), we find 



-...- 

1.2- " 

and, in order that the remainder R n should approach zero, it is first 
of all necessary that the series whose general term is 



(m 1) (m n + 1) 
1.2-.-W 



should converge. But the ratio of any term to the preceding is 

m n -\- 1 



x 



which approaches x as n increases indefinitely. Hence, exclud 
ing the case where m is a positive integer, which leads to the ele 
mentary binomial theorem, the series in question cannot converge 
unless \x < 1. Let us restrict ourselves to the case in which I a; I < 1. 



Ill, 50] TAYLOR S SERIES WITH A REMAINDER 105 

To show that the remainder approaches zero, let us write it in the 
Cauchy form : 



1.2... 

The first factor 

m(m 1) (m n~) n + l 
1.2..-W 

approaches zero since it is the general term of a convergent 
series. The second factor (1 #)/(! -f &c) is less than unity; and, 
finally, the last factor (1 + Ox) m ~ l is less than a fixed limit. For, 
if m - 1 > 0, we have (1 + ftc)" 1 < 2 "- 1 ; while if m - 1< 0, 
(1 + Ox) m ~ l < (1 ici)" 1 " 1 . Hence for every value of x between 
1 and + 1 we have the development 




-.- 



We shall postpone the discussion of the case where x = 1. 
In the same way we might establish the following formulae 

, 1 x a , 1 . 3 x 6 , 
arcsm* = * + -- + ^- + ... 

.5...(2r? -1) x 2 " + 1 






2.4.6--.2w 2w-|-l " 

X 8 iC 5 iC 7 CC 2fl + 1 

- + - y + ... + (-l)^ TI +---, 

which we shall prove later by a simpler process, and which hold 
for all values of x between 1 and + 1. 

Aside from these examples and a few others, the discussion of 
the remainder presents great difficulty on account of the increas 
ing complication of the successive derivatives. It would therefore 
seem from this first examination as if the application of Taylor s 
series for the development of a function in an infinite series were of 
limited usefulness. Such an impression would, however, be utterly 
false ; for these developments, quite to the contrary, play a funda 
mental role in modern Mathematical Analysis. In order to appre 
ciate their importance it is necessary to take another point of 
view and to study the properties of power series for their own 



106 TAYLOR S SERIES [III, 5J 

sake, irrespective cf their origin. We shall do this in several of 
the following chapters. 

Just now we will merely remark that the series 



may very well be convergent without representing the function 
f(x) from which it was derived. The following example is due to 
Cauchy. Let /(*) = e~ 1 *. Then / (*) = (2/x*)e- l "> ; and, in 
general, the nth derivative is of the form 



where P is a polynomial. All these derivatives vanish for x = 0, 
for the quotient of e~ l/3 * by any positive power of x approaches 
zero with x.* Indeed, setting x = 1/z, we may write 



and it is well known that e z */z m increases indefinitely with z, no 
matter how large m may be. Again, let <f> (x) be a function to which 
the formula (8) applies : 



Setting F(x) = <fr(x ) + e~ llx \ we find 
F(0) . <*> (0), F (0) - * (0), , F<">(0) - ^->(0), , 

and hence the development of F(x) by Maclaurin s series would 
coincide with the preceding. The sum of the series thus obtained 
represents an entirely different function from that from which the 
series was obtained. 

In general, if two distinct functions f(x) and <f> (a;), together with 
all their derivatives, are equal for x = 0, it is evident that the 

*It is tacitly assumed that /(O) = 0, which is the only assignment which would 
render/(:c) continuous at x = 0. But it should be noticed that no further assignment 
is necessary for / (a:), etc., at x = 0. For 

,,, m lim /(a) -/(O) _ ft 
= x = ^~~ - 

which defines / (x) at x and makes / (z) continuous at a: = 0, etc. TRANS. 



Ill, 61] TAYLOR S SERIES WITH A REMAINDER 107 

Maclaurin series developments for the two functions cannot both 
be valid, for the coefficients of the two developments coincide. 

51. Extension to functions of several variables. Let us consider, for 
definiteness, a function o> = f(x, y, z) of the three independent vari 
ables x, y, z, and let us try to develop f(x -f h, y -f k, z + I) accord 
ing to powers of h, k, I, grouping together the terms of the same 
degree. Cauchy reduced this problem to the preceding by the fol 
lowing device. Let us give x, y, z, h, k, I definite values and let 
us set 

<f> () =f(x + ht,y + kt, z + ft), 

where t is an auxiliary variable. The function <() depends on t 
alone ; if we apply to it Taylor s series with a remainder, we find 



(17) 



f *<">(0) + ? r^ -T; *<-+ >(*), 



where <(0), < (0), , < (n) (0) are the values of the function <f>(f) 
and its derivatives, for t 0; and where < ( " + 1) (0) is the value of 
the derivative of order n + 1 for the value &t, where lies between 
zero and one. But we may consider < (f) as a composite function of 
t, $() =/(w, v, w), the auxiliary functions 

u = x 4- ht, v = y + kt, w = K -f It 

being linear functions of t. According to a previous remark, the 
expression for the differential of order m, d m <f>, is the same as if t/, 
v, w were the independent variables. Hence we have the symbolic 
equation 



which may be written, after dividing by dt m , in the form 

A + | t + LC 

CV CW 

For t = 0, u, v, w reduce, respectively, to x, y, z, and the above 
equation in the same symbolism becomes 



108 TAYLOR S SERIES [III, 52 

Similarly, 



(n + l) 



where cc, y, z are to be replaced, after the expression is developed, by 

x + Oht, y + 6kt, z + Bit, 
respectively. If we now set t = 1 in (17), it becomes 



,dx 

(18)1 



1f\ Id -!/-> - I *-; 

. 2 n \cx cy cz 

The remainder R n may be written in the form 



n+1) 



where x, y, z are to be replaced by x + 6h, y + 6k, z + 01 after the 
expression is expanded.* 

This formula (18) is exactly analogous to the general formula 
(3). If for a. given set of values of cc, y, z, h, k, I the remainder R n 
approaches zero when n increases indefinitely, we have a develop 
ment of f(x + h, y + k, z + I) in a series each of whose terms is a 
homogeneous polynomial in h, k, I. But it is very difficult, in gen 
eral, to see from the expression for R n whether or not this remainder 
approaches zero. 

52. From the formula (18) it is easy to draw certain conclusions 
analogous to those obtained from the general formula (3) in the 
case of a single independent variable. For instance, let z =f(x, y) 
be the equation of a surface S. If the function f(x, y), together 
with all its partial derivatives up to a certain order n, is continuous 
in the neighborhood of a point (X Q , y ), the formula (18) gives 

/ df , 

/( + h, y + k) = f(x , y ~) + I h ^- -f fc 7 



1 . 2 

Restricting ourselves, in the second member, to the first two terms, 
then to the first three, etc., we obtain the equation of a plane, then 

* It is assumed here that all the derivatives used exist and are continuous. TRANS. 



Ill, 52] TAYLOR S SERIES WITH A REMAINDER 109 

that of a paraboloid, etc., which differ very little from the given sur 
face near the point (x 0) y )- The plane in question is precisely the 
tangent plane ; and the paraboloid is that one of the family 

* = Ax 2 + 2 Bxy + Cy* 



which most nearly coincides with the given surface S. 

The formula (18) is also used to determine the limiting value of 
a function which is given in indeterminate form. Let f(x, y) and 
< (#, T/) be two functions which both vanish for x = a, y = b, but 
which, together with their partial derivatives up to a certain order, 
are continuous near the point (a, ). Let us try to find the limit 
approached by the ratio 



when x and y approach a and b, respectively. Supposing, first, that 
the four first derivatives df/da, df/8b, d<f>/8a, 8<f>/db do not all 
vanish simultaneously, we may write 



k[T*-+<} + k(%- 

K) _ 



<(>(a + h,b + k) i /d<f> . \, , 

h \Ta + ^) + k \db 

where e, c , c,, e{ approach zero with h and &. When the point 
(x, y) approaches (a, b~), h and k approach zero ; and we will sup 
pose that the ratio k/h approaches a certain limit a, i.e. that the 
point (x, y) describes a curve which has a tangent at the point (a, b~). 
Dividing each of the terms of the preceding ratio by A, it appears 
that the fraction f(x, y)/<$>(x, y) approaches the limit 

o^~ "+" a ~oT 
ca do 



-Z -- P OL -^r- 

ca cb 

This limit depends, in general, upon a, i.e. upon the manner in 
which x and y approach their limits a and b, respectively. In order 
that this limit should be independent of a it is necessary that the 
relation 



_ 

da db db da 
should hold ; and such is not the case in general. 



110 TAYLOR S SERIES [HI, 53 

If the four first derivatives df/Sa, df/Sb, d<l>/da, 8<f>/db vanish 
simultaneously, we should take the terms of the second order in the 
formula (18) and write 



* 



f(a + h, b + K) _ 



where e, e , c", c u e/, e / are infinitesimals. Then, if a be given the 
same meaning as above, the limit of the left-hand side is seen to be 



V? + 2 f-fj a H- %-? a 2 

a <ya cb co 

which depends, in general, upon a. 

II. SINGULAR POINTS MAXIMA AND MINIMA 

53. Singular points. Let (x , ?/ ) be the coordinates of a point M 
of a curve C whose equation is F(x, y~) = 0. If the two first par 
tial derivatives 8F/dx, 8F/dy do not vanish simultaneously at this 
point, we have seen ( 22) that a single branch of the curve C passes 
through the point, and that the equation of the tangent at that 
point is 



where the symbol d p + q F /dx$ dyl denotes the value of the derivative 
fip + vp /8x p di/* for x = x , y = y . If dF/dx Q and dF /dy both van 
ish, the point (x , y ) is, in general, a singular point* Let us suppose 
that the three second derivatives do not all vanish simultaneously 
for x = x , y y , and that these derivatives, together with the third 
derivatives, are continuous near that point. Then the equation of 
the curve may be written in the form. 



* That is, the appearance of the curve is, in general, peculiar at that point. For an 
exact analytic definition of a singular point, see 192. TRANS. 



Ill, 53] SINGULAR POINTS MAXIMA AND MINIMA 111 



(19) 



;_ g \24-2 ^ 

1 [OF . 

I / y* 



cF 



where x and y are to be replaced in the third derivatives by 
X() + Q(x- B ) and ?/ + 0(y - y ), respectively. We may assume 
that the derivative d*F/dy$ does not vanish; for, at any rate, we 
could always bring this about by a change of axes. Then, setting 
y y = t (x ar ) and dividing by (x ic ) 2 , the equation (19) 
becomes 



(20) 

1 



fyo 



- 0, 



where P (x x , t*) is a function which remains finite when x 
approaches x . Now let ti and # 2 be the two roots of the equation 



If these roots are real and unequal, i.e. if 

2 



the equation (20) may be written in the form 

(t - ,) (t - * 2 ) + (x - x,) P = 0. 

For x = x the above quadratic has two distinct roots t = t l9 t = 2 . 
As x approaches x that equation has two roots which approach # t 
and 2 , respectively. The proof of this is merely a repetition of 
the argument for the existence of implicit functions. Let us set 
t = t l -\- u, for example, and write down the equation connecting x 
and u: 

where Q (x, u) remains finite, while x approaches x and u approaches 
zero. Let us suppose, for definiteness, that t l t z > ; and let M 
denote an upper limit of the absolute value of Q(x, ), and ra a 
lower limit of t t t 2 + u, when x lies between x h and x + h, 



112 TAYLOR S SERIES [III, 53 

and u between h and -f h, where h is a positive number less than 
ti t 2 . Now let c be a positive number less than h, and rj another 
positive number which satisfies the two inequalities 

m 
77 < h, rj < e. 

If a; be given such a value that x x \ is less than 77, the left-hand 
side of the above equation will have different signs if e and then 
-+- c be substituted for u. Hence that equation has a root which 
approaches zero as x approaches x , and the equation (19) has a 
root of the form 

V = !/o + (a - *o) (*i + a), 

where a approaches zero with x x . It follows that there is one 
branch of the curve C which is tangent to the straight line 

y - y<> = *i (* - z<>) 

at the point (x , T/ O ). 

In like manner it is easy to see that another branch of the 
curve passes through this same point tangent to the straight line 
y y Q = t 2 (x x ). The point M is called a double point; and 
the equation of the system of tangents at this point may be found 
by setting the terms of the second degree in (x x ), (y y ) in 
(19) equal to zero. 

If 



the point (cc , T/ O ) is called an isolated double point. Inside a suffi 
ciently small circle about the point Af as center the first member 
F(x, ?/) of the equation (19) does not vanish except at the point M 
itself. For, let us take 

x = x + p cos <f>, y = ?/ + p sin <f> 
as the coordinates of a point near M . Then we find 

,2 



" - ~ cos <^> sin 4 + - sin 2 ^ + P L 



where L remains finite when p approaches zero. Let H be an upper 
limit of the absolute value of L when p is less than a certain posi 
tive number r. For all values of < between and 2?r the expression 



c 2 F 
cos A sin d> + -T-T- sin 2 



Ill, 53] SINGULAR POINTS MAXIMA AND MINIMA 113 

has the same sign, since its roots are imaginary. Let m be a lower 
limit of its absolute value. Then it is clear that the coefficient 
of p 2 cannot vanish for any point inside a circle of radius p<m/H. 
Hence the equation F(x, y) = has no root other than p = 0, i.e. 
x = x , y = y , inside this circle. 
In case we have 



dx 8y 

the two tangents at the double point coincide, and there are, in gen 
eral, two branches of the given curve tangent to the same line, thus 
forming a cusp. The exhaustive study of this case is somewhat 
intricate and will be left until later. Just now we will merely 
remark that the variety of cases which may arise is much greater 
than in the two cases which we have just discussed, as will be seen 
from the following examples. 

The curve y 2 = x s has a cusp of the first kind at the origin, both 
branches of the curve being tangent to the axis of x and lying on 
different sides of this tangent, to the right of the y axis. The 
curve y 2 2x 2 y + x* x 5 = has a cusp of the second kind, both 
branches of the curve being tangent to the axis of x and lying on 
the same side of this tangent ; for the equation may be written 

y = x 2 z% 

and the two values of y have the same sign when x is very small, 
but are not real unless x is positive. The curve 



has two branches tangent to the x axis at the origin, which do not 
possess any other peculiarity ; for, solving for y, the equation becomes 

3 x 2 x 2 V8 - x 2 
y ~ 1+x 2 

and neither of the two branches corresponding to the two signs 
before the radical has any singularity whatever at the origin. 

It may also happen that a curve is composed of two coincident 
branches. Such is the case for the curve represented by the 
equation 



When the point (x, y) passes across the curve the first member F(x, y) 
vanishes without changing sign. 



11-4 TAYLOR S SERIES [III, 54 

Finally, the point (cc , y ) may be an isolated double point. Such 
is the case for the curve y 2 + x 4 + y 4 = 0, on which the origin is an 
isolated double point. 

54. In like manner a point M of a surface S, whose equation is 
F ( x > y> ) = 0, is, in general, a singular point of that surface if the 
three first partial derivatives vanish for the coordinates x , y , z of 
that point : 

dF _ ZF _ CF 

5 w, - 0, 7 = 0. 

CX Q CIJ^ CZ Q 

The equation of the tangent plane found above ( 22) then reduces 
to an identity ; and if the six second partial derivatives do not all 
vanish at the same point, the locus of the tangents to all curves on 
the surface S through the point M is, in general, a cone of the 
second order. For, let 



be the equations of a curve C on the surface S. Then the three 
functions f(), $(), \l/(t) satisfy the equation F(x, y, z) = 0, and 
the first and second differentials satisfy the two relations 

dF T cF cF 

-^ dx + 7T- dy + -5 dz = 0, 

ex cy cz 

y cF , dF Y 2) ()F cF dF 

dx + d,j + ~dz) +~d*x + ^-d 2 y + ~d*z = 0. 

cy cz I ex Cy J cz 

For the point x = x , y = y , z = z the first of these equations 
reduces to an identity, and the second becomes 



+ 2 r=- dxdy + 2 ^- dy dz + 2 -f- dx dz = 0. 

cx dy 0y d*t dx oz 

The equation of the locus of the tangents is given by eliminating 
dx, dy, dz between the latter equation and the equation of a tangent 
line 



dx dy dz 

which leads to the equation of a cone T of the second degree : 



in,M] SINGULAR POINTS MAXIMA AND MINIMA 115 



(21) 



c 2 F 

(V rV -I- ( Y 11 V 4- 

^A a%; -r- 8 , v,* y ; 



2 - (A r x n } ( Y 

3 o \ O/ \ 



c 2 F 2 s F 

2 = =- C^- ?/o) (Z - ) + 2 ^ ^~ 



On the other hand, applying Taylor s series with a remainder 
and carrying the development to terms of the third order, the equa 
tion of the surface becomes 



(22) ^ 1. 

7 , CF 



where x, y, in the ter)ns of the third order are to be replaced- by 
x + 0(x-x ), y + 8(y-yd, z + 0(z-s ), respectively. The 
equation of the cone T may be obtained by setting the terms of 
the second degree in x x , y y , z z in the equation (22) equal 
to zero. 

Let us then, first, suppose that the equation (21) represents a real 
non-degenerate cone. Let the surface 5 and the cone T be cut by a 
plane P which passes through two distinct generators G and G of 
the cone. In order to find the equation of the section of the sur 
face 5 by this plane, let us imagine a transformation of coordinates 
carried out which changes the plane P into a plane parallel to the 
xy plane. It is then sufficient to substitute z = z<> in the equation (22). 
It is evident that for this curve the point M is a double point with 
real tangents ; from what we have just seen, this section is composed 
of two branches tangent, respectively, to the two generators G, G . 
The surface S near the point M therefore resembles the two nappes 
of a cone of the second degree near its vertex. Hence the point M n 
is called a conical point. 

When the equation (21) represents an imaginary non-degenerate 
cone, the point M is an isolated singular point of the surface fi. 
Inside a sufficiently small sphere about such a point there exists no 
set of solutions of the equation F(x, y, z) = other than x = x , 
y = y , z = z . For, let M be a point in space near M , p the 



116 TAYLOR S SERIES [m, 55 

distance MM , and a, (3, y the direction cosines of the line M M. 
Then if we substitute 

X = X + pa, y = y + p } K = Z + py, 

the function F(x, y, 2) becomes 



where L remains finite when p approaches zero. Since the equation 
(21) represents an imaginary cone, the expression 

) 
or H ----- h 2 



cannot vanish when the point (a, /?, y) describes the sphere 

2 + yS 2 + y 2 = 1. 

Let w be a lower limit of the absolute value of this polynomial, 
and let H be an upper limit of the absolute value of L near the 
point Af . If a sphere of radius m/H be drawn about M as center, 
it is evident that the coefficient of p 2 in the expression for F(x, y, z) 
cannot vanish inside this sphere. Hence the equation 

F(x, y,z) = 
has no root except p = 0. 

When the equation (21) represents two distinct real planes, two 
nappes of the given surface pass through the point A/ , each of 
which is tangent to one of the planes. Certain surfaces have a 
line of double points, at each of which the tangent cone degenerates 
into two planes. This line is a double curve on the surface along 
which two distinct nappes cross each other. For example, the circle 
whose equations are z = 0, x 1 + y 2 = 1 is a double line on the surface 
whose equation is 

4 + 2z 2( x * + ,f) - (r* + ,f- l)^ = 0. 

When the equation (21) represents a system of two conjugate 
imaginary planes or a double real plane, a special investigation is 
necessary in each particular case to determine the form of the sur 
face near the point M . The above discussion will be renewed in 
the paragraphs on extrema. 

55. Extrema of functions of a single variable. Let the function f(x) 
be continuous in the interval (a, 6), and let c be a point of that 



Ill, 55] SINGULAR POINTS MAXIMA AND MINIMA 117 

interval. The function /(#) is said to have an extremum (i.e. a 
maximum or a minimum) for x = c if a positive number 77 can be 
found such that the difference f(c -\- A) f(c), which vanishes for 
h = 0, has the same sign for all other values of h between rj 
and + i). If this difference is positive, the function f(x) has a 
smaller value for x = c than for any value of x near c ; it is said 
to have a minimum at that point. On the contrary, if the differ 
ence f(c -f A) /(c) is negative, the function is said to have a 
maximum. 

If the function f(x) possesses a derivative for x = c, that deriva 
tive must vanish. For the two quotients 



h -h 

each of which approaches the limit / (c) when h approaches zero, have 
different signs ; hence their common limit / (c) must be zero. Con 
versely, let c be a root of the equation / (#) = which lies between 
a and b, and let us suppose, for the sake of generality, that the 
first derivative which does not vanish for x = c is that of order n, 
and that this derivative is continuous when x = c. Then Taylor s 
series with a remainder, if we stop with n terms, gives 






which may be written in the form 
/( + A) -/() = 



where c approaches zero with h. Let rj be a positive number such 
that |/ (n) ( c ) | is greater than e when x lies between c 77 and c + 77. 
For such values of x, / (n) (c) -f c has the same sign as f*- n) (c), and 
consequently /(c -f A) /(c) has the same sign as A n / (n) (c). If 
n is odd, it is clear that this difference changes sign with A, and 
there is neither a maximum nor a minimum at x = c. If n is even, 
f(c + A) /(c) has the same sign as/ (n) (c), whether A be positive 
or negative ; hence the function is a maximum if / ( (c) is negative, 
and a minimum if f (n) (c) is positive. It follows that the necessary 
and sufficient condition that the function f(x) should have a maximum 
or a minimum f or x = c is that the first derivative which does not 
vanish for x = c should be of even order. 



118 TAYLOR S SERIES [III, 56 

Geometrically, the preceding conditions mean that the tangent to 
the curve y =f(x) at the point A whose abscissa is c must be par 
allel to the axis of x, and moreover that the point A must not be 
a point of inflection. 

Notes. When the hypotheses which we have made are not satisfied 
the function f(x) may have a maximum or a minimum, although 
the derivative / (#) does not vanish. If, for instance, the derivative 
is infinite for x = c, the function will have a maximum or a mini 
mum if the derivative changes sign. Thus the function y = a^ is at 
a minimum for x = 0, and the corresponding curve has a cusp at the 
origin, the tangent being the y axis. 

When, as in the statement of the problem, the variable x is 
restricted to values which lie between two limits a and b, it may 
happen that the function has its absolute maxima and minima pre 
cisely at these limiting points, although the derivative / (x) does 
not vanish there. Suppose, for instance, that we wished to find 
the shortest distance from a point P whose coordinates are (a, 0) 
to a circle C whose equation is x z + y 2 R 2 0. Choosing for our 
independent variable the abscissa of a point M of the circle C, we 
find 

d 2 = PM Z = (x - a) 2 + y 2 = x 2 + y 2 - 2 ax + a 2 , 

or, making use of the equation of the circle, 
d 2 = R 2 + a* - 2 ax. 

The general rule would lead us to try to find the roots of the derived 
equation 2 a = 0, which is absurd. But the paradox is explained if 
we observe that by the very nature of the problem the variable x 
must lie between R and + R. If a is positive, d 2 has a minimum 
for x = R and a maximum for x = R. 

56. Extrema of functions of two variables. Let f(x, y) be a con 
tinuous function of x and y when the point M, whose coordinates 
are x and y, lies inside a region ft bounded by a contour C. The 
function f(x, y) is said to have an extremum at the point M (x 0) ?/ ) 
of the region O if a positive number rj can be found such that the 
difference 



which vanishes for h = k = 0, keeps the same sign for all other sets 
of values of the increments h and k which are each less than T in 



Ill, 50] SINGULAR POINTS MAXIMA AND MINIMA 119 

absolute value. Considering y for the moment as constant and 
equal to ?/o, becomes a function of the single variable x ; and, by 
the above, the difference 



cannot keep the same sign for small values of h unless the deriva 
tive df/dx vanishes at the point M . Likewise, the derivative df/dy 
must vanish at M Q ; and it is apparent that the only possible sets of 
values of x and y which can render the function f(x, y) an extre- 
mum are to be found among the solutions of the two simultaneous 
equations 

*=o, f=o. 

tix cy 

Let x = x , y = y Q be a set of solutions of these two equations. 
We shall suppose that the second partial derivatives of f(x, y) do 
not all vanish simultaneously at the point M whose coordinates 
are (x , y ), and that they, together with the third derivatives, are 
all continuous near M . Then we have, from Taylor s expansion, 



A = 



(23) 



1.2 



+ 6 



(3) 



We can foresee that the expression 



will, in general, dominate the whole discussion. 

In order that there be an extremum at M Q it is necessary and 
sufficient that the difference A should have the same sign when the 
point (X Q + h, y + k) lies anywhere inside a sufficiently small square 
drawn about the point M as center, except at the center, where 
A = 0. Hence A must also have the same sign when the point 
(x + h, y + k) lies anywhere inside a sufficiently small circle whose 
center is A/ ; for such a square may always be replaced by its 
inscribed circle, and conversely. Then let C be a circle of radius 
r drawn about the point M Q as center. All the points inside this 
circle are given by 



120 TAYLOR S SERIES [111, 56 

where < is to vary from to 2 TT, and p from r to + r. We might, 
indeed, restrict p to positive values, but it is better in what follows 
not to introduce this restriction. Making this substitution, the 
expression for A becomes 

2 S 

A = (A cos 2 < + 2 B sin <f> cos <f> + C sin 2 <) + ^- Z, 
where 



and where Z is a function whose extended expression it would be 
useless to write out, but which remains finite near the point (X Q , y ). 
It now becomes necessary to distinguish several cases according to 
the sign of B 2 - A C. 

First case. Let B z A C > 0. Then the equation 
A cos 2 < + 2 B sin < cos <p + C sin 2 </> = 

has two real roots in tan <, and the first member is the difference 
of two squares. Hence we may write 

2 8 

A o" E a ( a cos ^ + b sin ^) 2 "" P( a> cos ^ + b> sin ^) 2 ] + ^ L > 

where 

a > 0, > 0, aft fta =jfc 0. 

If <f> be given a value which satisfies the equation 
a cos < + b sin < = 0, 

A will be negative for sufficiently small values of p ; while, if < be 
such that a cos<f> + 6 sin< = 0, A will be positive for infinitesimal 
values of p. Hence no number r can be found such that the differ 
ence A has the same sign for any value of < when p is less than r. 
It follows that the function f(x, y) has neither a maximum nor a 
minimum for x = x , y = y . 

Second case. Let B 2 A C < 0. The expression 
A cos 2 </> + 2.Bcos 



cannot vanish for any value of <. Let m be a lower limit of its 
absolute value, and, moreover, let H be an upper limit of the abso 
lute value of the function L in a circle of radius R about (z , y ) as 



Ill, 57] SINGULAR POINTS MAXIMA AND MINIMA 121 

center. Finally, let r denote a positive number less than R and less 
than 3m/ H. Then inside a circle of radius r the difference A will 
have the same sign as the coefficient of p 2 , i.e. the same sign as A 
or C. Hence the function f(x, y~) has either a maximum or a mini 
mum for x = X Q , y = ?/o- 

To recapitulate, if at the point (x , y ) we have 

* *">(>, 



^dx dy 
there is neither a maximum nor a minimum. But if 



there is either a maximum or a minimum, depending on the sign of 
the two derivatives c 2 f/dx%, o^f/dyl. There is a maximum if these 
derivatives are negative, a minimum if they are positive. 

57. The ambiguous case. The case where B 2 A C = is not cov 
ered by the preceding discussion. The geometrical interpretation 
shows why there should be difficulty in this case. Let be the 
surface represented by the equation z = f(x, ?/). If the function 
f(x, y) has a maximum or a minimum at the point (X Q , y ), n ear 
which the function and its derivatives are continuous, we must have 



which shows that the tangent plane to the surface S at the point 
M , whose coordinates are (x 0) y , ), must be parallel to the xy 
plane. In order that there should be a maximum or a minimum it 
is also necessary that the surface S, near the point M , should lie 
entirely on one side of the tangent plane ; hence we are led to study 
the behavior of a surface with respect to its tangent plane near the 
point of tangency. 

Let us suppose that the point of tangency has been moved to the 
origin and that the tangent plane is the xy plane. Then the equa 
tion of the surface is of the form 

(24) z = ax 2 + 2 bxy + cy* + ax s + 3 /3x*y + 3 yxy 2 + Sy 8 , 

where a, b, c are constants, and where a, /8, y, 8 are functions of x 
and y which remain finite when x and y approach zero. This equa 
tion is essentially the same as equation (19), where x and y have 
been replaced by zeros, and h and k by x and y, respectively. 



122 



TAYLOR S SERIES 



[III, 57 



In order to see whether or not the surface S lies entirely on 
one side of the xy plane near the origin, it is sufficient to study the 
section of the surface by that plane. This section is given by the 
equation 

(25) ax* + 2bxy + cy 2 + ax* + = 0; 

hence it has a double point at the origin of coordinates. If b 2 ac 
is negative, the origin is an isolated double point ( 53), and the 
equation (25) has no solution except x = y = 0, when the point 
(x, y) lies inside a circle C of sufficiently small radius r drawn 
about the origin as center. The left-hand side of the equation (25) 
keeps the same sign as long as the point (x, y) remains inside this 
circle, and all the points of the surface S which project into the 
interior of the circle C are on the same side of the xy plane except 
the origin itself. In this case there is an extremum, and the por 
tion of the surface S near the origin resembles a portion of a sphere 
or an ellipsoid. 

If b 2 ac> 0, the intersection of the surface S by its tangent 
plane has two distinct branches C lf C z which pass through the 
origin, and the tangents to these two branches are given by the 

equation 

ax* + 2bxy + cy 2 = 0. 

Let the point (x, y) be allowed to move about in the neighborhood 
of the origin. As it crosses either of the two branches C x , C 2 , the 
left-hand side of the equation (25) vanishes and changes sign. 
Hence, assigning to each region of the plane in the neighborhood 
of the origin the sign of the left-hand side of the equation (25), we 
find a configuration similar to Fig. 7. Among the points of the 
surface which project into points inside a circle about the origin in 

the xy plane there are always some which 
lie below and some which lie above the 
xy plane, no matter how small the circle 
be taken. The general aspect of the sur 
face at this point with respect to its tan 
gent plane resembles that of an imparted 
hyperboloid or an hyperbolic paraboloid. 
The function f(x, y) has neither a maxi- 
mum nor a minimum at the origin. 
The case where b 2 ac = is the case in which the curve of 
intersection of the surface by its tangent plane has a cusp at the 
origin. We will postpone the detailed discussion of this case. If the 




FlG - 7 



Ill, 58] SINGULAR POINTS MAXIMA AND MINIMA 123 

intersection is composed of two distinct branches through the origin, 
there can be no extremum, for the surface again cuts the tangent 
plane. If the origin is an isolated double point, the function f(x, y~) 
has an extremum for x = y 0. It may also happen that the inter 
section of the surface with its tangent plane is composed of two 
coincident branches. For example, the surface K y* 2 x*y -f x* 
is tangent to the plane z = all along the parabola y = x 2 . The 
function ?/ 2 2 x 2 y -}- x 4 is zero at every point on this parabola, but is 
positive for all points near the origin which are not on the parabola. 

58. In order to see which of these cases holds in a given example it is neces 
sary to take into account the derivatives of the third and fourth orders, and some 
times derivatives of still higher order. The following discussion, which is usually 
sufficient in practice, is applicable only in the most general cases. When 
6 2 ac the equation of the surface may be written in the following form 
by using Taylor s development to terms of the fourth order: 

iW 



(26) z - f(x, y) = A(xsinu -y cos w) 2 + fa (x, y) + --[x + y , 

24 \ dx dy / 



ftr 



Let us suppose, for definiteness, that A is positive. In order that the surface S 
should lie entirely on one side of the xy plane near the origin, it is necessary that 
all the curves of intersection of the surface by planes through the z axis should 
lie on the same side of the xy plane near the origin. But if the surface be cut 
by the secant plane 

y = xtan 0, 

the equation of the curve of intersection is found by making the substitution 
x = p cos 0, y = p sin <j> 

in the equation (26), the new axes being the old z axis and the trace of the secant 
plane on the xy plane. Performing this operation, we find 

z = A p 2 (cos sin w cos w sin 0) 2 -f K p 3 + Lp*, 

where K is independent of p. If tan w ^ tan 0, z is positive for sufficiently small 
values of p ; hence all the corresponding sections lie above the xy plane near the 
origin. Let us now cut the surface by the plane 

y = x tan u. 
If the corresponding value of K is not zero, the development of z is of the form 



and changes sign with p. Hence the section of the surface by this plane has a 
point of inflection at the origin and crosses the xy plane. It follows that the 
function /(x, y) has neither a maximum nor a minimum at the origin. Such is 
the case when the section of the surface by its tangent plane has a cusp of the 
first kind, for instance, for the surface 

z = w 2 x 8 . 



124 TAYLOR S SERIES [HI, 58 

If K = for the latter substitution, we would carry the development out to 
terms of the fourth order, and we would obtain an expression of the form 



where K\ is a constant which may be readily calculated from the derivatives of 
the fourth order. We shall suppose that K\ is not zero. For infinitesimal val 
ues of p, z has the same sign as K\ ; if K\ is negative, the section in question lies 
beneath the xy plane near the origin, and again there is neither a maximum nor 
a minimum. Such is the case, for example, for the surface z = y 2 x 4 , whose 
intersection with the xy plane consists of the two parabolas y = x 2 . Hence, 
unless K = and K\ > at the same time, it is evidently useless to carry the 
investigation farther, for we may conclude at once that the surface crosses its 
tangent plane near the origin. 

But if K = and KI > at the same time, all the sections made by planes 
through the z axis lie above the xy plane near the origin. But that does not 
show conclusively that the surface does not cross its tangent plane, as is seen 
by considering the particular surface 

z - (y - x 2 ) (y - 2 x 2 ), 

which cuts its tangent plane in two parabolas, one of which lies inside the other. 
In order that the surface should not cross its tangent plane it is also necessary 
that the section of the surface made by any cylinder whatever which passes 
through the z axis should lie wholly above the xy plane. Let y = <f> (x) be the 
equation of the trace of this cylinder upon the xy plane, where <f> (x) vanishes for 
x = 0. The function F(x) =/[x, 0(x)] must be at a minimum for x = 0, what 
ever be the function (x). In order to simplify the calculation we will suppose 
that the axes have been so chosen that the equation of the surface is of the form 

z = Ay 2 + <f> 3 (x, !/) + , 
where A is positive. With this system of axes we have 

=0 ^>0 



8xo dy Q dx* dx dy 

at the origin. 

The derivatives of the function F(x) are given by the formulae 



F " x = 




+ 3 L $" ( X ) + 3 ^ W + ^ $ "(), 
dxdy dy 2 cy 

+4 -*-* fart 4. 6 ^ * *(x) | 1 ^ ^(z) | g*^ * 



^3 f 53 f 3 f 

6 -^- 0" (x) + 12 ^- + 6 -^ <f> 2 4>" 

dx 2 dy dxdy 2 dy 3 

d 2 f , d 2 f , , //2 8/ 

dx dy dy 2 dy 



Ill, 59] SINGULAR POINTS MAXIMA AND MINIMA 125 

from which, for x = y = 0, we obtain 



c !/0 

If tf> (0) does not vanish, the function F(x) has a minimum, as is also apparent 
from the previous discussion. But if < (()) = 0, we find the formulas 



Hence, in order that F(x) be at a minimum, it is necessary that d*f/x% vanish 
and that the following quadratic form in tf>"(0), 



r T~ 2 > 

dx* cx 2 dy dy 2 

be positive for all values of 0"(0). 

It is easy to show that these conditions are not satisfied for the above function 
z = y 2 3x 2 y + 2z 4 , but that they are satisfied for the function z = y 2 + x*. 
It is evident, in fact, that the latter surface lies entirely above the xy plane. 

We shall not attempt to carry the discussion farther, for it requires extremely 
nice reasoning to render it absolutely rigorous. The reader who wishes to exam 
ine the subject in greater detail is referred to an important memoir by Ludwig 
Scheffer, in Vol. XXXV of the Mathematische Annalen. 

59. Functions of three variables. Let u = f(x, y, z) be a continuous 
function of the three variables x, y, z. Then, as before, this func 
tion is said to have an extremum (maximum or minimum) for a set 
of values x , y , z if a positive number rj can be found so small 
that the difference 



which vanishes for h = k = I = 0, has the same sign for all other 
sets of values of h, k, I, each of which is less in absolute value 
than i]. If only one of the variables *, y, z is given an increment, 
while the other two are regarded as constants, we find, as above, 
that u cannot be at an extremum unless the equations 



are all satisfied, provided, of course, that these derivatives are con 
tinuous near the point (or , y 0) z ~). Let us now suppose that x , y , z 
are a set of solutions of these equations, and let M be the point 
whose coordinates are x w y Q , z . There will be an extremum if a 
sphere can be drawn about M so small that f(x, y, z) f(x 0) y , z ) 



126 TAYLOR S SERIES [III, 59 

has the same sign for all points (x, y, z) except M inside the sphere. 
Let the coordinates of a neighboring point be represented by the 
equations 

x = x + pa, y = y + pft, z = z + py, 

where a, ft, y satisfy the relation a 2 + /3 2 + y 2 = 1 ; and let us replace 
x x , y y , z z in Taylor s expansion of f(x, y, ) by pa, pft, 
py, respectively. This gives the following expression for A : 

A = p 2 [>O, ft, y) +.--], 

where <f>(a, ft, y) denotes a quadratic form in a, ft, y whose coeffi 
cients are the second derivatives of f(x, y, z), and where Z is a 
function which remains finite near the point M Q . The quadratic 
form may be expressed as the sum of the squares of three distinct 
linear functions of a, ft, y, say P, P , P", multiplied by certain con 
stant factors a, a , a", except in the particular case when the dis 
criminant of the form is zero. Hence we may write, in general, 



*(a, ft, y) = aP 2 + a P 2 + a"P" 2 , 

where a, a , a" are all different from zero. If the coefficients a, a , a" 
have the same sign, the absolute value of the quadratic form <f> will 
remain greater than a certain lower limit when the point a, ft, y 
describes the sphere 

2 + ft 2 + y 2 = 1, 

and accordingly A has the same sign as a, a , a" when p is less than 
a certain number. Hence the f imction f(x, y, z) has an extremum. 

If the three coefficients a, a , a" do not all have the same sign, 
there will be neither a maximum nor a minimum. Suppose, for 
example, that a > 0, a < 0, and let us take values of a, ft, y which 
satisfy the equations P = 0, P" = 0. These values cannot cause P 
to vanish, and A will be positive for small values of p. But if, on 
the other hand, values be taken for a, ft, y which satisfy the equa 
tions P = 0, P" = 0, A will be negative for small values of p. 

The method is the same for any number of independent variables : 
the discussion of a certain quadratic form always plays the prin 
cipal role. In the case of a function u = f(x, y, z) of only three 
independent variables it may be noticed that the discussion is 
equivalent to the discussion of the nature of a surface near a singu 
lar point. For consider a surface 2 whose equation is 

F(*> y, ) =f( x > y> *) -f(*o, y , O = 0; 



Ill, 60] SINGULAR POINTS MAXIMA AND MINIMA 127 

this surface evidently passes through the point M n whose coordi 
nates are (x , y , ), and if the function f(x, y, z) has an extremum 
there, the point M is a singular point of 2- Hence, if the cone of 
tangents at M is imaginary, it is clear that F(x, y, z) will keep the 
same sign inside a sufficiently small sphere about M as center, and 
/(# 2/j ) w iH surely have a maximum or a minimum. But if the 
cone of tangents is real, or is composed of two real distinct planes, 
several nappes of the surface pass through A/ , and F(x, y, z) 
changes sign as the point (x, y, z) crosses one of these nappes. 

60. Distance from a point to a surface. Let us try to find the maximum and the 
minimum values of the distance from a fixed point (a, b, c) to a surface S whose 
equation is F(x, y, z) = 0. The square of this distance, 

u = d* = (x - a)2 + (y - 6) + ( - c), 

is a function of two independent variables only, x and y, for example, if z be 
considered as a function of x and y defined by the equation F = 0. In order 
that u be at an extremum for a point (x, y, z) of the surface, we must have, for 
the coordinates of that point, 

1 du , dz 
= (x a) + (z c) = 0, 

2 dx ix 

1 du . dz 

_ = y _ & + (z - c) = 0. 
2 dy cy 

We find, in addition, from the equation F = 0, the relations 

dF dFdz dF dFdz rt 

-\ = U, 1 = U, 

dx dz dx dy dz dy 

whence the preceding equations take the form 

x a _ y b _ z c 
djr ~~ c_F_ " d_F_ 
dx dy dz 

This shows that the normal to the surface S at the point (x, y, z) passes through 
the point (a, 6, c). Hence, omitting the singular points of the surface S, the 
points sought for are the feet of normals let fall from the point (a, 6, c) upon the 
surface S. In order to see whether such a point actually corresponds to a maxi 
mum or to a minimum, let us take the point as origin and the tangent plane as 
the xy plane, so that the given point shall lie upon the axis of z. Then the func 
tion to be studied has the form 

u = x* + y 2 + (z - c)2, 

where z is a function of x and y which, together with both its first derivatives, 
vanishes for x = y = 0. Denoting the second partial derivatives of z by r, s, t, 
we have, at the origin, 

^ = 2(1 -or), -fiL 1* ^ = 2(l-cO, 
dx* dxdy dy 2 



128 TAYLOR S SERIES [III, 61 

and it only remains to study the polynomial 

A(C) = C 2 2 _ (1 _ cr) (1 _ ct ) = C 2( S 2 _ rt ) + (r + t ) c _ L 

The roots of the equation A (c) = are always real by virtue of the identity 
(r + ) 2 + 4 (s 2 rt) = 4 s 2 + (r t) 2 . There are now several cases which must 
be distinguished according to the sign of s 2 rt. 

First case. Let s 2 rt < 0. The two roots Ci and c 2 of the equation A (c) = 
have the same sign, and we may write A(c) = (s 2 rt) (c Ci) (c Cj). Let us 
now mark the two points A\ and A% of the z axis whose coordinates are c\ and c 2 . 
These two points lie on the same side of the origin ; and if we suppose, as is 
always allowable, that r and t are positive, they lie on the positive part of the 
z axis. If the given point A (0, 0, c) lies outside the segment AiA z , A(c) is 
negative, and the distance OA is a maximum or a minimum. In order to see 
which of the two it is we must consider the sign of 1 cr. This coefficient 
does not vanish except when c = 1 /r ; and this value of c lies between Ci and c 2 , 
since A (1/r) = s 2 /r 2 . But, for c = 0, 1 cr is positive ; hence 1 cr is posi 
tive, and the distance OA is a minimum if the point A and the origin lie on 
the same side of the segment A\A%. On the other hand, the distance OA i& a 
maximum if the point A and the origin lie on different sides of that segment. 
When the point A lies between AI and A 2 the distance is neither a minimum 
nor a maximum. The case where A lies at one of the points AI, A 2 is left in 
doubt. 

Second case. Let s 2 rt > 0. One of the two roots c\ and c 2 of A (c) = is 
positive and the other is negative, and the origin lies between the two points 
A\ and J. a . If the point A does not lie between A\ and A 2 , A(c) is positive 
and there is neither a maximum nor a minimum. If A lies between AI and 
A 2 , A (c) is negative, 1 cr is positive, and hence the distance OA is a minimum. 

Third case. Let s 2 - rt = 0. Then A(c) = (r + t) (c - cj), and it is easily 
seen, as above, that the distance OA is a minimum if the point A and the origin 
lie on the same side of the point AI, whose coordinates are (0, 0, Ci), and that 
there is neither a maximum nor a minimum if the point AI lies between the point 
A and the origin. 

The points AI and A 2 are of fundamental importance in the study of curva 
ture ; they are the principal centers of curvature of the surface S at the point 0. 

61. Maxima and minima of implicit functions. We often need to find 
the maxima and minima of a function of several variables which 
are connected by one or more relations. Let us consider, for 
example, a function to = f(x, y, z, tt) of the four variables x, y, z, u, 
which themselves satisfy the two equations 

/i (*, y, *, ) = 0, /,(*, y, z, M) = 0. 

For definiteness, let us think of x and y as the independent vari 
ables, and of z and u as functions of x and y defined by these equa 
tions. Then the necessary conditions that u> have an extremum are 



HI, 61] SINGULAR POINTS MAXIMA AND MINIMA 129 

2 + ^ + ^ = !/+? + 3?!? = 

dx dz dx du dx dy dz dy du dy 

and the partial derivatives dz/dx, du/dx, dz/dy, du/dy are given 
by the relations 

M_i.^^4.M^ = o 2 4. a * 4. ?!f = o 

dx dz dx^ du dx dx dz dx du dx 

0/i ,0/10*, 0/i0"_n 0/, , 0/,0* , 0/,0u 

"^ -- P "TT~ o -- P "5 ~^T~ ", "a P "o~ o P a "a~ U- 

^z/ ^* dy du dy cy oz cy cu cy 

The elimination of dz/dx, du/dx, dz/dy, du/dy leads to the new 
equations of condition 



.p _ ft 
/)(*, *, u) 

which, together with the relations / x = 0, / 2 = 0, determine the val 
ues of x, y, z, u, which may correspond to extrema. But the equa 
tions (27) express the condition that we can find values of \ and p. 
which satisfy the equations 



- -t- A -^- -I- w. IT- = "j a l ~ A "o " I* "a~ = w i 

c OX OX dy dy oy 

3 d Jl ^ 2 = o ^ + X^4 ^ 2 = 0- 

02 3s ds dw du du 

hence the two equations (27) may be replaced by the four equations 
(28), where X and p. are unknown auxiliary functions. 

The proof of the general theorem is self-evident, and we may 
state the following practical rule : 

Given a function 

Atm /m . *F \ * 

iCjj ^ 2 , , *, n ) 

of n variables, connected by h distinct relations 



in order to find the values of x^ a; 2 , , x n which may render this 
function an extremum we must equate to zero the partial derivatives 
of the auxiliary function 



regarding \ 1} X 2 , , X A as constants. 



130 TAYLOR S SERIES [III, 62 

62. Another example. We shall now take up another example, where the mini 
mum is not necessarily given by equating the partial derivatives to zero. Given 
a triangle ABC; let us try to find a point P of the plane for which the sum 
PA + PB + PC of the distances from P to the vertices of the triangle is a 
minimum. Let (01, 61), (a 2 , 6 2 ), (a 3 , 63) be respectively the coordinates of the 
vertices A, B, C referred to a system of rectangular coordinates. Then the func 
tion whose minimum is sought is 



(29) z = V(x - oi)a + (y- 6i) a + V(x - a 2 ) 2 + (y - 6 2 ) 2 + V(x - a,)* + (y - 



where each of the three radicals is to be taken with the positive sign. This equa 
tion (29) represents a surface S which is evidently entirely above the xy plane, 
and the whole question reduces to that of finding the point on this surface which 
is nearest the xy plane. From the relation (29) we find 



x a-2 x a s 

H -- , + 



s * V(x - m) a + (y - &!)2 V(x - a 2 ) 2 + (y - 6 2 )2 V(z - a s ) 2 + (y - 

dz _ y -bi y - 6 2 y -b 3 

1 



ft V(z _ ai )2 + (y - &!)2 V(x - as) 2 + (y - btf V(x - a 3 ) 2 + (y - 6 3 ) 2 

and it is evident that these derivatives are continuous, except in the neighbor 
hood of the points A, B, C, where they become indeterminate. The surface S, 
therefore, has three singular points which project into the vertices of the given 
triangle. The minimum of z is given by a point on the surface where the tan 
gent plane is parallel to the xy plane, or else by one of these singular points. In 
order to solve the equations cz/cx = 0, cz/dy = 0, let us write them in the 
form 

x 0,1 x a 2 x a 3 

i : 



V(x - ai) + (y- &i) 2 V(x - a 2 ) 2 + (y - 6 2 ) 2 V(x - a s ) 2 + (y - 
y-E>i y - &2 _ y - b 3 



V( X - fll )2 + (y - &!) V( X - aa) 2 +(y- 6 2 )2 

Then squaring and adding, we find the condition 



V(x - a 2 ) 2 + (y - 



_ 



The geometrical interpretation of this result is easy : denoting by a and /3 the 
cosines of the angles which the direction PA makes with the axes of x and j/, 
respectively, and by a and /3 the cosines of the angles which PB makes with the 
same axes, we may write this last condition in the form 

1 + 2 (aa + flS 7 ) = 0, 
or, denoting the angle APE by o>, 

2 cos u + 1 = 0. 

Hence the condition in question expresses that the segment AB subtends an 
angle of 120 at the point P. For the same reason each of the angles BPC and 
CPA must be 120.* It is clear that the point P must lie inside the triangle 

* The reader is urged to draw the figure. 



Ill, 63] SINGULAR POINTS MAXIMA AND MINIMA 131 

ABC, and that there is no point which possesses the required property if any 
angle of the triangle ABC is equal to or greater than 120. In case none of the 
angles is as great as 120, the point P is uniquely determined by an easy con 
struction, as the intersection of two circles. In this case the minimum is given 
by the point P or by one of the vertices of the triangle. But it is easy to show 
that the sum PA + PB + PC is less than the sum of two of the sides of the tri 
angle. For, since the angles APB and APC are each 120, we find, from the 
two triangles PAG and PBA, the formulae 



AB = Vi + b 2 + ab, AC = Va 2 + c 2 + oc, 
where PA = a, PB = 6, PC = c. But it is evident that 



Vo 2 + 6 2 + 06 > 6+-, Va 2 -)- c 2 + ac> c + ~, 
2 2 

and hence 

AB + AC > a + b + c. 

The point P therefore actually corresponds to a minimum. 

When one of the angles of the triangle ABC is equal to or greater than 120 
there exists no point at which each of the sides of the triangle ABC subtends an 
angle of 120, and hence the surface S has no tangent plane which is parallel to 
the xy plane. In this case the minimum must be given by one of the vertices of 
the triangle, and it is evident, in fact, that this is the vertex of the obtuse angle. 
It is easy to verify this fact geometrically. 

63. D Alembert s theorem. Let F(x, y) be a polynomial in the two variables 
x and y arranged into homogeneous groups of ascending order 

F(x, y) = H + <f> P (x, y) + <t> p + i (x, y) + + m (x, y), 

where H is a constant. If the equation <f> p (x, y) = 0, considered as an equation 
in y/Xj has a simple root, the function F(x, y) cannot have a maximum or a mini 
mum for x = y = 0. For it results from the discussion above that there exist sec 
tions of the surface z + H = F(x, y) made by planes through the z axis, some 
of which lie above the xy plane and others below it near the origin. From this 
remark a demonstration of d Alembert s theorem may be deduced. For, let/(z) 
be an integral polynomial of degree m, 

/(z) = A 9 + AIZ + A 2 z* + + A m z m , 

where the coefficients are entirely arbitrary. In order to separate the real and 
imaginary parts let us write this in the form 

f(x + iy) = a + ib + (ai + t&i) (x + iy) + + (a m + ib m ) (x + iy) m , 
where OQ, &o i, &ii m, &m are real. We have then 

f(z) = P+iQ, 
where P and Q have the following meanings : 

P = o + etix - biy -\ ---- , 
Q = & + bix + a^y + ; 
and hence, finally, 



132 TAYLOR S SERIES [III, 63 

We will first show that |/(z) |, or, what amounts to the same thing, that 
ps -j. Q2 f cannot be at a minimum for z = y = except when a = 60 = 0. For 
this purpose we shall introduce polar coordinates p and 0, and we shall suppose, 
for the sake of generality, that the first coefficient after A which does not 
vanish is A p . Then we may write the equations 

P = o + (a p cos p<f> b p sin p<f>) pp + , 
Q = b + (b p cos p0 + dp sin p<j>) pp + , 
P 2 + Q 2 = of, + 6g + 2/>p [(aoOp + b b p ) cosptf. + (b a p - a b p ) sinp0] + , 

where the terms not written down are of degree higher than p with respect to p. 
But the equation 

(aoap + b bp) cosp<j> + (b a p a b p ) smp<J> = 

gives tan p$ = K, which determines p straight lines which are separated by 
angles each equal to 2 n /p. It is therefore impossible by the above remark that 
P 2 + Q 2 should have a minimum for z = y = unless the quantities 

aoa p -f &o&p boa,p (tobp 

both vanish. But, since a 2 + ft 2 is not zero, this would require that a = 60 = ; 
that is, that the real and the imaginary parts of /(z) should both vanish at the 
origin. 

If |/(z) | has a minimum for z = a, y /3, the discussion may be reduced to 
the preceding by setting z = a + i/3 + z . It follows that \f(z) \ cannot be at a 
minimum unless P and Q vanish separately for x= a, y = p. 

The absolute value of /(z) must pass through a minimum for at least one 
value of z, for it increases indefinitely as the absolute value of z increases indefi 
nitely. In fact, we have 



where the terms omitted are of degree less than 2 m in p. This equation may be 
written in the form 



where t approaches zero as p increases indefinitely. Hence a circle may be 
drawn whose radius R is so large that the value of VP 2 + Q 2 is greater at every 
point of the circumference than it is at the origin, for example. It follows that 
there is at least one point 

x = a, y = 

inside this circle for which Vp + Q 2 is at a minimum. By the above it fol 
lows that the point x = a, y = /3 is a point of intersection of the two curves 
P = 0, Q = 0, which amounts to saying that z = a + /3i is a root of the equation 
/(*)=. 

In this example, as in the preceding, we have assumed that a function of the 
two variables x and y which is continuous in the interior of a limited region 
actually assumes a minimum value inside or on the boundary of that region. 
This is a statement which will be readily granted, and, moreover, it will be 
rigorously demonstrated a little later (Chapter VI). 



Ill, EM.] EXERCISES 133 

EXERCISES 

1. Show that the number 0, which occurs in Lagrange s form of the re 
mainder, approaches the limit l/(n + 2) as A approaches zero, provided that 
/( + 2J(a) i s not zero. 

2. Let F(x) be a determinant of order n, all of whose elements are functions 
of x. Show that the derivative F (x) is the sum of the n determinants obtained 
by replacing, successively, all of the elements of a single line by their deriva 
tives. State the corresponding theorem for derivatives of higher order. 

3. Find the maximum and the minimum values of the distance from a fixed 
point to a plane or a skew curve ; between two variable points on two curves ; 
between two variable points on two surfaces. 

4. The points of a surface S for which the sum of the squares of the dis 
tances from n fixed points is an extremum are the feet of the normals let fall 
upon the surface from the center of mean distances of the given n fixed points. 

5. Of all the quadrilaterals which can be formed from four given sides, that 
which is inscriptible in a circle has the greatest area. State the analogous 
theorem for polygons of n sides. 

6. Find the maximum volume of a rectangular parallelepiped inscribed in 
an ellipsoid. 

7. Find the axes of a central quadric from the consideration that the vertices 
are the points from which the distance to the center is an extremum. 

8. Solve the analogous problem for the axes of a central section of an ellipsoid. 

9. Find the ellipse of minimum area which passes through the three vertices 
of a given triangle, and the ellipsoid of minimum volume which passes through 
the four vertices of a given tetrahedron. 

10. Find the point from which the sum of the distances to two given straight 
lines and the distance to a given point is a minimum. 

[JOSEPH BERTRAND.] 

11. Prove the following formulae : 

log (3 + 2) = 2 log(z + 1) - 2 log (x-l) + log(x - 2) 



- _ - 

z 3 -3z 3Vx 3 -3z/ 6\z 3 -3z 

[BORDA S Series.] 

log(x + 6) = log(x -f 4) + log(x + 3) - 2 logx 

+ log(z - 3) + log(z - 4) - log(x - 5) 

of 72 If 72 y -1 

|_x* - 25z 2 + 72 3 \z* - 26z 2 + 72/ J 

[HARO S Series.] 



CHAPTER IV 
DEFINITE INTEGRALS 



I. SPECIAL METHODS OF QUADRATURE 

64. Quadrature of the parabola. The determination of the area 
bounded by a plane curve is a problem which has always engaged 
the genius of geometricians. Among the examples which have 
come down to us from the ancients one of the most celebrated is 
Archimedes quadrature of the parabola. We shall proceed to 
indicate his method. 

Let us try to find the area bounded by the arc A CB of a parabola 
and the chord A 13. Draw the diameter CD, joining the middle 
point D of AB to the point C, where the tangent is parallel to AB. 
Connect AC and BC, and let E and E be the points where the 

tangent is parallel to .BC and 
AC, respectively. We shall 
first compare the area of the 
triangle BEC, for instance, 
with that of the triangle ABC. 
Draw the tangent ET, which 
cuts CD at T. Draw the diam 
eter EF, which cuts CB at F; 




and, finally, draw EK and FH 
parallel to the chord AB. By 
an elementary property of the 
parabola TC = CK. Moreover, 
CT EF = KH, and hence 
EF= CH/2 = CD/ 4. The 
areas of the two triangles BCE 
and BCD, since they have the 
same base BC, are to each other as their altitudes, or as EF is 
to CD. Hence the area of the triangle BCE is one fourth the area 
of the triangle BCD, or one eighth of the area 5 of the triangle ABC. 
The area of the triangle A CE is evidently the same. Carrying out 
the same process upon each of the chords BE, CE, CE , E A, we 

134 



FIG. 8 



IV, 65] 



SPECIAL METHODS 



135 



obtain four new triangles, the area of each of which is S/8 2 , and so 
forth. The nih operation gives rise to 2" triangles, each having the 
area S/8 n . The area of the segment of the parabola is evidently 
the limit approached by the sum of the areas of all these triangles 
as n increases indefinitely ; that is, the sum of the following descend 
ing geometrical progression : 



and this sum is 4 5/3. It follows that the required area is equal to 
two thirds of the area of a parallelogram whose sides are AB and CD. 
Although this method possesses admirable ingenuity, it must be 
admitted that its success depends essentially upon certain special 
properties of the parabola, and that it is lacking in generality. The 
other examples of quadratures which we might quote from ancient 
writers would only go to corroborate this remark : each new curve 
required some new device. But whatever the device, the area to be 
evaluated was always split up into elements the number of which 
was made to increase indefinitely, and it was necessary to evaluate 
the limit of the sum of these partial areas. Omitting any further 
particular cases,* we will proceed at once to give a general method 
of subdivision, which will lead us naturally to the Integral Calculus. 

65. General method. For the sake of definiteness, let us try to 
evaluate the area 5 bounded by a curvilinear arc A MB, an axis xx 
which does not cut that arc, and two perpendiculars AA and BB let 
fall upon xx from 
the points A and B. 
We will suppose 
further that a par 
allel to these lines 
AA , BB cannot 
cut the arc in more 
than one point, as 
indicated in Fig. 9. 
Let us divide the segment A B into a certain number of equal or 
unequal parts by the points P l9 P 2 , , P n .\, and through these 
points let us draw lines PiQi, P 2 Q 2 , , P n _iQ H _i parallel to AA 
and meeting the arc AB in the points Qi, Q 2 > j Q n -i> respectively. 




FIG. 9 



* A large number of examples of determinations of areas, arcs, and volumes by 
the methods of ancient writers are to be found in Duhamel s TraiM. 



136 DEFINITE INTEGRALS [IV, 65 

Now draw through A a line parallel to xx , cutting P t Q t at q ; 
through Qi a parallel to xx , cutting P 2 Q 2 at q 2 ; and so on. We 
obtain in this way a sequence of rectangles RI, R 2 , , R t , , R n . 
Each of these rectangles may lie entirely inside the contour AB A , 
but some of them may lie partially outside that contour, as is 
indicated in the figure. 

Let a ( denote the area of the rectangle R { , and /^ the area bounded 
by the contour P i _ l P i Q i Q i _ l . In the first place, each of the ratios 
fii/ a D ^2/ a 2> > A/ a i> approaches unity as the number of 
points of division increases indefinitely, if at the same time each 
of the distances A P l} PiP 2 , , P^P,, approaches zero. For 
the ratio /?,/<*,, for example, evidently lies between ,-/ /*,-_! Q,-_i and 
L i /P i _ l Q i _ l , where l f and L { are respectively the minimum and the 
maximum distances from a point of the arc Q i _ l Q i to the axis xx . 
But it is clear that these two fractions each approach unity as the 
distance P t _ l P i approaches zero. It therefore follows that the ratio 

a-! + or 2 H ----- h a n 



A + & + + & 

which lies between the largest and the least of the ratios tfi//3i, 
a 2//?2> > a n/ Pm w iH a ^ so approach unity as the number of the 
rectangles is thus indefinitely increased. But the denominator of 
this ratio is constant and is equal to the required area S. Hence 
this area is also equal to the limit of the sum a x + a 2 + + a n , as 
the number of rectangles n is indefinitely increased in the manner 
specified above. 

In order to deduce from this result an analytical expression for 
the area, let the curve AB be referred to a system of rectangular 
axes, the x axis Ox coinciding with xx , and let y =f(x) be the 
equation of the curve AB. The function f(x) is, by hypothesis, a 
continuous function of x between the limits a and b, the abscissae 
of the points A and B. Denoting by x 1} x 2 , , x n _ l the abscissas 
of the points of division P 1} P 2 , , P n _j, the bases of the above 
rectangles are x a, x. 2 x ly , x t x^^ , b x n _ l , and their 
altitudes are, in like manner, f(a) t f(x^ t -, /(<_,), , /(_i). 
Hence the area S is equal to the limit of the following- sum : 

(1) ( Xl - a)f(a) + (x 2 - *,)/(*!) + - - + (b - *_,)/(*_,), 

as the number n increases indefinitely in such a way that each of 
the differences x l a, x 2 x l} approaches zero. 



SPECIAL METHODS 137 

66. Examples. If the base AB be divided into n equal parts, each 
of length h (b a = nh), all the rectangles have the same base h, 
and their altitudes are, respectively, 

/(a), /(a + h), f(a + 2 A), -, /[ + ( - 1) A]. 
It only remains to find the limit of the sum 

h !/() +/( + 7 +/( + 2 7 + 

where 



as the integer w increases indefinitely. This calculation becomes 
easy if we know how to find the sum of a set of values f(x) corre 
sponding to a set of values of x which form an arithmetic progres 
sion ; such is the case if f(x) is simply an integral power of x, or, 
again, if t /(o;)= s mmx or /"(#)= cosmx, etc. 

Let us reconsider, for example, the parabola x* = 2py, and let us try 
to find the area enclosed by an arc OA of this parabola, the axis of x, 
and the straight line x = a which passes through the extremity A. 
The length being divided into n equal parts of length h (nh = a), we 
must try to find by the above the limit of the sum 



The quantity inside the parenthesis is the sum of the squares of the 
first (n 1) integers, that is, n(n 1) (2 n l)/6; and hence the 
foregoing sum is equal to 



As n increases indefinitely this sum evidently approaches the limit 
a*/6p = (1/3) (a. a 2 /2p), or one third of the rectangle constructed 
upon the two coordinates of the point A, which is in harmony with 
the result found above. 

In other cases, as in the following example, which is due to 
Fermat, it is better to choose as points of division points whose 
abscissae are in geometric progression. 

Let us try to find the area enclosed by the curve y = Ax*, the 
axis of x, and the two straight lines x a, x = I (0 < a < b), where 



138 DEFINITE INTEGRALS [IV, 66 

the exponent /* is arbitrary. In order to do so let us insert between 
a and b, n 1 geometric means so as to obtain the sequence 



where the number a satisfies the condition a (1 4- a)" = b. Tak 
ing this set of numbers as the abscissae of the points of division, the 
corresponding ordinates have, respectively, the following values : 



ay, Aa* (I + a) 2 * 1 , 
and the area of the pth rectangle is 
[a (1 + a)" a (1 + a)*" 1 ] Aa*(l + a) <-* = Aa +l 
Hence the sum of the areas of all the rectangles is 



If /i + 1 is not zero, as we shall suppose first, the sum inside the 
parenthesis is equal to 



or, replacing a (1 + a)" by i, the original sum may be written in the 
form 



\^ I . v 

As a approaches zero the quotient [(1 + a) M + 1 !]/<* approaches 
as its limit the derivative of (1 + a)^ + 1 with respect to a for a = 0, 
that is, /i + 1 ; hence the required area is 



If p. = 1, this calculation no longer applies. The sum of the 
areas of the inscribed rectangles is equal to nAa, and we have to 
find the limit of the product na where n and a are connected by the 
relation 

a(l -I- a)" = b. 

From this it follows that 

, b a . b 1 

na = log - r = log - 

& alog(l + ) h a 



, ,4 
log(l 



IV, 67] 



SPECIAL .METHODS 



139 




where the symbol log denotes the Naperian logarithm. As a 
approaches zero, (1 + a) 1 /* approaches the number e, and the prod 
uct na approaches log (b fa). Hence the required area is equal to 
vl log (&/) 

67. Primitive functions. The invention of the Integral Calculus 
reduced the problem of evaluating a plane area to the problem of 
finding a function whose derivative is known. Let y =f(xj be the 
equation of a curve referred to two rectangular axes, where the 
function f(x) is continuous. Let us consider the area enclosed by 
this curve, the axis of x, a fixed ordinate M P , and a variable 
ordinate MP, as a function of the abscissa x of the variable ordinate. 
In order to include all pos 
sible cases let us agree to 
denote by A the sum of the 
areas enclosed by the given 
curve, the x axis, and the 
straight lines M P , MP, 
each of the portions of 
this area being affected 
by a certain sign : the 
sign + for the portions to 
the right of M P and above Ox, the sign 
right of M P and below Ox, and the opposite convention for por 
tions to the left of M^Pg. Thus, if MP were in the position M P , we 
would take A equal to the difference 

JI/ P C - M P C; 

and likewise, if MP were at M"P", A = M"P"D - M P D. 

With these conventions we shall now show that the derivative of 
the continuous function A, defined in this way, is precisely /(#). As 
in the figure, let us take two neighboring ordinates MP, NQ, whose 
abscissae are x and x -f Ax. The increment of the area A.4 evidently 
lies between the areas of the two rectangles which have the same 
base PQ, and whose altitudes are, respectively, the greatest and the 
least ordinates of the arc M N. Denoting the maximum ordinate by 
H and the minimum by h, we may therefore write 

AAz < <\A < 7/Ax, 

or, dividing by Ax, h < A/l /Ax < //. As Ax approaches zero, // and 
h approach the same limit MP, or /(x), since /(x) is continuous. 



FIG. 10 



for the portions to the 



140 DEFINITE INTEGRALS [IV, <;8 

Hence the derivative of A is f(x). The proof that the same result 
holds for any position of the point .17 is left to the reader. 

If we already know a primitive function of f(x), that is, a function 
F(x) whose derivative is/(z), the difference A F(x) is a constant, 
since its derivative is zero ( 8). In order to determine this con 
stant, we need only notice that the area A is zero for the abscissa 
x = a of the line MP. Hence 

A =F(x)-F(a). 

It follows from the above reasoning, first, that the determination 
of a plane area may be reduced to the discovery of a primitive func 
tion; and, secondly (and this is of far greater importance for us), 
that every continuous function f(x) is the derivative of some other 
function. This fundamental theorem is proved here by means of 
a somewhat vague geometrical concept, that of the area under a 
plane curve. This demonstration was regarded as satisfactory for a 
long time, but it can no longer be accepted. In order to have a stable 
foundation for the Integral Calculus it is imperative that this theo 
rem should be given a purely analytic demonstration which does not 
rely upon any geometrical intuition whatever. In giving the above 
geometrical proof the motive was not wholly its historical interest, 
however, for it furnishes us with the essential analytic argument of 
the new proof. It is, in fact, the study of precisely such sums as 
(1) and sums of a slightly more general character which will be 
of preponderant importance. Before taking up this study we must 
first consider certain questions regarding the general properties of 
functions and in particular of continuous functions.* 

II. DEFINITE INTEGRALS ALLIED GEOMETRICAL CONCEPTS 

68. Upper and lower limits. An assemblage of numbers is said to 
have an upper limit (see ftn., p. 91) if there exists a number N so 
large that no member of the assemblage exceeds N. Likewise, an 
assemblage is said to have a lower limit if a number N exists than 
which no member of the assemblage is smaller. Thus the assem 
blage of all positive integers has a lower limit, but no upper limit ; 

* Among the most important works on the general notion of the definite integral 
there should be mentioned the memoir by Riemann : fiber die Mb glichkeit, eine Func 
tion durch eine trigonometrische Reihe darzustellen (Werke, 2d ed., Leipzig, 1892, 
p. 239 ; and also French translation by Laugel, p. 225) ; and the memoir by Darboux, to 
which we have already referred : Sur les fonctions discontinues (Annales de VEcole 
Normals Suptrieure, 2d series, Vol. IV). 



IV, 68] ALLIED GEOMETRICAL CONCEPTS 141 

the assemblage of all integers, positive and negative, has neither ; 
and the assemblage of all rational numbers between and 1 has 
both a lower and an upper limit. 

Let (E) be an assemblage which has an upper limit. With 
respect to this assemblage all numbers may be divided into two 
classes. We shall say that a number a belongs to the first class if 
there are members of the assemblage (7?) which are greater than a, 
and that it belongs to the second class if there is no member of the 
assemblage (7?) greater than a. Since the assemblage (7?) has an 
upper limit, it is clear that numbers of each class exist. If A be 
a number of the first class and B a number of the second class, it 
is evident that A < B ; there exist members of the assemblage (7?) 
which lie between A and B, but there is no member of the assem 
blage (7?) which is greater than B. The number C = (A -f jB)/2 
may belong to the first or to the second class. In the former case 
we should replace the interval (A, B*) by the interval (C, 7?), in the 
latter case by the interval (A, C). The new interval (.4^ 7^) is half 
the interval (^4, B) and has the same properties : there exists at least 
one member of the assemblage (7) which is greater than A 1} bnt none 
which is greater than B. Operating upon (A l} B^) in the same way 
that we operated upon (A, B}, and so on indefinitely, we obtain an 
unlimited sequence of intervals (A, 73), (A lf 7^), (A 2 , 7? 2 ), j each 
of which is half the preceding and possesses the same property 
as (A, B} with respect to the assemblage (/?). Since the numbers 
A, AI, A z , , A n never decrease and are always less than B, they 
approach a limit A ( 1). Likewise, since the numbers B, B 1} B 2 , 
never increase and are always greater than A, they approach a limit X . 
Moreover, since the difference B n A n = (B A ) /2" approaches zero 
as n increases indefinitely, these limits must be equal, i.e. A = A. 
Let L be this common limit ; then L is called the iqjper limit of the 
assemblage (7?). From the manner in which we have obtained it, 
it is clear that L has the following two properties : 

1) No member of the assemblage (7i) is greater than L. 

2) There always exists a member of the assemblage (7?) which is 
greater than L e, where c is any arbitrarily small positive number. 

For let us suppose that there were a member of the assemblage 
greater than L, say L + h (Ji > 0). Since B n approaches L as n 
increases indefinitely, B n will be less than L -f h after a certain 
value of n. But this is impossible since B n is of the second class. 
On the other hand, let e be any positive number. Then, after a 



142 DEFINITE INTEGRALS [IV, 69 

certain value of n, A n will be greater than L e ; and since there are 
members of (E) greater than A n , these numbers will also be greater 
than L e. It is evident that the two properties stated above can 
not apply to any other number than L. 

The upper limit may or may not belong to the assemblage (). 
In the assemblage of all rational numbers which do not exceed 2, 
for instance, the number 2 is precisely the upper limit, and it belongs 
to the assemblage. On the other hand, the assemblage of all irra 
tional numbers which do not exceed 2 has the upper limit 2, but 
this upper limit is not a member of the assemblage. It should be 
particularly noted that if the upper limit L does not belong to the 
assemblage, there are always an infinite number of members of () 
which are greater than L c, no matter how small e be taken. For if 
there were only a finite number, the upper limit would be the largest 
of these and not L. When the assemblage consists of n different 
numbers the upper limit is simply the largest of these n numbers. 

It may be shown in like manner that there exists a number L\ 
in case the assemblage has a lower limit, which has the following 
two properties : 

1) No member of the assemblage is less than L . 

2) There exists a member of the assemblage which is less than 
L -\- (., where e is an arbitrary positive number.* 

This number L is called the lower limit of the assemblage. 

69. Oscillation. Let/() be a function of x defined in the closed f 
interval (a, ) ; that is, to each value of x between a and b and to each 
of the limits a and b themselves there corresponds a uniquely deter 
mined value of f(x}. The function is said to be finite in this closed 
interval if all the values which it assumes lie between two fixed 
numbers A and B. Then the assemblage of values of the function 
has an upper and a lower limit. Let M and m be the upper and 
lower limits of this assemblage, respectively ; then the difference 



* Whenever all numbers can be separated into two classes A and B, according to 
any characteristic property, in such a way that any number of the class A is less than 
any number of the class B, the upper limit L of the numbers of the class A is at the 
same time the lower limit of the numbers of the class B. It is clear, first of all, that 
any number greater than L belongs to the class B. And if there were a number L <L 
belonging to the class B, then every number greater than L would belong to the class B. 
Hence every number less than L belongs to the class A, every number greater than L 
belongs to the class B, and L itself may belong to either of the two classes. 

t The word " closed " is used merely for emphasis. See 2. TRANS. 



IV, 70J ALLIED GEOMETRICAL CONCEPTS 143 

A = M m is called the oscillation of the function f(x) in the 
interval (a, b). 

These definitions lead to several remarks. In order that a func 
tion be finite in a closed interval (a, b~) it is not sufficient that it 
should have a finite value for every value of x. Thus the function 
defined in the closed interval (0, 1) as follows : 

= 0, /(aj) = l/aj for x > 0, 



has a finite value for each value of x ; biit nevertheless it is not 
finite in the sense in which we have defined the word, for/(ce) > A 
if we take x<l / A. Again, a function which is finite in the closed 
interval (a, b) may take on values which differ as little as we please 
from the upper limit M or from the lower limit m and still never 
assume these values themselves. For instance, the function /(#), 
defined in the closed interval (0, 1) by the relations 



= 0, f(x) = l-x for 0<x<l, 
has the upper limit M = 1, but never reaches that limit. 

70. Properties of continuous functions. We shall now turn to the 
study of continuous functions in particular. 

THEOREM A. Letf(x) be a function which is continuous in the closed 
interval (a, b) and e an arbitrary positive number. Then we can 
always break up the interval (a, ft) into a certain number of partial 
intervals in such a way that for any two values of the variable 
whatever, x and x", which belong to the same partial interval, we 
always have \f(x ) f( x ") \ < 

Suppose that this were not true. Then let c=(a + ft)/2; at 
least one of the intervals (a, c), (c, ft) would have the same prop 
erty as (a, ft); that is, it would be impossible to break it up into 
partial intervals which would satisfy the statement of the theorem. 
Substituting it for the given interval (a, ft) and carrying out the 
reasoning as above ( 68), we could form an infinite sequence of 
intervals (a, ft), (a l} b^, (a 2 , ft 2 ), -, each of which is half the preced 
ing and has the same property as the original interval (a, ft). For 
any value of n we could always find in the interval ( n , ft n ) two 
numbers x and x" such that |/(V)~/( X ")I would be larger than e. 
Now let X be the common limit of the two sequences of numbers 
a, a 1} a 2 > " an ^ b, b^ b 2 , . Since the function /(#) is continuous 
for x = X, we can find a number rj such that |/(.r) /(X)| < e/2 



144 DEFINITE INTEGRALS 



[IV, 70 



whenever ja; A| is less than rj. Let us choose n so large that both 
a n and b n differ from A by less than 77. Then the interval (a n , b n } 
will lie wholly within the interval (A - rj, A + 77) ; and if * and a;" 
are any two values whatever in the interval (a n , Z> B ), we must have 



and hence |/<V) -/(.x") | < . It follows that the hypothesis made 
above leads to a contradiction ; hence the theorem is proved. 

Corollary I. Let a, x lt x 2 , , x p _ 1} b be a method of subdivision 
of the interval (a, i) into p subintervals, which satisfies the con 
ditions of the theorem. In the interval (a, a^) we shall have 
!/(*) \ < I/O) I + 5 and, in particular, \f( Xl ) \ < |/(a) | + e . Like 
wise, in the interval (x l} a-,) we shall have [/(*) I < j/(*i) + e, 
and, a fortiori, \f(x) \ < |/(a) | + 2 c ; in particular, for x = xj, 
|/(0 I < I /() | + 2 e ; and so forth. For the last interval we shall 
have 

I/(*)|<|/(P-I) | + <|/(a) 1 + pe. 

Hence the absolute value of f(x) in the interval (a, b~) always 
remains less than |/(a) | + pe. It follows that every function which 
is continuous in a closed interval (a, b) is finite in that interval. 

Corollary II. Let us suppose the interval (a, b) split up into 7? sub- 
intervals (a, x^, (x lt x 2 ), ..., (x p _v b) such that \f(x ) f(x")\< e /2 
for any two values of x which belong to the same closed subinterval. 
Let 77 be a positive number less than any of the differences ^ a, 
x 2 Xi , b x p _ l . Then let us take any two numbers whatever 
in the interval (a, b) for which \x - x" < ^ and let us try to find 
an upper limit for (/(* ) -/(*")). If the two numbers x and x" 
fall in the same subinterval, we shall have \f(x ) f(x")\< e /2. 
If they do not, x and x" must lie in two consecutive intervals, 
and it is easy to see that |/<V) -f(x") \ < 2 ( e /2) = c . Hence cor 
responding to any positive number c another positive number rj can be 
found such that 

I/W-/CO.I<* 

where x and x" are any two numbers of the interval (a, b) for which 
, [** *"!< >? This property is also expressed by saying that the 
function /(x) is uniformly continuous in the interval (a, b). 

THEOREM B. A function f(x) which is continuous in a closed 
interval (a, b) takes on every value between /(a) and f(b) at least 
once for some value of x which lies between a and b. 



IV, 70] ALLIED GEOMETRICAL CONCEPTS 145 

Let us first consider a particular case. Suppose that f(a) and 
/() have opposite signs, that /(a) < and/(6) > 0, for instance. 
We shall then show that there exists at least one value of x between 
a and b for which f(x) = 0. Now/(x) is negative near a and posi 
tive near b. Let us consider the assemblage of values of x between 
a and b for which /(#) is positive, and let \ be the lower limit of 
this assemblage (a < A < b*). By the very definition of a lower 
limit /(A A) is negative or zero for every positive value of h. 
Hence /(A.), which is the limit of /(A A), is also negative or zero. 
But /(A) cannot be negative. For suppose that /(A) = m, where 
m is a positive number. Since the function /(x ) is continuous for 
x = A, a number rj can be found such that )/(#) /(A) < m when 
ever \x A, | < rj, and the function f(x) would be negative for all 
values of x between A. and A + rj. Hence A could not be the lower 
limit of the values of x for which /(ic) is positive. Consequently 
/(A) - 0. 

Now let N be any number between /(a) and /(>). Then the 
function <(#) =f(x) N is continuous and has opposite signs for 
x = a and x b. Hence, by the particular case just treated, it 
vanishes at least once in the interval (a, &). 

THEOREM C. Every function which is continuous in a closed inter 
val (a, b) actually assumes the value of its upper and of its lower 
limit at least once. 

In the first place, every continuous function, since we have 
already proved that it is finite, has an upper limit M and a lower 
limit m. Let us show, for instance, that f(x) M for at least one 
value of x in the interval (a, 5). 

Taking c = (a + b)/2, the upper limit of f(x) is equal to M for 
at least one of the intervals (a, e), (c, b). Let us replace (a, b) 
by this new interval, repeat the process upon it, and so forth. 
Reasoning as we have already done several times, we could form 
an infinite sequence of intervals (a, b), (a u & t ), (o 2 , & 2 ), , each of 
which is half the preceding and in each of which the upper limit of 
f(x) is M. Then, if A is the common limit of the sequences a, a if 
, a n , and b, b 1} , b n , , /(A) is equal to M. For suppose that 
/(A) = M h, where h is positive. We can find a positive number 
rj such that f(x) remains between /(A) + h/2 and /(A) h/ 2, and 
therefore less than M h/2 as long as x remains between A rj 
and A -f rj- Let us now choose n so great that a n and b n differ from 
their common limit A by less than 77. Then the interval (a,,, &) lies 



146 DEFINITE INTEGRALS [IV, 71 

wholly inside the interval (A. 77, A. 4- *;), and it follows at once 
that the upper limit of f(x) in the interval (a n , b n ) could not be 
equal to M. 

Combining this theorem with the preceding, we see that any func 
tion which is continuous in a closed interval (a, ft) assumes, at least 
once, every value between its upper and its lower limit. Moreover 
theorem A may be stated as follows : Given a function which is 
continuous in a closed interval (a, ft), it is possible to divide the inter 
val into such small subreyions that the oscillation of the function in 
any one of them will be less than an arbitrarily assigned positive 
number. For the oscillation of a continuous function is equal to 
the difference of the values of /(x) for two particular values of the 
variable. 

71. The sums S and s. Let /(#) be a finite function, continuous 
or discontinuous, in the interval (a, ft), where a < b. Let us sup 
pose the interval (a, b) divided into a number of smaller partial 
intervals (a, o^), (a; u a; 2 ), , (x p _ l , b), where each of the numbers 
x lt x 2 , , # p _i is greater than the preceding. Let M and m be the 
limits of f(x) in the original interval, and M { and m i the limits 
in the interval (a^-i, #,-), and let us set 

S = M, (x, - a) + M 2 (x 2 -x 1 )+--- + M p (b - z p _ t ), 
s = m l (x l a) -f- 77*2(3-2 x i)-\ H m P (* X P -I)- 

To every method of division of (a, b) into smaller intervals there 
corresponds a sum S and a smaller sum s. It is evident that none 
of the sums 5 are less than m(b a), for none of the numbers M i 
are less than m ; hence these sums S have a lower limit /.* Like 
wise, the sums s, none of which exceed M(b a) have an upper 
limit / . We proceed to show that / is at most equal to I. For this 
purpose it is evidently sufficient to show that s^S and s 5j S, where 
S, s and S , s are the two sets of sums which correspond to any 
two given methods of subdivision of the interval (a, b). 

In the first place, let us suppose each of the subintervals (a, a^), 
(#1, a 2 ), redivided into still smaller intervals by new points of 
division and let 



* If f(x) is a constant, S = s, M = m, and, in general, all the inequalities mentioned 
become equations. TRANS. 



IV, 72] ALLIED GEOMETRICAL CONCEPTS 147 

be the new suite thus obtained. This new method of subdivision 
is called consecutive to the first. Let 2 and cr denote the sums anal 
ogous to S and s with respect to this new method of division of the 
interval (a, b), and let us compare S and s with 2 and a. Let us 
compare, for example, the portions of the two sums 5 and 2 which 
arise from the interval (a, a^). Let M[ and m[ be the limits of 
f(x) in the interval (a, y^, M[ and m^ the limits in the interval 
G/i> 1/2)9 " M k an( i m k the limits in the interval (y t -i> %i)- Then 
the portion of 2 which comes from (a, a^) is 



and since the numbers M{, M%, -, M/. cannot exceed M lt it is clear 
that the above sum is at most equal to 3/j (x l a). Likewise, the 
portion of 2 which arises from the interval (x l} a: 2 ) is at most equal 
to M 2 (x 2 ,), and so on. Adding all these inequalities, we find 
that 2 = S, and it is easy to show in like manner that a- ^ s. 

Let us now consider any two methods of subdivision whatever, 
and let S, s and S , s be the corresponding sums. Superimposing 
the points of division of these two methods of subdivision, we get a 
third method of subdivision, which may be considered as consecu 
tive to either of the two given methods. Let 2 and <r be the sums 
with respect to this auxiliary division. By the above we have the 
relations 

2<S, (r>s, 2<S , o->s ; 

and, since 2 is not less than a-, it follows that s ^ S and s^ S . Since 
none of the sums S are less than any of the sums s, the limit 7 
cannot be less than the limit / ; that is, / ^ / . 

72. Integrable functions. A function which is finite in an inter 
val (a, b~) is said to be integrable in that interval if the two sums 
S and 5 approach the same limit when the number of the partial 
intervals is indefinitely increased in such a way that each of those 
partial intervals approaches zero. 

The necessary and sufficient condition that a function be integrable 
in an interval is that corresponding to any positive number e another 
number rj exists such that S s is less than c whenever each of the 
partial intervals is less than -r\. 

This condition is, first, necessary, for if S and s have the same 
limit 7, we can find a number ^ so small that | S T\ and js 7| are 



148 DEFINITE INTEGRALS [iv, 72 

each less than e/2 whenever each of the partial intervals is less 
than 77. Then, a fortiori, S s is less than e. 

Moreover the condition is sufficient, for we may write * 

s-s = s-i + i- r + r - s , 

and since none of the numbers S I, I I , I s can be negative, 
each of them must be less than e if their sum is to be less than e. 
But since I I is a fixed number and e is an arbitrary positive 
number, it follows that we must have / = 7. Moreover S I < e 
and / s < e whenever each of the partial intervals is less than 77, 
which is equivalent to saying that S and s have the same limit 7. 

The function /(#) is then said to be integrable in the interval 
(a, ), and the limit 7 is called a definite integral. It is represented 
by the symbol 



= ff(x)dx, 

*J a 



which suggests its origin, and which is read " the definite integral 
from a to b of f(x) dx." By its very definition 7 always lies between 
the two sums S and s for any method of subdivision whatever. 
If any number between S and s be taken as an approximate value 
of 7, the error never exceeds S s. 

Every continuous function is integrable. 

The difference S s is less than or equal to (b a), where 
w denotes the upper limit of the oscillation of f(x~) in the partial 
intervals. But 77 may be so chosen that the oscillation is less than 
a preassigned positive number in any interval less than 77 ( 70). 
If then 77 be so chosen that the oscillation is less than (./(b a), 
the difference S s will be less than e. 

Any monotonically increasing or monotonically decreasing function 
in an interval is integrable in that interval. 

A f unction /"( x) is said to increase monotonically in a given interval 
(a, 6) if for any two values x , x" in that interval /(# ) >/(") when 
ever x > x". The function may be constant in certain portions of the 
interval, but if it is not constant it must increase with x. Dividing 
the interval (a, b) into n subintervals, each less than 77, we may write 

S =/<X> (*! - a) +/(* 2 ) (x 2 _ aJt) + +/() (b - *_!), 
S =/(a)(ar 1 - a) +/(.T 1 )(* 2 - xj -\- +f(x n _,} (b - *_,), 



*For the proof that I and / exist, see 73, which may be read before 72. TRANS. 



IV, 72] ALLIED GEOMETRICAL CONCEPTS 149 

for the upper limit of f(x) in the interval (a, a^), for instance, 
is precisely f(x\), the lower limit /"(); and so on for the other 
subintervals. Hence, subtracting, 



-s = (x 1 - a) [/(zj) -/(a)] + (* 2 - 

-/(*._,)]. 



None of the differences which occur in the right-hand side of this 
equation are negative, and all of the differences x l a, x 2 x l} 
are less than t] ; consequently 



or 

-<*[/(*) -/()], 

and we need only take 



in order to make 5 <s < e. The reasoning is the same for a mono- 
tonically decreasing function. 

Let us return to the general case. In the definition of the inte 
gral the sums S and s may be replaced by more general expres 
sions. Given any method of subdivision of the interval (a, i) : 



let i* s> *,(> be values belonging to these intervals in order 
(z, _ i = 4- = x i) Then the sum 



(2) . 



evidently lies between the sums S and s, for we always have 
7w f ^/(,.) 5 Af,-. If the function is integrable, this new sum has the 
limit /. In particular, if we suppose that 1? 2 , , ^ n coincide 
with a, ajj, , a; n _ 1 , respectively, the sum (2) reduces to the sum 
(1) considered above ( 65). 

There are several propositions which result immediately from the 
definition of the integral. We have supposed that a < b ; if we now 
interchange these two limits a and b, each of the factors x t x { _! 
changes sign; hence 



Cf(x)dx = - Cf(x)dx. 

Ja Jb 



150 DEFINITE INTEGRALS [IV, 72 

It also evidently follows from the definition that 
f(x)dx = C f(x)dx + f(x}dx, 



f 

Jo. 



at least if c lies between a and b; the same formula still holds when 
I) lies between a and c, for instance, provided that the function f(x) 
is integrable between a and c, for it may be written in the form 

ff(x)dx = Cf(x)dx~ f f(x}dx = Cf(x)dx + f C f(x)dx. 

Jo. Ja Jc J a Jb 

If f(x) = A<f>(x) -f B\j/(x), where A and B are any two constants, 
we have 

/> b /->h s*b 

I f(x)dx = A I <f>(x)dx + B I ij/(x*)dx, 

J a J a. \J a 

and a similar formula holds for the sum of any number of functions. 
The expression /(,-) in (2) may be replaced by a still more gen 
eral expression. The interval (-, I) being divided into n sub- 
intervals (a, a^), , (#/_!, a;,-), , let us associate with each of the 
subintervals a quantity ,, which approaches zero with the length 
x t x i _ l of the subinterval in question. We shall say that , 
approaches zero uniformly if corresponding to every positive num 
ber c another positive number -rj can be found independent of i and 
such that j,- < e whenever a\ x f-1 is less than 77. We shall now 
proceed to show that the sum 



approaches the definite integral j^ffx^dx as its limit provided 
that ,- approaches zero uniformly. For suppose that rj is a number 
so small that the two inequalities 



are satisfied whenever each of the subintervals x i x i _ 1 is less 
than 17. Then we may write 



/(*-i)(i-*-i)- f 



IV, 73] ALLIED GEOMETRICAL CONCEPTS 151 

and it is clear that we shall have 

< e + c(b a) 



S - C f(x)dx 

\J a 



whenever each of the subintervals is less than 77. Thus the theorem 
is proved.* 

73. Darboux s theorem. Given any function f(x) which is finite in an inter 
val (a, 6); the sums S and s approach their limits I and / , respectively, when 
the number of subintervals increases indefinitely in such a way that each of 
them approaches zero. Let us prove this for the sum S, for instance. We 
shall suppose that a<6, and that/(x) is positive in the interval (a, 6), which can 
be brought about by adding a suitable constant to/(x), which, in turn, amounts 
to adding a constant to each of the sums S. Then, since the number / is the 
lower limit of all the sums S, we can find a particular method of subdivision, say 

a, zi, x 2 , -, Xp-i, 6, 

for which the sum S is less than I + e/2, where e is a preassigned positive num 
ber. Let us now consider a division of (a, 6) into intervals less than r;, and let us 
try to find an upper limit of the corresponding sum S . Taking first those inter 
vals which do not include any of the points x lt x 2 , , Xp_i, and recalling the 
reasoning of 71, it is clear that the portion of S which comes from these inter 
vals will be less than the original sum S, that is, less than I + e/2. On the other 
hand, the number of intervals which include a point of the set Xi, x 2 , , Xp_j 
cannot exceed p 1, and hence their contribution to the sum S cannot exceed 
(p 1) Mil, where M is the upper limit of /(x). Hence 

S <I+e/2 + (p-l)Mr,, 

and we need only choose r) less than e/2 M (p - 1) in order to make S less than 
I + f. Hence the lower limit I of all the sums /S is also the limit of any sequence 
of <S s which corresponds to uniformly infinitesimal subintervals. 

It may be shown in a similar manner that the sums s have the limit / . 
If the function /(x) is any function whatever, these two limits I and 7 are in 
general different. In order that the function be integrable it is necessary and 
sufficient that 7 = I. 

74. First law of the mean for integrals. From now on we shall P 
assume, unless something is explicitly said to the contrary, that 
the functions under the integral sign are continuous. 

* The above theorem can be extended without difficulty to double and triple inte 
grals ; we shall make use of it in several places ( 80, 95, 97, 131, 144, etc.). 

The proposition is essentially only an application of a theorem of Duhamel s 
according to which the limit of a sum of infinitesimals remains unchanged when 
each of the infinitesimals is replaced by another infinitesimal which differs from the 
given infinitesimal by an infinitesimal of higher order. (See an article by W. F. 
Osgood, Annals of Mathematics, 2d series, Vol. IV, pp. 161-178 : The Integral as 
the Limit of a Sum and a Theorem of Duhamel s.) 



152 DEFINITE INTEGRALS [IV, 74 

Let f(x) and < (x) be two functions which are each continuous 
in the interval (a, b), one of which, say <(:*), has the same sign 
throughout the interval. And we shall suppose further, for the 
sake of definiteness, that a < b and <f> (x) > 0. 

Suppose the interval (a, b~) divided into subintervals, and let 
i> 2, > o be values of x which belong to each of these 
smaller intervals in order. All the quantities /(,) lie between the 
limits M and m of f(x) in the interval (a, b) : 



Let us multiply each of these inequalities by the factors 



respectively, which are all positive by hypothesis, and then add 
them together. The sum S/(&) <(&) fa z f _,) evidently lies 
between the two sums ra 2 <() fa o^) and 3/2 <() fa ,._,). 
Hence, as the number of subintervals increases indefinitely, we 
have, in the limit, 

m f $ (x} dx <, C /( $ (x} dx < M C $ (x) dx, 

Ja Jo. Ja 

which may be written 

Xb s*b 

f(x}$(x}dx = p. \ $(x)dx, 
J u. 

where /x lies between m and M. Since the function f(x) is con 
tinuous, it assumes the value /t for some value of the variable 
which lies between a and b ; and hence we may write the preceding 
equation in the form 



(3) f /(*) <(> (x) dx = /(*) C $ (x) dx, 

J a / 



where lies between a and b.* If, in particular, <(#) =1, the 
integral JH* dx reduces to (b a) by the very definition of an inte 
gral, and the formula becomes 

(4) f /(*)<& (*-a)/tf). 



* The lower sign holds in the preceding relations only when / (a-) = k. It is evident 
that the formula still holds, however, and that a< < b in any case. TRANS. 



IV, 75] ALLIED GEOMETRICAL CONCEPTS 153 

75. Second law of the mean for integrals. There is a second formula, due to 
Bonnet, which he deduced from an important lemma of Abel s. 

Lemma. Let e , ei, , e p be a set of monotonically decreasing positive quanti 
ties, and MO , MI , , Up the same number of arbitrary positive or negative quantities. 
If A and B are respectively the greatest and the least of all of the sums s = u , 
s\ = UQ + i , i p = w + MI + + u t> , the sum 

S e W + eiWi + + e p u p 
will lie between Ae Q and -Be ) i-e. Aeo > S^ Bf . 
For we have 

UQ SO, Ui=Si S , , U p = S p S p ^i, 

whence the sum S is equal to 

So (*0 i) + Si (ei e 2 ) + + Sp-i (e p -i - t p ) + S p e p . 

Since none of the differences e ei, ei e 2 , , f p ~i f p are negative, two 
limits for S are given by replacing s , i , , s p by their upper limit A and then 
by their lower limit B. In this way we find 

S < A (e ei + ei e 2 + -f ep_i e p + e v ) = At , 

and it is likewise evident that S ^ Be . 

Now let/(x) and <j> (x) be two continuous functions of x, one of which, (a;), 
is a positive monotonically decreasing function in the interval a < x < b. Then 
the integral f^f(x) <j>(x)dx is the limit of the sum 

/(a) <f> (a) (xi - a) + f(xi) <j> (xj) (x 2 - x a ) + . . . . 



The numbers 0(a), 0(xi), form a set of monotonically decreasing positive 
numbers; hence the above sum, by the lemma, lies between A<f>(a) and B<f>(a), 
where A and B are respectively the greatest and the least among the following 
sums : 

/(a) (X! - a) , 

/(a) (xi - a) +/(xi) (x 2 - xi) , 



/(a) (X! - a) +/(xj) (x 2 - Xi) + +f(x n - l ) (b - x n ^). 

Passing to the limit, it is clear that the integral in question must lie between 
Ai<f,(a) and #10 (a), where AI and BI denote the maximum and the minimum, 
respectively, of the integral f^f(x)dx, as c varies from a to b. Since this inte 
gral is evidently a continuous function of its upper limit c ( 76), we may write 
the following formula : 



(5) f /(x)0(x)cZx = 0(a) T 

J a Ja 



When the function 0(x) is a monotonically decreasing function, without 
being always positive, there exists a more general formula, due to Weierstrass. 
In such a case let us set <f> (x) = <f> (b) + \fs (x). Then f (x) is a positive monoton 
ically decreasing function. Applying the formula (5) to it, we find 

f f(x)dx. 

J a 



154 DEFINITE INTEGRALS [IV, 76 



From this it is easy to derive the formula 



C f(x)+(x)dx = C f(x)+(b)dx + [0(o) - 0(6)] fV(x) 

c/a J a Ja 

f /(x) 0(z) dx = 0(a) f %s) <*z + 0(6) f b f(x) dx . 

i/a "u 



Similar formulae exist for the case when the function 0(x) is increasing. 

76. Return to primitive functions. We are now in a position to 
give a purely analytic proof of the fundamental existence theorem 
( 67). Let/(x) be any continuous function. Then the definite integral 



where the limit a is regarded as fixed, is a function of the upper 
limit x. We proceed to show that the derivative of this function 
isf(x). In the first place, we have 

>x + A 

f(t)dt, 



= f 

Jx 



or, applying the first law of the mean (4), 



where lies between x and x + h. As h approaches zero, 
approaches /(x) ; hence the derivative of the function F(x) is /(x), 
which was to be proved. 

All other functions which have this same derivative are given 
by adding an arbitrary constant C to F(x). There is one such 
function, and only one, which assumes a preassigned value T/ O for 
x = a, namely, the function 



When there is no reason to fear ambiguity the same letter x is 
used to denote the upper limit and the variable of integration, and 
/*/"(*) d x i g written in place of f*f(t) dt. But it is evident that 
a definite integral depends only upon the limits of integration and 
the form of the function under the sign of integration. The letter 
which denotes the variable of integration is absolutely immaterial. 

Every function whose derivative is /(x) is called an indefinite 
integral of /(x), or & primitive function of /"(x), and is represented 
by the symbol 

r 

f(x)dx, 



IV, 70] ALLIED GEOMETRICAL CONCEPTS 155 

the limits not being indicated. By the above we evidently have 



Conversely, if a function F(x) whose derivative is /(x) can be 
discovered by any method whatever, we may write 

f(x)dx = F(aj)+ C. 

In order to determine the constant C we need only note that the 
left-hand side vanishes for x = a. Hence C = F(a), and the 
fundamental formula becomes 



(6) f(x)dx = F(*)-F(a). 

C/ U 

If in this formula /(a- ) be replaced by F (x), it becomes 
F(a)-F(a)*= f F (x)dx, 

*J a 

or, applying the first law of the mean for integrals, 



where lies between a and x. This constitutes a new proof of the 
law of the mean for derivatives ; but it is less general than the one 
given in section 8, for it is assumed here that the derivative F (a:) is 
continuous. 

We shall consider in the next chapter the simpler classes of func 
tions whose primitives are known. Just now we will merely state 
a few of those which are apparent at once : 

A(x a) a dx = A - (- ( , a-fl^O; 

dx 



I cos x dx = sin x -f C ; I sin x dx = cos x -f- C ; 

x dx = h C, m = 0; 



150 DEFINITE INTEGRALS [IV, 76 







= log/(x) + C. 

The proof of the fundamental formula (6) was based upon the 
assumption that the function f(x) was continuous in the closed inter 
val (a, b). If this condition be disregarded, results may be obtained 
which are paradoxical. Taking f(x) == I/a; 2 , for instance, the for 
mula (6) gives 

f"^ = i_i. 

} x 2 a I 

\J a 

The left-hand side of this equality has no meaning in our present 
system unless a and b have the same sign ; but the right-hand side 
has a perfectly determinate value, even when a and b have different 
signs. We shall find the explanation of this paradox later in the 
study of definite integrals taken between imaginary limits. 
Similarly, the formula (6) leads to the equation 



r 

1 



^ 



/(*> /() 



If /(a) and/() have opposite signs, f(x) vanishes between a and b, 
and neither side of the above equality has any meaning for us at 
present. We shall find later the signification which it is convenient 
to give them. 

Again, the formula (6) may lead to ambiguity. Thus, if 
/() = !/(!+ * 2 ), we find 

= arc tan b arc tan a. 



Here the left-hand side is perfectly determinate, while the right- 
hand side has an infinite number of determinations. To avoid this 
ambiguity, let us consider the function 



This function F(x) is continuous in the whole interval and van 
ishes with x. Let us denote by arc tan x, on the other hand, an 
angle between - Tr/2 and + Tr/2. These two functions have the 



iv, 77] ALLIED GEOMETRICAL CONCEPTS 157 

same derivative and they both vanish for x = 0. It follows that 
they are equal, and we may write the equality 

r b dx r b dx r a dx 

\ : = I ; | -. = arc tan b arc tan a, 

J m 1 + x* J 1 + x* J 1 + x* 

where the value to be assigned the arctangent always lies between 
-7T/2 and +7T/2. 

In a similar manner we may derive the formula 

>b dx 



f 

I/O 



= arc sin b arc sin a, 



where the radical is to be taken positive, where a and b each lie 
between 1 and + 1, and where arc sin x denotes an angle which 
lies between Tr/2 and + Tr/2. 

77. Indices. In general, when the primitive F(x) is multiply determinate, we 
should choose one of the initial values F(a) and follow the continuous variation 
of this branch as x varies from a to b. Let us consider, for instance, the integral 



fobzar*. f-^w- 
J. ^+<? J. i +/ (*) 



where 



and where P and Q are two functions which are both continuous in the interval 
(a, b) and which do not both vanish at the same time. If Q does not vanish 
between a and 6, /(x) does not become infinite, and arc tan/(x) remains between 
7f/2 and + rt/2. But this is no longer true, in general, if the equation Q = 
has roots in this interval. In order to see how the formula must be modified, let 
us retain the convention that arc tan signifies an angle between if/2 and + if/2, 
and let us suppose, in the first place, that Q vanishes just once between a and b 
for a value x = c. We may write the integral in the form 

b 



r 
J a 



f (x)dx 



where e and e are two very small positive numbers. Since /(x) does not become 
infinite between a and c e, nor between c + e and 6, this may again be written 

f dx 

= arc tan/(c e) - arc tan/(a) 






fC+t 

+ arc tan/ (6) - arc tan/(c + e ) + I 

Jc-e 



Several cases may now present themselves. Suppose, for the sake of definite- 
ness, that/(x) becomes infinite by passing from + oo to oo. Then/(c e) will 
be positive and very large, and arc tan /(c e) will be very near to ir/2; while 



158 DEFINITE INTEGRALS [IV, 78 

/(c + e ) will be negative and very large, and arc tan/(c + e 7 ) will be very near 
7T/2. Also, the integral J^lV wil1 be verv small in absolute value; and, 
passing to the limit, we obtain the formula 



f 



f(x)dx 

= TT + arc tan/(6) arctan/(a). 



Similarly, it is easy to show that it would be necessary to subtract n if /(x) 
passed from co to + . In the general case we would divide the interval 
(a, 6) into subintervals in such a way that /(x) would become infinite just once 
in each of them. Treating each of these subintervals in the above manner and 
adding the results obtained, we should find the formula 





*s a 



f (x) dx 

. T = arc tan/(6) - arc tan/(o) + (K - K ) x, 
i + j (x) 



where K denotes the number of times that /(x) becomes infinite by passing from 
+ oo to co, and K the number of times that /(x) passes from oo to + oo. 
The number K K is called the index of the function /(x) between a and 6. 

When/(x) reduces to a rational function Vi/V, this index may be calculated 
by elementary processes without knowing the roots of V. It is clear that we 
may suppose Vi prime to and of less degree than V, for the removal of a poly 
nomial does not affect the index. Let us then consider the series of divisions 
necessary to determine the greatest common divisor of Fand FI, the sign of the 
remainder being changed each time. First, we would divide V by FI, obtaining 
a quotient Qi and a remainder F 2 . Then we would divide FI by F 2 , obtaining a 
quotient Q 2 and a remainder Vs ; and so on. Finally we should obtain a con 
stant remainder F + 1. These operations give the following set of equations : 

F = FiQi - F 2 , 
F! = F 2 Q 2 - F 8 , 



The sequence of polynomials 

(7) F, FI, F a , -, Vr.it r r , F r + 1 , -.., F n , F n + 1 

has the essential characteristics of a Sturm sequence : 1) two consecutive poly 
nomials of the sequence cannot vanish simultaneously, for if they did, it could 
be shown successively that this value of x would cause all the other polynomials 
to vanish, in particular V n + \; 2) when one of the intermediate polynomials FI, 
Pai ,Vn vanishes, the number of changes of sign in the series (7) is not altered, 
for if F r vanishes for x = c, F r _i and V r + \ have different signs for x c. It 
follows that the number of changes of sign in the series (7) remains the same, 
except when x passes through a root of F = 0. If Fi/F passes from + oo to oo, 
this number increases by one, but it diminishes by one on the other hand if 
V\/V passes from co to +00. Hence the index is equal to the difference of 
the number of changes of sign in the series (7) for x = 6 and x = a. 

78. Area of a curve. We can now give a purely analytic definition 
of the area bounded by a continuous plane curve, the area of the 
rectangle only being considered known. For this purpose we need 



IV, 78] ALLIED GEOMETRICAL CONCEPTS 159 

only translate into geometrical language the results of 72. Let 
f(x) be a function which is continuous in the closed interval (a, b), 
and let us suppose for definiteness that a < b and that f(x) > in 
the interval. Let us consider, as above (Fig. 9, 65), the portion of 
the plane bounded by the contour AMBB A , composed of the seg 
ment A B of the x axis, the straight lines AA and BB parallel to 
the y axis, and having the abscissae a and b, and the arc of the curve 
A MB whose equation is y =f(x). Let us mark off on A B Q a certain 
number of points of division P l} P 2 , , P,_i, P i} , whose abscissae 
are x 1} x 2 , , #;_!, x i} , and through these points let us draw 
parallels to the y axis which meet the arc A MB in the points 
Qi, #2> , Qi-D 0>i> ) respectively. Let us then consider, in 
particular, the portion of the plane bounded by the contour 
Qi-iQiPiPi-iQi-i, an( i l et us m ark upon the arc Q.-^Q,- the highest 
and the lowest points, that is, the points which correspond to the 
maximum M { and to the minimum m { of f(x) in the interval 
(#,_!, a;,-). (In the figure the lowest point coincides with <2,-_j.) 
Let Rf be the area of the rectangle P i - l P i s i s i _ 1 erected upon the 
base Pi-i-P, with the altitude JJ/ f , and let r { be the area of the 
rectangle Pf-iP^Q,--! erected upon the base P,-^^ with the alti 
tude m t . Then we have 



and the results found above ( 72) may now be stated as follows : 
whatever be the points of division, there exists a fixed number / 
which is always less than 2A\ and greater than 2r,., and the two 
sums 2Ri and 2r f approach / as the number of sabintervals P i ^ 1 P i 
increases in such away that each of them approaches zero. We shall 
call this common limit I of the two sums 2-Ri and 2r { the area of 
the portion of the plane bounded by the contour AMBB A A. Thus 
the area under consideration is defined to be equal to the definite 
integral I j (<) ax, 

This definition agrees with the ordinary notion of the area of a 
plane curve. For one of the clearest points of this rather vague 
notion is that the area bounded by the contour P i -iPiQ i n i Q i _ ] P i _ l 
lies between the two areas R f and r ( of the two rectangles Pj_iP,-*,- s -i 
and Pi-iPiiiQi-i; hence the total area bounded by the contour 
AMBB A A must surely be a quantity which lies between the two 
sums 2/? f and 2r,-. But the definite integral / is the only fixed quan 
tity which always lies between these two sums for any mode of 
subdivision of A Q B , since it is the common limit of 2-R, and 2r<. 



160 



DEFINITE INTEGRALS 



[IV, 79 



where & is any value whatever in the interval (x t _ 1} a;,). 
element 



The given area may also be defined in an infinite number of other 
ways as the limit of a sum of rectangles. Thus we have seen that 
the definite integral / is also the limit of the sum 



But the 



of this sum represents the area of a rectangle whose base is P.-.jPj 
and whose altitude is the ordinate of any point of the arc Q i _ l n i Q i . 
It should be noticed also that the definite integral / represents 
the area, whatever be the position of the arc AMR with respect to 
the x axis, provided that we adopt the convention made in 67. 
Every definite integral therefore represents an area ; hence the calcu 
lation of such an integral is called a quadrature. 

The notion of area thus having been made rigorous once for all, 
there remains no reason why it should not be used in certain 
arguments which it renders nearly intuitive. For instance, it is 
perfectly clear that the area considered above lies between the areas 
of the two rectangles which have the common base A B , and which 
have the least and the greatest of the ordinates of the arc A MB, 
respectively, as their altitudes. It is therefore equal to the area of 
a rectangle whose base is A B and whose altitude is the ordinate 
of a properly chosen point upon the arc AMB, which is a restate 
ment of the first law of the mean for integrals. 

79. The following remark is also important. Let f(x) be a func 
tion which is finite in the interval (a, b) and which is discontinuous 

in the manner described below for 
a finite number of values between 
a and b. Let us suppose that /(a-) 
is continuous from c to c + &(&>0), 
and that f(c -f c ) approaches a cer 
tain limit, which we shall denote 
f( c + 0), as e approaches zero 
through positive values ; and like 
wise let us suppose that f(x) is 
continuous between c k and c and that/(c - c) approaches a limit 
f(c - 0) as e approaches zero through positive values. If the two 
limits f(c + 0) and f(c - 0) are different, the function f(x) is dis 
continuous for x = c. It is usually agreed to take for /(c) the 



n 




FIG. 11 



IV, 80] ALLIED GEOMETRICAL CONCEPTS 161 

value [f(c 4- 0) +f(c 0)]/2. If the function /(a;) has a certain 
number of points of discontinuity of this kind, it will be repre 
sented graphically by several distinct arcs AC, C D, D B. Let c 
and d, for example, be the abscissae of the points of discontinuity. 
Then we shall write 

Xb /~> c s*d s*b 

f(x)dx = I f(x)dx + I f(x)dx + I f(x)dx, 
i/a *J c i/a* 

in accordance with the definitions of 72. Geometrically, this definite 
integral represents the area bounded by the contour A CC DD BB A A. 
If the upper limit b now be replaced by the variable x, the definite 
integral 



is still a continuous function of x. In a point x where f(x) is con 
tinuous we still have F (x )=f(x ). For a point of discontinuity, 
x = c for example, we shall have 

S>C+fl 

F(c + K) - F(c) = I f(x) dx = hf(c + BK), < $ < 1, 

and the ratio \_F(c + A) F(c)]/h approaches f(c + 0) or f(c 0) 
according as h is positive or negative. This is an example of a 
function F(x) whose derivative has two distinct values for certain 
values of the variable. 

80. Length of a curvilinear arc. Given a curvilinear arc AB; let us 
take a certain number of intermediate points on this arc, m 1 , m 2) 
, m n -\, and let us construct the broken line A?n 1 m 2 m n _ l B by 
connecting each pair of consecutive points by a straight line. 

If the length of the perimeter of this broken line approaches a 
limit as the number of sides increases in such a way that each of 
them approaches zero, this limit is defined to be the length of the 
arc AB. 

Let 



be the rectangular coordinates of a point of the arc AB expressed 
in terms of a parameter t, and let us suppose that as t varies from 
a to b (a < b ) the functions /, <, and \j/ are continuous and possess 
continuous first derivatives, and that the point (x, y, z) describes 
the arc AB without changing the sense of its motion. Let 



DEFINITE INTEGRALS [IV, 80 

be the values of t which correspond to the vertices of the broken 
line. Then the side c t is given by the formula 



or, applying the law of the mean to x { #,_!, -, 



where ,., 77,, & lie between ,._! and t t . When the interval (,._!, /,) 
is very small the radical differs very little from the expression 



In order to estimate the error we may write it in the form 

A .-- .)] + 



But we have 

!/ () + 1 / (**- ! 

and consequently 



Hence, if each of the intervals be made so small that the oscillation 
of each of the functions / (*), <f> (t), ^ (*) is less than c/3 in any 
interval, we shall have 



where 

M<J 

and the perimeter of the broken line is therefore equal to 



The supplementary term 2e,-(,- ,-_,) is less in absolute value 
than e2(fc #,- ._,), that is, than c(/> a). Since e may be taken as 
small as we please, provided that the intervals be taken sufficiently 
small, it follows that this term approaches zero ; hence the length S 
of the arc AB is equal to the definite integral 



(8) s=C 

c/ (l 



This definition may be extended to the case where the derivatives 
/ , <f> , $ are discontinuous in a finite number of points of the arc AB, 



IV, 80] ALLIED GEOMETRICAL CONCEPTS 163 

which occurs when the curve has one or more corners. We need only 
divide the arc AB into several parts for each of which/ , < , ^ are 
continuous. 

It results from the formula (8) that the length S of the arc 
between a fixed point A and a variable point M, which corresponds 
to a value t of the parameter, is a function of t whose derivative is 



whence, squaring and multiplying by dt 2 , we find the formula 
(9) dS 2 = dx 2 + dy* + dz 2 , 

which does not involve the independent variable. It is also easily 
remembered from its geometrical meaning, for it means that dS is 
the diagonal of a rectangular parallelepiped whose adjacent edges are 
dx, dy, dz. 

Note. Applying the first law of the mean for integrals to the 
definite integral which represents the arc M M 1} whose extremities 
correspond to the values t , ^ of the parameter (^ > * ), we find 

s = arc JUoJ/i = (t, - t ) V/ 2 (0) + 4, 2 (0) + ,/,"(0), 

where lies in the interval (t , ^). On the other hand, denoting 
the chord M n M^ by c, we have 



c2 = [/CO -/Co)] 2 

Applying the law of the mean for derivatives to each of the differ 
ences f(ti)f(t n ), , we obtain the formula 



where the three numbers , rj, belong to the interval ( , ^). By 
the above calculation the difference of the two radicals is less than e, 
provided that the oscillation of each of the functions/ ^), < (), 
is less than e/3 in the interval (# , ^). Consequently we have 



or, finally, 



1- 

s 



If the arc M Q M l is infinitesimal, t l t approaches zero; hence c, 
and therefore also 1 c/s, approaches zero. It follows that the ratio 
of an infinitesimal arc to its chord approaches unity as its limit. 



164 DEFINITE INTEGRALS [IV, 81 

Example. Let us find the length of an arc of a plane curve whose 
equation in polar coordinates is p = /(w). Taking <o as independent 
variable, the curve is represented by the three equations x = p cos w, 
y = p sin o>, z = ; hence 

ds 2 = dx 2 + dy 2 = (cos to dp p sin o> e?a>) 2 + (sin wdp -}- p cos to c?w) 2 , 

or, simplifying, 

ds 2 = dp 2 + P 2 d^ 2 . 

Let us consider, for instance, the cardioid, whose equation is 

p = R + R cos a). 
By the preceding formula we have 

ds 2 = R 2 dta 2 [sin 2 w + (1 + cos o>) 2 ] = 4 R 2 cos 2 ^ do 2 , 
or, letting o> vary from to TT only, 

ds = 2 R cos f/o) ; 

U 

and the length of the arc is 

f 
&R sin 



where w and u^ are the polar angles which correspond to the extrem 
ities of the arc. The total length of the curve is therefore 8 R. 

81. Direction cosines. In studying the properties of a curve we are 
often led to take the arc itself as the independent variable. Let us 
choose a certain sense along the curve as positive, and denote by s 
the length of the arc AM between a certain fixed point A and a vari 
able point M, the sign being taken + or according as M lies in 
the positive or in the negative direction from A. At any point M 
of the curve let us take the direction of the tangent which coincides 
with the direction in which the arc is increasing, and let a, ft, y be 
the angles which this direction makes with the positive directions 
of the three rectangular axes Ox, Oy, OK. Then we shall have the 
following relations : 

COS a _ COS ft _ COS y 1 1 

dx dy dz - Vrfz 2 + dif + dz 2 ~ ds 

To find which sign to take, suppose that the positive direction of 
the tangent makes an acute angle with the x axis ; then x and s 
increase simultaneously, and the sign + should be taken. If the 
angle a is obtuse, cos a is negative, x decreases as s increases, dx/ds 



IV, 82] ALLIED GEOMETRICAL CONCEPTS 165 

is negative, and the sign -f- should be taken again. Hence in any 
case the following formulae hold : 

dx dy dz 

(10) cos a = > cos B = ~r cos y = - > 
as as ds 

where dx, dy, dz, ds are differentials taken with respect to the same 
independent variable, which is otherwise arbitrary. 

82. Variation of a segment of a straight line. Let MM^ be a segment 
of a straight line whose extremities describe two curves C, C x . On 
each of the two curves let us choose a 
point as origin and a positive sense of 
motion, and let us adopt the follow 
ing notation : s, the arc AM; s 1} the arc 
A l M l , the two arcs being taken with 
the same sign ; I, the length M M l ; B, the 
angle between M M^ and the positive di 
rection of the tangent MT; 1} the angle 

between AT, M and the positive direction 

FIG 12 
of the tangent M^ 7\. We proceed to 

try to find a relation between $, 6 1 and the differentials ds, ds l} dl. 
Let (x, y, z), (x 1} y lt z^ be the coordinates of the points M, M ly 
respectively, a, ft, y the direction angles of MT, and a^, fa, y l the 
direction angles of M^ 7\. Then we have 

P = (x - arO" + (y - y^ + (z - ztf, 
from which we may derive the formula 
ldl = (x- x^ (dx dxj + (y - y,) (dy - dyj + (z - z^ (dz dzj, 

which, by means of the formulae (10) and the analogous formulae 
for C lf may be written in the form 




_ 

ds 



Ix x, 11 ?/, z z, \ 

dl = I - - C03 a + J Jl COS (3 H -- j 1 COS y I 

\ V C l> / 

/x, x ?/, 11 z, z 

+ I - L ^ cos <*! -f >/i cos ft + -*-j cos 

\ 6 (/ l> 

But (a; x^)/l, (y y\) fl, (z z-^/l are the direction cosines of 
M M, and consequently the coefficient of ds is cos 0. Likewise 
the coefficient of ds l is cos ^; hence the desired relation is 

(10 ) dl = ds cos 6 ds! cos 0,. 

We shall make frequent applications of this formula ; one such we 
proceed to discuss immediately. 




166 DEFINITE INTEGRALS [IV, 83 

83. Theorems of Graves and of Chasles. Let E and E be two confocal ellipses, 
and let the two tangents MA, MB to the interior ellipse E be drawn from a point 

M, which lies on the exterior ellipse E . The 
difference MA + MB arc ANB remains con 
stant as the point M describes the ellipse E . 

Let and s denote the arcs OA and OB, 
<r the arc O M, I and I the distances AM and 
BM, 6 the angle between MB and the positive 
direction of the tangent M T. Since the ellipses 
are confocal the angle between MA and M T is 
FlQ 13 equal to it 6. Noting that AM coincides 

with the positive direction of the tangent at A, 
and that BM is the negative direction of the tangent at B, we find from the 
formula (10 ), successively, 

dl = ds + d<r cos 6 , 
dl ds do- cos 6 
whence, adding, 

d(l + l )=d (s 1 -s)=d (arc ANB), 

which proves the proposition stated above. 

The above theorem is due to an English geometrician, Graves. The following 
theorem, discovered by Chasles, may be proved in a similar manner. Given an 
ellipse and a confocal hyperbola which meets it at N. If from a point M on that 
branch of the hyperbola which passes through N the two tangents MA and MB 
be drawn to the ellipse, the difference of the arcs NA NB will be equal to the 
difference of the tangents MA MB. 

III. CHANGE OF VARIABLE INTEGRATION BY PARTS 

A large number of definite integrals which cannot be evaluated 
directly yield to the two general processes which we shall discuss 
in this section. 

84. Change of variable. If in the definite integral /*/(*) dx the 

variable x be replaced by a new independent variable t by means 
of the substitution x = <f>(t), a new definite integral is obtained. 
Let us suppose that the function <f>(t) is continuous and possesses a 
continuous derivative between a and ft, and that <f>(f) proceeds from 
a to b without changing sense as t goes from a to ft. 

The interval (a, ft) having been broken up into subintervals by 
the intermediate values a, t v , t, , t n _ l , ft, let a, x l} x z , , x n _ l} b 
be the corresponding values of x == <f>(t). Then, by the law of the 
mean, we shall have 



where B t lies between t i _ l and ?,-. Let ,- <(0,-) be the corresponding 
value of x which lies between x i _ l and x ( . Then the sum 



IV. 5*1 CHANGE OF VARIABLE 167 



(x, - a) + /(&) (x, - * t ) + +/(,) (6 - x n _ 

approaches the given definite integral as its limit. But this sum 
may also be written 



and in this form we see that it approaches the new definite integral 

C 

Ja 

as its limit. This establishes the equality 

C b 

(\Y\ I f(x~\dx 

\- L - L J t ,/ v v " 

v a ^ 



which is called the formula for the change of variable. It is to 
be observed that the new differential under the sign of integration 
is obtained by replacing x and dx in the differential f(x}dx by their 
values <f>(t) and <}> (t)dt, while the new limits of integration are the 
values of t which correspond to the old limits. By a suitable choice 
of the function <() the new integral may turn out to be easier to 
evaluate than the old, but it is impossible to lay down any definite 
rules in the matter. 

Let us take the definite integral 



/ 

Jo 



dx 



(x - a)* + p* 

for instance, and let us make the substitution x = a -f- fit. It 
becomes 

dx 1 r dt I a 

tan * + arc tan 




or, returning to the variable x, 

1 / x a a 

- ; l arc tan - -- \- arc tan - 

Xot all the hypotheses made in establishing the formula (11) were 
necessary. Thus it is not necessary that the function <() should 
always move in the same sense as t varies from a to f3. For defi- 
niteness let us suppose that as t increases from a to y (y < /8), <() 
steadily increases from a to c (c > i) ; then as t increases from y to 
/3, <() decreases from c to I. If the f unction /(x) is continuous in 
the interval (a, c), the formula may be applied to each of the inter 
vals (a, c), (c, b), which gives 



168 DEFINITE INTEGRALS L*v,$84 



or, adding, 



On the other hand, it is quite necessary that the function <f>(t) 
should be uniquely denned for all values of t. If this condition be 
disregarded, fallacies may arise. For instance, if the formula be 
applied to the integral f_ l dx, using the transformation x = 1? /2 , 
we should be led to write 



/ + i r l 3 

*-J j 



which is evidently incorrect, since the second integral vanishes. In 
order to apply the formula correctly we must divide the interval 
( 1, -f- 1) into the two intervals ( 1, 0), (0, 1). In the first of 
these we should take x = Vr and let t vary from 1 to 0. In the 
second half interval we should take x = ~\/t s and let t vary from 
to 1. We then find a correct result, namely 



X + i <~i 

dx = 3 I 
1 t/O 



Note. If the upper limits b and ft be replaced by x and t in the 
formula (11), it becomes 



which shows that the transformation x = <() carries a function 
F(x), whose derivative is /(#), into a function <() whose derivative 
is /[<()]< (). This also follows at once from the formula for the 
derivative of a function of a function. Hence we may write, in 
general, 



which is the formula for the change of variable in indefinite 
integrals. 



ivr, 85] INTEGRATION BY PARTS 169 

85. Integration by parts. Let u and v be two functions which, 
together with their derivatives u and v , are continuous between a 
and b. Then we have 

d(uv) _ dv du 
dx dx dx 

whence, integrating both sides of this equation, we find 

C dhm) C b dv C b du 

\ dx = I u dx+ \ v ----- dx. 
J a d x J a dx J a dx 

This may be written in the form 

f*b /*b 

(12) / u dv = \_uv~\l - I v du, 

\J a *J a 

where the symbol [F(x)] denotes, in general, the difference 



If we replace the limit b by a variable limit x, but keep the limit a 
constant, which amounts to passing from definite to indefinite inte 
grals, this formula becomes 



(13) | u dv = uv I v du. 



Thus the calculation of the integral / u dv is reduced to the cal 
culation of the integral fvdu, which may be easier. Let us try, 
for example, to calculate the definite integral 

r 

I x m logxdx, ra + 1^0. 

\J ct 

Setting u = logic, v x m + l /(m + 1), the formula (12) gives 

c\ n^+ iogarr i r b 

logx.x m dx=\ I x m dx 

Ja L m + 1 J in + 1 J a 

" 1 + 1 log X X m + * 1 6 



_ 

m+1 ~(m+l) 2 a 

This formula is not applicable if m + 1 = ; in that particular 
case we have 



It is possible to generalize the formula (12). Let the succes 
sive derivatives of the two functions u and v be represented by 
u , M", -.., w ( " + 1 >; v , v", -, v (n + 1 \ Then the application of the 



170 DEFINITE INTEGRALS [IV, 85 

formula (12) to the integrals fudv, fu dv<*-", leads to the 
following equations : 



/>6 s*b 

I uv (n + 1) dx= I udv^ = [>w (n) ]* 

Ja Ja 

s*\> r>t> f b 

I u v^dx =1 M ^ ( -- = |> r<" - I*- I wV 

Ja Ja J* 

~b /-.b 

/ u<*>v dx =1 u^do =[ ( "]a 

Ja Ja. 

Multiplying these equations through by + 1 and 1 alternately, 
and then adding, we find the formula 




+ l C (n 
Ja 

which reduces the calculation of the integral fuv^ n+l) dx to the cal 
culation of the integral f*i< H + l) vdx. 

In particular this formula applies when the function under the 
integral sign is the product of a polynomial of at most the wth 
degree and the derivative of order (n + 1) of a known function v. 
For then w (M + 1) = 0, and the second member contains no integral 
signs. Suppose, for instance, that we wished to evaluate the definite 
integral 

fW(*)*, 

\J a 

where /(x) is a polynomial of degree n. Setting u =/(z), v = e wi /u) n+v , 
the formula (14) takes the following form after e" x has been taken 
out as a factor : 



The same method, or, what amounts to the same thing, a series of 
integrations by parts, enables us to evaluate the definite integrals 

I c,osmxf(x)dx, I sinmxf(x)dx, 

J a */m 

where f(x) is a polynomial. 



IV, 86] INTEGRATION BY PARTS 171 

86. Taylor s series with a remainder. In the formula (14) let us 
replace u by a function F(x) which, together with its first n + 1 
derivatives, is continuous between a and b, and let us set v = (b x) n . 
Then we have 

v = n(b a;)"- 1 , v" = n(n - l)(b -re)"- 2 , -., 
v<> = (-!)!. 2 .--W, y ( " + 1 > = 0, 

and, noticing that v, v , v", , i/"- 1 ) vanish for a: = b, we obtain the 
following equation from the general formula : 

= (-!) n\F(b) n\F(a) -n\F (a)(b -a) 

ri\ ~] 

-jF t (a) (b -a) 2 ---- F<> (a) (b - a) 



] 



which leads to the equation 



n, a y 
- - ^ 



i r b 

-7 / ^ i + I )(a-)(&-a;) n ^. 
?i !i/ a 



Since the factor (i x) n keeps the same sign as x varies from a to 
b, we may apply the law of the mean to the integral on the right, 
which gives 

I F + l \x)(b - x)dx = F< + 1 (f) f (b- x} n dx 

Ja Ja 



where lies between a and b. Substituting this value in the preced 
ing equation, we find again exactly Taylor s formula, with Lagrange s 
form of the remainder. 

87. Transcendental character of e . From the formula (15) we can prove a 
famous theorem due to Hermite : The number e is not a root of any algebraic 
equation whose coefficients are all integers.* 

Setting a = and w = 1 in the formula (15), it becomes 



JT - 



* The present proof is due to D. Hilbert, who drew his inspiration from the method 
used by Hermite. 



172 DEFINITE INTEGRALS [IV, 87 

where 

F(x) =/(x) +/ (z) 

and this again may be written in the form 
(16) F(b) = 

Now let us suppose that e were the root of an algebraic equation whose coeffi 
cients are all integers : 

c + c\e + c 2 e 2 + ---- 1- c m e m = 0. 

Then, setting b = 0, 1, 2, , wi, successively, in the formula (16), and adding 
the results obtained, after multiplying them respectively by c , c l5 , c m , we 
obtain the equation 



(17) 



e ~* dx = 



where the index i takes on only the integral values 0, 1, 2, , m. We proceed 
to show that such a relation is impossible if the polynomial /(x), which is up to 
the present arbitrary, be properly chosen. 
Let us choose it as follows : 

/( X ) - - 1 - XP~ I (X - l)p(x-2)P---(x -m)P, 

(P - I) 1 - 

where p is a prime number greater than m. This polynomial is of degree 
m p -|- p i ? and all of the coefficients of its successive derivatives past the pth 
are integral multiples of p, since the product of p successive integers is divisible 
by p!. Moreover /(x), together with its first (p - 1) derivatives, vanishes for 
x = 1, 2, -, m, and it follows that F(l), F(2), , F(m) are all integral mul 
tiples of p. It only remains to calculate F(0), that is, 



=/(0) 

In the first place, /(O) =/(0) = = /O- 2 >(0) = 0, while /Cp>(0), /^ + 1 >(0), 
are all integral multiples of p, as we have just shown. To find /</ - 1) (0) we need 
only multiply the coefficient of XP~ I in/(x) by (p - 1) !, which gives (1 . 2 m)p. 
Hence the sum 

c F(0) 



is equal to an integral multiple of p increased by 

i- c (l . 2 m)p. 

If p be taken greater than either m or c , the above number cannot be divisible 
by p ; hence the first portion of the sum (17) will be an integer different from zero. 
We shall now show that the sum 



can be made smaller than any preassigned quantity by taking p sufficiently large. 
As x varies from to i each factor of /(x) is less than m ; hence we have 



IV, 88] INTEGRATION BY PARTS 173 



u 



f(x)e-*dx 



. m mp+ P -\ | e~ x dx< 






(p-1)! Jo (P-1)I 

from which it follows that 



2<f/(x)< 

/o 



where M is an upper limit of | c + | Ci | + + | c m | . As p increases indefi 
nitely the function 0(p) approaches zero, for it is the general term of a conver 
gent series in which the ratio of one term to the preceding approaches zero. It 
follows that we can find a prime number p so large that the equation (17) is 
impossible ; hence Hermite s theorem is proved. 

88. Legendre s polynomials. Let us consider the integral 



where P n (x) is a polynomial of degree n and Q is a polynomial of degree less 
than n, and let us try to determine P n (x) in such a way that the integral van 
ishes for any polynomial Q. We may consider P n (x) as the nth derivative of a 
polynomial R of degree 2n, and this polynomial R is not completely determined, 
for we may add to it an arbitrary polynomial of degree (n 1) without changing 
its nth derivative. We may therefore set P n = d n R/dx n , where the polynomial E, 
together with its first (n 1) derivatives, vanishes for x = a. But integrating 
by parts we find 



rQ dnE dx- 
J a Q ^ - 



and since, by hypothesis, 

E(o)=0, B (a) = 0, -, B(-J)(o)=0, 
the expression 

Q (6) R( - 1) (6) - Q (b) fi( - 2) (6) + Qf.* ~ (b) R (b) 

must also vanish if the integral is to vanish. 

Since the polynomial Q of degree n 1 is to be arbitrary, the quantities 
Q(&)i Q (&)> > Qf- n ~ l )(b) are themselves arbitrary; hence we must also have 

B(6) = 0, R (b) = 0, -, E<-i)(6) = 0. 

The polynomial R (x) is therefore equal, save for a constant factor, to the product 
(x - a) n (x - b) n ; and the required polynomial P n (x) is completely determined, 
save for a constant factor, in the form 



If the limits a and 6 are 1 and + 1, respectively, the polynomials P n are 
Legendre s polynomials. Choosing the constant C with Legendre, we will set 

(18) X n = - - - [(x 2 - !)]. 

2.4.6...2nax LV 



174 DEFINITE INTEGRALS [IV, 88 

If we also agree to set X 1, we shall have 

y i y- r 3x *~ l r 5x3 -3x 

-AO = 1, -<M = x, JL Z = X s = > 

"2i 2i 

In general, X n is a polynomial of degree n, all the exponents of x being even or 
odd with n. Leibniz formula for the nth derivative of a product of two factors 
( 17) gives at once the formulae 

(19) -Z(l) = l, -T.(-l) = (-!)" 

By the general property established above, 



(20) C + X, t <t>(x)dx = 0, 

/ i 



where tf> (x) is any polynomial of degree less than n. In particular, if m and n 
are two different integers, we shall always have 



(21) C + 

J-i 



This formula enables us to establish a very simple recurrent formula between 
three successive polynomials X n . Observing that any polynomial of degree n 
can be written as a linear function of X , Xi, , X n , it is clear that we may set 



where C , Ci, C 2 , are constants. In order to find C 3 , for example, let us 
multiply both sides of this equation by ^T n _ 2 , and then integrate between the 
limits 1 and + 1. By virtue of (20) and (21), all that remains is 

C + 2 
3 J_1 "" "~ 2 

and hence C 3 = 0. It may be shown in the same manner that C = 0, 5 = 0, . 
The coefficient Ci is zero also, since the product xX n does not contain x". Finally, 
to find Co and C 2 we need only equate the coefficients of x n + 1 and then equate 
the two sides for x = 1. Doing this, we obtain the recurrent formula 

(22) (n + l)X n + l - (2n + l)xX n + nX,,_i = 0, 

which affords a simple means of calculating the polynomials X n successively. 
The relation (22) shows that the sequence of polynomials 

/oo\ ~TT ~V ~Y~ ~y 

\&) -^-Oi -**-!} -"-2i " i -"-n 

possesses the properties of a Sturm sequence. As x varies continuously from 1 
to + 1, the number of changes of sign in this sequence is unaltered except when 
x passes through a root of X n = 0. But the formulse (19) show that there are n 
changes of sign in the sequence (23) f or x = 1, and none for x = 1. Hence 
the equation X n = has n real roots between 1 and -f 1, which also readily 
follows from Rolle s theorem. 



IV, 89] IMPROPER AND LINE INTEGRALS 175 

IV. GENERALIZATIONS OF THE IDEA OF AN INTEGRAL 
IMPROPER INTEGRALS LINE INTEGRALS* 

89. The integrand becomes infinite. Up to the present we have sup 
posed that the integrand remained finite between the limits of inte 
gration. In certain cases, however, the definition may be extended 
to functions which become infinite between the limits. Let us first 
consider the following particular case : f(x) is continuous for every 
value of x which lies between a and b, and for x = b, but it becomes 
infinite for x = a. We will suppose for definiteness that a < b. 
Then the integral of f(x) taken between the limits a + e and 
b (e > 0) has a definite value, no matter how small e be taken. If 
this integral approaches a limit as e approaches zero, it is usual and 
natural to denote that limit by the symbol 



Jf 



/(*) dx. 

If a primitive of /(cc), say F(x~), be known, we may write 



C 

Ja + 



and it is sufficient to examine F(a -f- e) for convergence toward a 
limit as c approaches zero. We have, for example, 

Mdx 



r 

lL 



If fj. > 1, the term l/c^" 1 increases indefinitely as e approaches zero. 
But if /u, is less than unity, we may write l/e?~ l = e 1 " 1 , and it is 
clear that this term approaches zero with c. Hence in this case 
the definite integral approaches a limit, and we may write 



C" Mdx 

I Tr ^~ 

Ja ( X a ) 

If fi = 1, we have; 



/: 

/a4 



M dx (b a 

= M log 



and the right-hand side increases indefinitely when e approaches zero. 
To sum up, the necessary and sufficient condition that the given inte 
gral should approach a limit is that /x should be less than unity. 

*It is possible, if desired, to read the next chapter before reading the closing sec 
tions of this chapter. 



176 DEFINITE INTEGRALS [IV, 89 

The straight line x = a is an asymptote of the curve whose equa 
tion is 

U 



if p. is positive. It follows from the above that the area bounded by 
the x axis, the fixed line x = b, the curve, and its asymptote, has a 
finite value provided that //,<!. 

If a primitive of f(x) is not known, we may compare the given 
integral with known integrals. The above integral is usually taken 
as a comparison integral, which leads to certain practical rules which 
are sufficient in many cases. In the first place, the upper limit b 
does not enter into the reasoning, since everything depends upon the 
manner in which f(x} becomes infinite for x = a. We may therefore 
replace b by any number whatever between a and b, which amounts 
to writing f* +f = f a +( + f* . In particular, unless f(x) has an infi 
nite number of roots near x = a, we may suppose that f(x) keeps 
the same sign between a and c. 

We will first prove the following lemma : 

Let $(x) be a function which is positive in the interval (a, b), 
and suppose that the integral f* ^ <f> (x) dx approaches a limit as e 
approaches zero. Then, if \f(x) \<<j>(x) throughout the whole inter 
val, the definite integral f a + t f(x)dx also approaches a limit. 

Hf( x ) is positive throughout the interval (a, b), the demonstration 
is immediate. For, since f(x) is less than </> (x), we have 

/ f(x)dx < I ^(x}dx. 

*J a + e J a + e 

Moreover f* +f f(x)dx increases as c diminishes, since all of its ele 
ments are positive. But the above inequality shows that it is con 
stantly less than the second integral ; hence it also approaches a 
limit. If f(x} were always negative between a and b, it would 
be necessary merely to change the sign of each element. Finally, 
if the function f(x) has an infinite number of roots near x = a, we 
may write down the equation 

f f(x) dx = f [/(*) + | /(or) \\dx-f \f(x) | dx. 

*Ja + f c/a + e <Ja + s 

The second integral on the right approaches a limit, since 
|/(*)| <$(*). Now the function f(x) + \f(x)\ is either positive 



IV, 89] IMPROPER AND LINE INTEGRALS 177 



or zero between a and b, and its value cannot exceed 2 <(#); hence 
the integral 



f 

i/a + 



also approaches a limit, and the lemma is proved. 

It follows from the above that if a function f(x) does not approach 
any limit whatever for x = a, but always remains less than a fixed 
number, the integral approaches a limit. Thus the integral 
f Q l sin(l/x)dx has a perfectly definite value. 

Practical rule. Suppose that the function /(#) can be written in 
the form 



where the function ij/(x) remains finite when x approaches a. 

If fjL < 1 and the function \]/ (x~) remains less in absolute value than 
a fixed number M, the integral approaches a limit. But if /JL ^ 1 and 
the absolute value of ty(x) is greater than a positive number ra, the 
integral approaches no limit. 

The first part of the theorem is very easy to prove, for the abso 
lute value of f(x) is less than M/(x aY, and the integral of the 
latter function approaches a limit, since p, < 1. 

In order to prove the second part, let us first observe that ^(a*) 
keeps the same sign near x = a, since its absolute value always 
exceeds a positive number m. We shall suppose that \(/ (x) > 
between a and b. Then we may write 



X 
4- 



m dx 



and the second integral increases indefinitely as e decreases. 

These rules are sufficient for all cases in which we can find an 
exponent p such that the product (x ^/(.i") approaches, for 
x = a, a limit A different from zero. If /* is less than unity, the 
limit b may be taken so near a that the inequality 



holds inside the interval (a, J), where L is a positive number greater 



178 DEFINITE INTEGRALS [IV, 89 

than I K \. Hence the integral approaches a limit. On the other hand, 
if p ^ 1, b may be taken so near to a that 



I./ V"V I " /, r n \t>. 

\ c a ) 

inside the interval (a, >), where I is a positive number less than | K\. 
Moreover the function f(x), being continuous, keeps the same sign ; 
hence the integral f b f(x)dx increases indefinitely in absolute 
value.* 

Examples. Let/(x) = P/Q be a rational function. If a is a root 
of order m of the denominator, the product (x a) m /(x) approaches 
a limit different from zero for x = a. Since m is at least equal to 
unity, it is clear that the integral f f(x)dx increases beyond all 
limit as e approaches zero. But if we consider the function 

/(*) = 

where P and R are two polynomials and -R(ar) is prime to its deriv 
ative, the product (x a) 1/2 /(z) approaches a limit for x = a if a 
is a root of R(x), and the integral itself approaches a limit. Thus 
the integral 

dx 



f 

J -\ 



approaches 7r/2 as e approaches zero. 

Again, consider the integral f f l \ogxdx. The product # 1/2 loga; 
has the limit zero. Starting with a sufficiently small value of x, we 
may therefore write log x < Mx~ 1/2 , where M is a positive number 
chosen at random. Hence the integral approaches a limit. 

Everything which has been stated for the lower limit a may be 
repeated without modification for the upper limit b. If the function 
f (cc) is infinite for x b,we would define the integral J a /(#) dx to be 
the limit of the integral /J*" / ( x ) dx as c approaches zero. If /(#) 
is infinite at each limit, we would define f f (x~) dx as the limit of 
the integral C b " e f(x)dx as c and e both approach zero independ 
ently of each other. Let c be any number between a and b. Then 
we may write 

*The first part of the proposition may also be stated as follows: the integral has 
a limit if an exponent /JL can be found (0 < /* < 1) such that the product (x a)>*f(x) 
approaches a limit A as a; approaches a, the case where A = not being excluded. 



IV, 90] IMPROPER AND LINE INTEGRALS 179 



r /(*)<& . f f(x)dx + c 

U a + f <J a- + f *J c 



and each of the integrals on the right should approach a limit in 
this case. 

Finally, if f(x) becomes infinite for a value c between a and b, 
we would define the integral */(%) dx as the sum of the limits of 
the two integrals f~ e f(x)dx, f c b +f f(x)dx, and we would proceed 
in a similar manner if any number of discontinuities whatever lay 
between a and b. 

It should be noted that the fundamental formula (6), which was 
established under the assumption that f(x) was continuous between 
a and b, still holds when f(x) becomes infinite between these limits, 
provided that the primitive function F(x~) remains continuous. For 
the sake of definiteness let us suppose that the f unction f(x) becomes 
infinite for just one value c between a and b. Then we have 

I f(x) dx = lim I f(x) dx + lim I f(x) dx ; 

t) a e =0t/a e = 0c/c + e 

and if F(x") is a primitive of /(x), this may be written as follows : 

Xft 
f(x) dx = lim F(c - c ) - F(a) + F(b) - lim F(c + e). 
=0 f=0 

Since the function F(x) is supposed continuous for x = c, F(c + e) 
and F(c e ) have the same limit F(c), and the formula again 
becomes 



f 

I j (xj ix 

*J a 

The following example is illustrative : 

J + 1 dx 
-J- 
i x 



If the primitive function F(x) itself becomes infinite between a and 
b, the formula ceases to hold, for the integral on the left has as yet 
no meaning in that case. 

The formulae for change of variable and for integration by parts 
may be extended to the new kinds of integrals in a similar manner 
by considering them as the limits of ordinary integrals. 

90. Infinite limits of integration. Let/(x) be a function of x which 
is continuous for all values of x greater than a certain number . 
Then the integral f l f(x) dx, where I > a, has a definite value, no 



180 DEFINITE INTEGRALS [IV, 90 

matter how large I be taken. If this integral approaches a limit 
as I increases indefinitely, that limit is represented by the symbol 



f 



f(x)dx. 



If a primitive of f(x) be known, it is easy to decide whether the 
integral approaches a limit. For instance, in the example 

dx 



f 

Jo 



= arc tan I 



the right-hand side approaches Tr/2 as I increases indefinitely, and 
this is expressed by writing the equation 



/ 

Ja 



7T 
2 



Likewise, if a is positive and //, 1 is different from zero, we have 
kdx k / 1 1 



I 

c/a 



If /A is greater than unity, the right-hand side approaches a limit as 
I increases indefinitely, and we may write 

kdx k 



On the other hand, if /i is less than one, the integral increases indefi 
nitely with I. The same is true for p. = 1, for the integral then 
results in a logarithm. 

When no primitive of /(#) is known, we again proceed by com 
parison, noting that the lower limit a may be taken as large as we 
please. Our work will be based upon the following lemma : 

Let <f> (x) be a function which is positive for x > a, and suppose that 
the integral JJ <f> (x) dx approaches a limit. Then the integral f l f(x) dx 
also approaches a limit provided that \f(x) \ ^ < () for all values of 
x greater than a. 

The proof of this proposition is exactly similar to that given above. 
If the function f(x) can be put into the form 

/() = *< 



where the function ty(x) remains finite when x is infinite, the follow 
ing theorems can be demonstrated, but we shall merely state them 



IV, 91] IMPROPER AND LINE INTEGRALS 181 

If the absolute value of ^ (x~) is less than a fixed number M and 
p. is greater than unity, the integral approaches a limit. 

If the absolute value of (]/ (x) is greater than a positive number m 
and p. is less than or equal to unity, the integral approaches no limit. 

For instance, the integral 

/ 
cos ax 

iT^ dx 

approaches a limit, for the integrand may be written 
cos ax 1 cos ax 

i+*-**rT| 

and the coefficient of 1/x 2 is less than unity in absolute value. 

The above rule is sufficient whenever we can find a positive num 
ber p, for which the product x*f(x) approaches a limit different from 
zero as x becomes infinite. The integral approaches a limit if p, is 
greater than unity, but it approaches no limit if p. is less than or 
equal to unity.* 

For example, the necessary and sufficient condition that the inte 
gral of a rational fraction approach a limit when the upper limit 
increases indefinitely is that the degree of the denominator should 
exceed that of the numerator by at least two units. Finally, if we 
take 



where P and R are two polynomials of degree p and r, respectively, 
the product x r/2 ~"f(x) approaches a limit different from zero when 
x becomes infinite. The necessary and sufficient condition that the 
integral approach a limit is that p be less than r/2 1. 

91. The rules stated above are not always sufficient for determin 
ing whether or not an integral approaches a limit. In the example 
f(x) = (sin x)/x } for instance, the product x*f(x) approaches zero if 
p. is less than one, and can take on values greater than any given 
number if p, is greater than one. If p. = 1, it oscillates between + 1 
and 1. None of the above rules apply, but the integral does ap 
proach a limit. Let us consider the slightly more general integral 

* The integral also approaches a limit if the product x^f(x) (where M> 1) approaches 
zero as x becomes infinite. 



182 DEFINITE INTEGRALS [IV, 91 



/i 
-ax S1 
e-~- 



sin a; , 

dx, a>0. 



The integrand changes sign for x = kir. We are therefore led to 
study the alternating series 

(24) - i + 2 - a s + ... + (_ !) + . . ., 

where the notation used is the following: 



/ I 

/ 

a = I 
Jo 



27T 

sm x [ sin cc , 

e~ ax -- -dx, 



sin x 



BIT X 

Substituting y + nir for x, the general term a n may be written 



y + WT 

It is evident that the integrand decreases as n increases, and hence 
a n + i< a n- Moreover the general term a n is less than f*(l/mr)dy, 
that is, than 1/n. Hence the above series is convergent, since the 
absolute values of the terms decrease as we proceed in the series, 
and the general term approaches zero. If the upper limit I lies 
between mr and (n + 1) TT, we shall have 



-dx = S n 6a n , 0<9<1, 

where S n denotes the sum of the first n terms of the series (24). As 
I increases indefinitely, n does the same, a n approaches zero, and the 
integral approaches the sum S of the series (24). 

In a similar manner it may be shown that the integrals 



r + * r + 

I sino; 2 ^x, I 

Jo Jo 



which occur in the theory of diffraction, each have finite values. 
The curve y = sin a; 2 , for example, has the undulating form of a sine 
curve, but the undulations become sharper and sharper as we go out, 
since the difference ^/(n + I)TT ^/n7^ of two consecutive roots of 
sin x 2 approaches zero as n increases indefinitely. 

Remark. This last example gives rise to an interesting remark. As x increases 
indefinitely sin 2 oscillates between 1 and + 1. Hence an integral may 
approach a limit even if the integrand does not approach zero, that is, even if 



IV, 92J IMPROPER AND LINE INTEGRALS 183 

the x axis is not an asymptote to the curve y = /(x). The following is an example 
of the same kind in which the function /(x) does not change sign. The function 



l + x 6 sin 2 x 

remains positive when x is positive, and it does not approach zero, since 
f(kn) = kit. In order to show that the integral approaches a limit, let us con 
sider, as above, the series 

flo + <*i + + o + i 
where 

a = 



l + x 6 sin 2 x 

As x varies from nit to (n + 1) TT, x 6 is constantly greater than n 6 7t 6 , and we may 
write 

l)rr dx 



A primitive function of the new integrand is 

== arc tan ( V 1 + n n 6 tan x), 
f n*7t 6 

and as x varies from mt to (n + 1) TT, tan x becomes infinite just once, passing 
from + co to oo. Hence the new integral is equal ( 77) to 7T/V1+ n s 7f 6 , and 
we have 

* 2 (n + 1) 
- - 



n , _ 

Vl + n TT" n 3 it 

It follows that the series 2a is convergent, and hence the integral J^ /(x) dx 
approaches a limit. 

On the other hand, it is evident that the integral cannot approach any limit 
if /(x) approaches a limit h different from zero when x becomes infinite. For 
beyond a certain value of x, /(x) will be greater than | h/2 \ in absolute value 
and will not change sign. 

The preceding developments bear a close analogy to the treatment of infinite 
series. The intimate connection which exists between these two theories is 
brought out by a theorem of Cauchy s which will be considered later (Chapter 
VIII). We shall then also find new criteria which will enable us to determine 
whether or not an integral approaches a limit in more general cases than those 
treated above. 

92. The function T(a). The definite integral 

(25) T(a)= f + V- 1 e-*dx 

Jo 

has a determinate value provided that a is positive. 
For, let us consider the two integrals 



r i r i 

I x a - e- x dx, I - l -*<fe, 



184 DEFINITE INTEGRALS 



[IV, 93 



where t is a very small positive number and I is a very large positive number. 
The second integral always approaches a limit, for past a sufficiently large value 
of x we have x a - l e~ x < 1/x 2 , that is, e x >x a + l . As for the first integral, the 
product x 1 - a f(x) approaches the limit 1 as x approaches zero, and the necessary 
and sufficient condition that the integral approach a limit is that 1 - a be less 
than unity, that is, that a be positive. Let us suppose this condition satisfied. 
Then the sum of these two limits is the function T(a), which is also called Euler s 
integral of the second kind. This function T(a) becomes infinite as a approaches 
zero, it is positive when a is positive, and it becomes infinite with a. It has 
a minimum for = 1.4616321-.., and the corresponding value of T(a) is 
0. 8856032- -. 

Let us suppose that a> 1, and integrate by parts, considering e~ x dx as the 
differential of er x . This gives 



but the product x a ~ l e- x vanishes at both limits, since a > 1, and there remains 
only the formula 

(26) r(o) = (a - l)T(o - 1). 

The repeated application of this formula reduces the calculation of Y(a) to 
the case in which the argument a lies between and 1. Moreover it is easy to 
determine the value of T(a) when a is an integer. For, in the first place, 



and the foregoing formula therefore gives, for a = 2, 3, 



and, in general, if n is a positive integer, 

(27) r(n) = 1.2.3...(n-l) = (n-l)l. 

93. Line integrals. Let AB be an arc of a continuous plane curve, 
and let P (x, y) be a continuous function of the two variables x and 
y along AB, where x and y denote the coordinates of a point of AB 
with respect to a set of axes in its plane. On the arc AB let us 
take a certain number of points of division m li m z , , m { , , whose 
coordinates are (x lt y^, (x 2 , y z ), , (x i; y.), ., and then upon each 
of the arcs m i _^rn i let us choose another point n { (,., ^.) at random. 
Finally, let us consider the sum 

(28) 

, *,*! -*,_, -.. 

extended over all these partial intervals. When the number of points 
of division is increased indefinitely in such a way that each of the 
differences x i x i _ l approaches zero, the above sum approaches a 



IV, 93] 



IMPROPER AND LINE INTEGRALS 



185 



limit which is called the line integral of P(x, y) extended over the 
arc AB, and which is represented by the symbol 



JAB 



P(x, y)dx. 



In order to establish the existence of this limit, let us first sup 
pose that a line parallel to the y axis cannot meet the arc AB in 
more than one point. Let a and b be the abscissae of the points A 
and B, respectively, and let y = <f>(x) be the equation of the curve AB. 
Then <(#) is a continuous function of x in the interval (a, b), by 
hypothesis, and if we replace y by <f>(x~) in the function P(x, y), the 
resulting function $(cc) = P[x, <(X)] is also continuous. Hence we 
have 



and the preceding sum may therefore be written in the form 
*(,) to - a) 



It follows that this surn approaches as its limit the ordinary definite 
integral 

I &(x)dx= I P[x, <t>(x)~\dx, 

i/a t/a 

and we have finally the formula 

I P(x, y}dx = I P[x, t(x)-]dx. 
JAB Ja 

If a line parallel to the y axis can meet the arc AB in more than 
one point, we should divide the arc 
into several portions, each of which 
is met in but one point by any line 
parallel to the y axis. If the given 
arc is of the form A CDB (Fig. 14), 
for instance, where C and D are 
points at which the abscissa has an 
extremum, each of the arcs A C, CD, 
DB satisfies the above condition, and 
we may write 

I P(x, y)dx= I P(x, y)dx + I P(x, y)dx + f P(x,y)dx. 
JACDB J,T JCD JOB 

But it should be noticed that in the calculation of the three integrals 




FIG. 14 



186 DEFINITE INTEGRALS [IV, 93 

on the right-hand side the variable y in the function P(x, y) 
must be replaced by three different functions of the variable x, 
respectively. 

Curvilinear integrals of the form J AR Q(x, y)dy may be denned 
in a similar manner. It is clear that these integrals reduce at once 
to ordinary definite integrals, but their usefulness justifies their 
introduction. We may also remark that the arc AB may be com 
posed of portions of different curves, such as straight lines, arcs of 
circles, and so on. 

A case which occurs frequently in practice is that in which the 
coordinates of a point of the curve AB are given as functions of a 
variable parameter 



where <j>(t) and \(/(t), together with their derivatives < () and 
are continuous functions of t. We shall suppose that as t varies 
from a to ft the point (x, y) describes the arc AB without changing 
the sense of its motion. Let the interval (a, /?) be divided into a 
certain number of subintervals, and let t i _ l and t f be two consecu 
tive values of t to which correspond, upon the arc AB, two points 
m,...! and m { whose coordinates are (#,_!, y f _i) and (x,-, y t ), respec 
tively. Then we have 



where 0,. lies between t i _ 1 and t { . To this value 0, there corresponds 
a point (,-, 17,) of the arc m i _ l m i ; hence we may write 



or, passing to the limit, 
P(x, 



f 

/.4 



An analogous formula for JQdy may be obtained in a similar manner. 
Adding the two, we find the formula 



(29) f P<& -f Qdy = f 

J^l / 



which is the formula for change of variable in line integrals. Of 
course, if the arc AB is composed of several portions of different 
curves, the functions <f>(t) and \fr() will not have the same form 
along the whole of AB, and the formula should be applied in that 
case to each portion separately. 



IV, t)4] IMPROPER AND LINE INTEGRALS 187 

94. Area of a closed curve. We have already defined the area of a 
portion of the plane bounded by an arc A MB, a straight line which 
does not cut that arc, and the two perpendiculars AA Q , BB let fall 
from the points A and B upon the straight line ( 65, 78, Fig. 9). 
Let us now consider a continuous closed curve of any shape, by 
which we shall understand the locus described by a point M whose 
coordinates are continuous functions x =f(), y = <f>(t) of a param 
eter t which assume the same values for two values t and T of 
the parameter t. The functions f(f) and <j>(t~) may have several 
distinct forms between the limits t and T; such will be the case, 
for instance, if the closed contour C be composed of portions of 
several distinct curves. Let M , M lt J/ 2 , -, M { _ u M if , M n _ lt M 
denote points upon the curve C corresponding, respectively, to the 
values t , t l} t 2 , -, t i _ l , t t , , t n _ j, T of the parameter, which 
increase from t to T. Connecting these points in order by straight 
lines, we obtain a polygon inscribed in the curve. The limit 
approached by the area of this polygon, as the number of sides is 
indefinitely increased in such a way that each of them approaches 
zero, is called the area of the closed curve C.* This definition is 
seen to agree with that given in the particular case treated above. 
For if the polygon A A(2 1 Q 2 BB A (Fig. 9) be broken up into 
small trapezoids by lines parallel to AA , the area of one of these 
trapezoids is (*,. - *,._,) [/(a-,-) + f(x { _ x )]/2, or (a\ -*<_,)/&), 
where ,. lies between x^-^ and cc f . Hence the area of the whole 
polygon, in this special case, approaches the definite integral 
ff(x)dx. 

Let us now consider a closed curve C which is cut in at most two 
points by any line parallel to a certain fixed direction. Let us 
choose as the axis of y a line parallel to this direction, and as the 
axis of a; a line perpendicular to it, in such a way that the entire 
curve C lies in the quadrant xOy (Fig. 15). 

The points of the contour C project into a segment ab of the axis 
Ox, and any line parallel to the axis of y meets the contour C in at 
most two points, m^ and m z . Let y v = ^(cc) and ?/ 2 = tl/ 2 (x) be the 
equations of the two arcs Am v B and Am z B, respectively, and let 
us suppose for simplicity that the points A and B of the curve C 
which project into a and b are taken as two of the vertices of the 



* It is supposed, of course, that the curve under consideration has no double point, 
and that the sides of the polygon have been chosen so small that the polygon itself 
has no double point. 



188 



DEFINITE INTEGRALS 



[IV, 94 



polygon. The area of the inscribed polygon is equal to the differ 
ence between the areas of the two polygons formed by the lines A a, 
ab, bB with the broken lines inscribed in the two arcs Am 2 B and 
AmiB, respectively. Passing to the limit, it is clear that the area 
of the curve C is equal to the difference between the two areas 
bounded by the contours Am^BbaA and Am^BbaA, respectively, that 

is, to the difference between 
the corresponding definite in 
tegrals 




X6 /tb 

\l/ z (x)dx I ^ 
*J a 



FIG. 15 



These two integrals represent 
the curvilinear integral fydx 
taken first along Am 2 B and 
then along Am-^B. If we 
agree to say that the contour 
C is described in the positive 
sense when an observer standing upon the plane and walking around 
the curve in that sense has the enclosed area constantly on his left 
hand (the axes being taken as usual, as in the figure), then the above 
result may be expressed as follows : the area O enclosed by the 
contour C is given by the formula 



(30) 



r 

J(C) 



where the line integral is to be taken along the closed contour C in 
the positive sense. Since this integral is unaltered when the origin 
is moved in any way, the axes remaining parallel to their original 
positions, this same formula holds whatever be 
the position of the contour C with respect to 
the coordinate axes. 

Let us now consider a contour C of any form 
whatever. We shall suppose that it is possible 
to draw a finite number of lines connecting 
pairs of points on C in such a way that the 
resulting subcontours are each met in at most 
two points by any line parallel to the y axis. 
Such is the case for the region bounded by the 
contour C in Fig. 16, which we may divide into three subregions 
bounded by the contours amba, abndcqa, cdpc, by means of the 




FIG. 16 



IV, 95] 



IMPROPER AND LINE INTEGRALS 



189 



transversals ab and cd. Applying the preceding formula to each 
of these subregions and adding the results thus obtained, the line 
integrals which arise from the auxiliary lines ab and cd cancel each 
other, and the area bounded by the closed curve C is still given by 
the line integral fydx taken along the contour C in the positive 
sense. 

Similarly, it may be shown that this same area is given by the 
formula 



(31) 



n = I 

J(C 



x dy\ 



and finally, combining these two formulae, we have 



(32) 



=- f 

2 J<c 



xdy ydx, 



where the integrals are always taken in the positive sense. This 
last formula is evidently independent of the choice of axes. 
If, for instance, an ellipse be given in the form 



its area is 



x = a cos t, y = b sin t, 

1 C 2 " 

fi = I ab(cos 2 t -{- sm 2 f)dt = Trab. 

2 Jo 



95. Area of a curve in polar coordinates. Let us try to find the 
area enclosed by the contour OAMBO (Fig. 17), which is composed 
of the two straight lines OA, OB, and the arc A MB, which is 
met in at most one 

point by any radius ,,-/ -^37 

vector. Let us take 
as the pole and a 
straight line Ox as / 
the initial line, and I 
let p = /(o>) be the \ 
equation of the arc 
A MB. 

Inscribing a polygon in the arc A MB, with A and B as two of 
the vertices, the area to be evaluated is the limit of the sum of such 
triangles as OMM . But the area of the triangle OMM is 





FIG. 17 



1 P 

- p(p + Ap) sin Aw = Aw I 



190 DEFINITE INTEGRALS [IV, 95 

where approaches zero with Aw. It is easy to show that all the 
quantities analogous to c are less than any preassigried number rj 
provided that the angles Aw are taken sufficiently small, and that 
we may therefore neglect the term cAw in evaluating the limit. 
Hence the area sought is the limit of the sum 2p 2 Aw/2, that is, it 
is equal to the definite integral 



where w t and w 2 are the angles which the straight lines OA and OB 
make with the line Ox. 

An area bounded by a contour of any form is the algebraic sum 
of a certain number of areas bounded by curves like the above. If 
we wish to find the area of a closed contour surrounding the point 
0, which is cut in at most two points by any line through 0, for 
example, we need only let w vary from to 2?r. The area of a con 
vex closed contour not surrounding O (Fig. 17) is equal to the dif 
ference of the two sectors 0AM BO and OANBO, each of which may 
be calculated by the preceding method. In any case the area is 
represented by the line integral 



taken over the curve C in the positive sense. This formula does 
not differ essentially from the previous one. For if we pass from 
rectangular to polar coordinates we have 

x = p cos w, y p &m w > 

dx = cos w dp p sin w c?w, dy = sin w dp + p cos w c?w, 
x dy y dx = p 2 dta. 

Finally, let us consider an arc AMB whose equation in oblique 
coordinates is y =f(x~). In order to find the area bounded by this 
arc AMB, the x axis, and the two lines AA , BB , which are parallel 
to the y axis, let us imagine a polygon inscribed in the arc AMB, and 
let us break up the area of this polygon into small trapezoids by 
lines parallel to the y axis. The area of one of these trapezoids is 



IV, 96] 



IMPROPER AND LINE INTEGRALS 



191 



which may be written in the form (z,--i ~ #,)/() sin 0, where | 
lies in the interval (x t _ l; x^. Hence the area in question is equal 
to the definite integral 



sin $ f(i 



x) dx, 



where x and A" denote the abscissae 
of the points A and B, respectively. 

It may be shown as in the similar 
case above that the area bounded by 
any closed contour C whatever is given 
by the formula 




A B 

FIG. 18 



x dy y dx. 
(O 

Note. Given a closed curve C (Fig. 15), let us draw at any point 
M the portion of the normal which extends toward the exterior, 
and let a, ft be the angles which this direction makes with the axes 
of x and y, respectively, counted from to TT. Along the arc Am^B 
the angle ft is obtuse and dx = ds cos ft. Hence we may write 



I y dx = \ y cos ft ds. 

\J (Am^B) J 



Along Bm z A the angle ft is acute, but dx is negative along Bm 2 A 
in the line integral. If we agree to consider ds always as positive, 
we shall still have dx = ds cos ft. Hence the area of the closed 
curve may be represented by the integral 



y cos ft ds, 



where the angle ft is defined as above, and where ds is essentially 
positive. This formula is applicable, as in the previous case, to a 
contour of any form whatever, and it is also obvious that the same 
area is given by the formula 



x cos a ds. 



These statements are absolutely independent of the choice of axes. 

96. Value of the integral /xdy ydx. It is natural to inquire what will 
be represented by the integral fxdy ydx, taken over any curve whatever, 
closed or unclosed. 



192 DEFINITE INTEGRALS [TV, 97 

Let us consider, for example, the two closed curves OAOBO and 
ApBqCrAsBtCuA (Fig. 19) which have one and three double points, respec 
tively. It is clear that we may replace either of these curves by a combination 
of two closed curves without double points. Thus the closed contour OA OBO 

is equivalent to a combination of the 
two contours 0-40 and OBO. The 
integral taken over the whole contour 
is equal to the area of the portion 
0.40 less the area of the portion 
OBO. Likewise, the other contour 
may be replaced by the two closed 
curves ApBqCrA and AsBtCuA, and 
the integral taken over the whole con 
tour is equal to the sum of the areas of ApBsA, BtCqB, and ArCuA, plus twice 
the area of the portion AsBqCuA. This reasoning is, moreover, general. Any 
closed contour with any number of double points determines a certain number 
of partial areas <TI, <r 2 , , <r p , of each of which it forms all the boundaries. 
The integral taken over the whole contour is equal to a sum of the form 




where mi, m 2 , , m p are positive or negative integers which may be found by 
the following rule : Given two adjacent areas <r, <r , separated by an arc ab of the 
contour C, imagine an observer walking on the plane along the contour in the sense 
determined by the arrows ; then the coefficient of the area at his left is one greater 
than that of the area at his right. Giving the area outside the contour the coeffi 
cient zero, the coefficients of all the other portions may be determined successively. 
If the given arc AB is not closed, we may transform it into a closed curve by 
joining its extremities to the origin, and the preceding formula is applicable to 
this new region, for the integral fxdy ydx taken over the radii vectores OA 
and OB evidently vanishes. 



V. FUNCTIONS DEFINED BY DEFINITE INTEGRALS 

97. Differentiation under the integral sign. We frequently have to 
deal with integrals in which the function tcr43e integrated depends 
not only upon the variable of integration but also upon one or more 
other variables which we consider as parameters. Let f(x, a) be a 
continuous function of the two variables x and a when x varies from 
x to X and a varies between certain limits and a^ We proceed 
to study the function of the variable a which is defined by the 

definite integral 

>,r 



= Cf(x,a)dx, 

Jx n 



where a is supposed to have a definite value between a and a lf and 
where the limits x and X are independent of a. 



IV, 97] FUNCTIONS DEFINED BY INTEGRALS 193 

We have then 
(33) F(a + Aa) - F(a) = f [/(*, a + Aa) -f(x, a)] dx. 

JjT 

Since the function f(x, a) is continuous, this integrand may be made 
less than any preassigned number c by taking Aa sufficiently small. 
Hence the increment AF(a) will be less than e\X x in absolute 
value, which shows that the function F(a) is continuous. 

If the function f(x, a) has a derivative with respect to a, let us 
write 

f(x, a + Aa) - f(x, a) = Aa [/. (x, ) + e] , 

where e approaches zero with Aa. Dividing both sides of (33) by 
Aa, we find 






and if -q be the upper limit of the absolute values of c, the absolute 
value of the last integral will be less than ri\X x \. Passing to 
the limit, we obtain the formula 

(34) ^ 

da 

In order to render the above reasoning perfectly rigorous we must 
show that it is possible to choose Aa so small that the quantity c 
will be less than any preassigned number rj for all values of x between 
the given limits x and X. This condition will certainly be satisfied 
if the derivative f a (x, a) itself is continuous. For we have from 
the law of the mean 

f(x, a + Aa) -f(x, a) = Aa/, (x, a + 0Aa), < $ < 1, 

and hence 



If the function f a is continuous, this difference e will be less than 77 
for any values of x and a, provided that | Aa j is less than a properly 
chosen positive number h (see Chapter VI, 120). 

Let us now suppose that the limits X and x ti are themselves func 
tions of a. If A.Y and Aa: denote the increments which correspond 
to an increment Aa, we shall have 



194 DEFINITE INTEGRALS [IV, 97 

F(a + Aa)- F(a) = f \_f(x, a + A,r) -f(x, a) 

Jx 

f*X+&X 

+ / /(a, a + Aa) dx 

Jx 

r x o + ^ 
I f(x, a + 

Jx 



or, applying the first law of the mean for integrals to each of the 
last two integrals and dividing by Aa, 

F(a 4- Aa) F() _ C A /(x, a 4- Aa-) f(x, a) , 
Aa J, Aa 



AT 

-^/(^o + ^A^, nr + Aa). 

As Aa- approaches zero the first of these integrals approaches the 
limit found above, and passing to the limit we find the formula 

(35) ^ = 

which is the general formula for differentiation under the integral 
sign. 

Since a line integral may always be reduced to a sum of ordinary 
definite integrals, it is evident that the preceding formula may be 
extended to line integrals. Let us consider, for instance, the line 
integral 



= f P(x, y, 

JAB 



F(a) = I P(x, y, a} dx + Q(x, y, a) dy 
JAB 

taken over a curve AB which is independent of a. It is evident that 
we shall have 



F\a) = I P a (x, y, a)dx + Q a (x, y, 
JAB 



where the integral is to be extended over the same curve. On the 
other hand, the reasoning presupposes that the limits are finite and 
that the function to be integrated does not become infinite between 
the limits of integration. We shall take up later (Chapter VIII, 
175) the cases in which these conditions are not satisfied. 



IV, 98] FUNCTIONS DEFINED BY INTEGRALS 195 

The formula (35) is frequently used to evaluate certain definite 
integrals by reducing them to others which are more easily calcu 
lated. Thus, if a is positive, we have 



/ 

Jo 



1 * 

= arc tan p. 
va vo 



whence, applying the formula (34) n 1 times, we find 

<-i r -.i:t...<.-i>r 7S 

Jo (^ 

98. Examples of discontinuity. If the conditions imposed are not satisfied for 
all values between the limits of integration, it may happen that the definite inte 
gral defines a discontinuous function of the parameter. Let us consider, for 
example, the definite integral 

f + 1 *<*** 

J_ l l-2xcosa + x 2 

This integral always has a finite value, for the roots of the denominator are 
imaginary except when a = kit, in which case it is evident that F(a) = 0. Sup 
posing that sin a ^ and making the substitution x = cos a + t sin a, the indefi 
nite integral becomes 

/sin a dx f dt . 
- - = I = arc tan t. 
1 - 2x cos a + x 2 J 1 + t 2 

Hence the definite integral F(a) has the value 

(1 cos a\ I 1 cos a\ 
] arc tan ( - ) 
sin a I \ sin a / 

where the angles are to be taken between n/2 and x/2. But 

1 cos a 1 cos a 

- x - = -1, 

sin a sin a 

and hence the difference of these angles is n/2. In order to determine the 
sign uniquely we need only notice that the sign of the integral is the same as 
that of sin a. Hence F(a) = n/2 according as sin a is positive or negative. 
It follows that the function F(a) is discontinuous for all values of a of the form 
kit. This result does not contradict the above reasoning in the least, however. 
For when x varies from 1 to + 1 and a varies from e to + e, for example, 
the function under the integral sign assumes an indeterminate form for the sets 
of values a = 0, x = - 1 and a = 0, x = + 1 which belong to the region in ques 
tion for any value of e. 

It would be easy to give numerous examples of this nature. Again, consider 
the integral 

n -4- a: 

sin mx 



f 

Ux 



, 
dx. 



196 DEFINITE INTEGRALS [IV, 99 

Making the substitution mx = y, we find 



X 




sin mx 



y 



where the sign to be taken is the sign of m, since the limits of the transformed 
integral are the same as those of the given integral if m is positive, but should 
be interchanged if m is negative. We have seen that the integral in the second 
member is a positive number N ( 91). Hence the given integral is equal to N 
according as m is positive or negative. If m = 0, the value of the integral is 
zero. It is evident that the integral is discontinuous for m = 0. 



VI. APPROXIMATE EVALUATION OF DEFINITE INTEGRALS 

99. Introduction. When no primitive of f(x) is known we may 
resort to certain methods for finding an approximate value of the 
definite integral f*f(x) dx. The theorem of the mean for integrals 
furnishes two limits between which the value of the integral must 
lie, and by a similar process we may obtain an infinite number of 
others. Let us suppose that $(x) <f(x) < ty(x) for all values of x 
between a and b (a < b). Then we shall also have 



s*t> s*b /-i 

I <j>(x}dx< I f(x)dx< I 

<J a J a c/a 



If the functions <(#) and ^(a;) are the derivatives of two known 
functions, this formula gives two limits between which the value of 
the integral must lie. Let us consider, for example, the integral 



C dx 

"Jo Vl-a; 4 



Now Vl-z 4 = Vl - x 2 Vl + x\ and the factor Vl + z 2 lies 
between 1 and V2 for all values of x between zero and unity. 
Hence the given integral lies between the two integrals 



r } dx i r l dx 

Jo Vl-cc 2 V2Jo Vl x a 



that is, between Tr/2 and 7r/(2V2). Two even closer limits may 
be found by noticing that (1 + z 2 )- 1 / 2 is greater than 1 x 2 /2, 
which results from the expansion of (1 -f 7/)- 1/2 by means of Taylor s 
series with a remainder carried to two terms. Hence the integral 
I is greater than the expression 



dx 1 r } x 2 dx 



IV, y] APPROXIMATE EVALUATION 197 

The second of these integrals has the value Tr/4 ( 105) ; hence / 
lies between Tr/2 and 3 7r/8. 

It is evident that the preceding methods merely lead to a rough 
idea of the exact value of the integral. In order to obtain closer 
approximations we may break up the interval (a, 6) into smaller 
subintervals, to each of which the theorem of the mean for inte 
grals may be applied. For definiteness let us suppose that the 
function /(a:) constantly increases as x increases from a to b. Let 
us divide the interval (a, b~) into n equal parts (b a nh). Then, 
by the very definition of an integral, ^f(x}dx lies between the 
two sums 



s = h\f(a) 

S = h\f(a + h) +f(a + 2A) -f +/(a + nh)\. 

If we take (S + s)/2 as an approximate value of the integral, the 
error cannot exceed S-s\/2= [(i - ) /2 n] [/(i) -/(a)] . The 
value of (S + s)/2 may be written in the form 

(/(a) + /O 



| , 

2 

/[a + (ro - 1) A] +f(a + nh) ) 
2 ) 



Observing that \f(a + ih) +/[a + (i +1) h~\\h/2 is the area of 
the trapezoid whose height is h and whose bases are /(a + ih) and 
f(a + ih + 7t), we may say that the whole method amounts to 
replacing the area under the curve y = f(x) between two neighbor 
ing ordinates by the area of the trapezoid whose bases are the two 
ordinates. This method is quite practical when a high degree of 
approximation is not necessary. 

Let us consider, for example, the integral 



/ 

C/O 



dx 



Taking n = 4, we find as the approximate value of the integral 



and the error is less than 1/16 = .0625.* This gives an approxi 
mate value of TT which is correct to one decimal place, 3.1311 

* Found from the formula \S s\/2. In fact, the error is about .00260, the exact 
value being rr/4. TRANS. 



198 DEFINITE INTEGRALS [IV, 100 

If the function f(x) does not increase (or decrease) constantly as 
x increases from a to b, we may break up the interval into sub- 
intervals for each of which that condition is satisfied. 

100. Interpolation. Another method of obtaining an approximate 
value of the integral f f(x)dx is the following. Let us determine 
a parabolic curve of order n, 

y = <(z) = a + a-iX + 1- a n x n , 

which passes through (n -f- 1) points B , B l} , B n of the curve 
y =f(x) between the two points whose abscissae are a and b. 
These points having been chosen in any manner, an approximate 
value of the given integral is furnished by the integral f b <j>(x}dx, 
which is easily calculated. 

Let (x , T/O), (#!, 7/i), , (x n , ?/) be the coordinates of the (n +1) 
points B Q , B!, , B n . The polynomial <f)(x) is determined by 
Lagrange s interpolation formula in the form 

<(*) = Z/o A -f ijt A*! -\ h i/i Xf -\ \-y n X n , 

where the coefficient of y t is a polynomial of degree n, 

x _ (^ - EQ) (x-x i _ l }(x- x i + } } (x - ay) ^ 

(X- - X o) (*i ~ Xi- l) (X -*<+!) (*< - Xn) 

which vanishes for the given values x , x, , x n , except for x = x f , 
and which is equal to unity when x x { . Hence we have 



/ 

U a 



Q : 

The numbers x i are of the form 

x = a -f (b a), a?! = a -f- $i(& ) , # = a + #(& ) 

where < < 6 t < < 6 H 5 1. Setting x = a + (b a) t, the ap 
proximate value of the given integral takes the form 

(36) (b a) (A" y + K \y\ H h A .y,,), 

where K i is given by the formula 

KI = I 77 ^ "^^ dt . 

If we divide the main interval (a, b) into subintervals whose 
ratios are the same constants for any given function /(x) whatever, 
the numbers , U , d n , and hence also the numbers K { , are inde 
pendent of f(x). Having calculated these coefficients once for all, 



IV, 101] APPROXIMATE EVALUATION 199 

it only remains to replace y , yi, , y n by their respective values 
in the formula (36). 

If the curve f(x) whose area is to be evaluated is given graph 
ically, it is convenient to divide the interval (a, 6) into equal parts, 
and it is only necessary to measure certain equidistant ordinates of 
this curve. Thus, dividing it into halves, we should take = 0, 
61 = 1/2, 2 = 1, which gives the following formula for the approxi 
mate value of the integral : 

b a 
I = Q- u/o + 4yi + y 2 ) 

Likewise, for n = 3 we find the formula 

1 = C ? /0 + 3#! + 3// 2 + 7/s) , 
o 

and for n = 4 

7 = ~9o ~ (7z/o 

Ir 

The preceding method is due to Cotes. The following method, 
due to Simpson, is slightly different. Let the interval (a, b~) be 
divided into 2n equal parts, and let y , y 1} ?/ 2 , , y 2n be the ordi 
nates of the corresponding points of division. Applying Cotes 
formula to the area which lies between two ordinates whose indices 
are consecutive even numbers, such as y and y 2 , ?/ 2 and y t , etc., we 
find an approximate value of the given area, in the form 



whence, upon simplification, we find Simpson s formula : 
1 = -- [2/0 + 2/ 2 + 2(y 2 + y, H ---- -f ?/ 2n _ 2 ) 



101. Gauss method. In Gauss method other values are assigned 
the quantities #,. The argument is as follows: Suppose that we 
can find polynomials of increasing degree which differ less and less 
from the given integrand f(x~) in the interval (a, &). Suppose, 
for instance, that we can write 

/(*) = a + a,x + a 2 x* + + a^ X 2n ~ l + 7? 2n (x) , 
where the remainder R 2H (x~) is less than a fixed number ^ for all 



200 DEFINITE INTEGRALS [IV, 101 

values of x between a and b* The coefficients a, will be in gen 
eral unknown, but they do not occur in the calculation, as we shall 
see. Let x , x 1} -, x n _ 1 be values of x between a and b, and let 
<(ce) be a polynomial of degree n 1 which assumes the same 
values as does /(ce) for these values of x. Then Lagrange s inter 
polation formula shows that this polynomial may be written in the 
form 



where <f m and ty k are at most polynomials of degree n I. It is 
clear that the polynomial <f> m (x) depends only upon the choice of 
x o> x i> "> x n-i- O n th e other hand, this polynomial <,(#) must 
assume the same values as does x m for x = x , x = x l} , x = x a _ l . 
For, supposing that all the a s except a m and also R^(x) vanish, 
f(x) reduces to a m x m and </>(a:) reduces to a m $ in (x). Hence the 
difference x m <f> m (x) must be divisible by the product 

P n (x) = (x - XQ) (x - a-i) (x - a^). 

It follows that a 1 " <f> m (x)= P n Q m _ n (x~), where Q m _ n (x) is a poly 
nomial of degree m n, if ra > n ; and that x m <J> m (x) = if m < n 1. 
The error made in replacing f f(x~) dx by J a <f> (x~) dx is evidently 
given by the formula 



^ r 
*" M j" 

_ V 7? ( Y \ 1 \b (x 
4 H 2,i ( x i) 1 *i \ x 

i=0 ^o 



The terms which depend upon the coefficients a , a 1 , -, a n _ l vanish 
identically, and hence the error depends only, upon the coefficients 
<*> a n + i> "> a -2n-\ an( l the remainder R 2n (x ). But this remain 
der is very small, in general, with respect to the coefficients 
a n , o- n + 1 , , a. 2n _ l . Hence the chances are good for obtaining a 
high degree of approximation if we can dispose of the quantities 
x , x lt , x n _ l in such a way that the terms which depend upon 
a > a n + i> > a 2n-i a ^ so vanish identically. For this purpose it is 
necessary and sufficient that the n integrals 



s*b f*\> nb 

/ P n Q u dx, I P^.dx, -, I 

Ja Jo. fc/a 



* This is a property of any function which is continuous in the interval (a, 
according to a theorem due to Weierstrass (see Chapter IX, 199). 



IV, 102] APPROXIMATE EVALUATION 201 

should vanish, where Q { is a polynomial of degree i. We have 
already seen ( 88) that this condition is satisfied if we take P n of 
the form 

**.-; [<*-) <*-*)]- 

It is therefore sufficient to take for x , x i} , x n _ l the n roots of 
the equation P n = 0, and these roots all lie between a and b. 

We may assume that a = 1 and b + 1, since all other cases 
may be reduced to this by the substitution x = (b + ) /2 + 2 (b a) /2. 
In the special case the values of x , x lf -, _! are the roots of 
Legendre s polynomial X n . The values of these roots and the 
values of K i for the formula (36), up to n = 5, are to be found to 
seven and eight places of decimals in Bertrand s Traite de Calcul 
integral (p. 342). 

Thus the error in Gauss method is 



C R*(x)dx -"j^R^x,) f *, 

Jo, t = *^ a 



where the functions ^ { (x) are independent of the given integrand. 
In order to obtain a limit of error it is sufficient to find a limit of 
R^(x), that is, to know the degree of approximation with which 
the function f(x) can be represented as a polynomial of degree 
2n 1 in the interval (a, &). But it is not necessary to know 
this polynomial itself. 

Another process for obtaining an approximate numerical value of 
a given definite integral is to develop the function f(x) in series and 
integrate the series term by term. We shall see later (Chapter VIII) 
under what conditions this process is justifiable and the degree of 
approximation which it gives. 

102. Amsler s planimeter. A great many machines have been invented to 
measure mechanically the area bounded by a closed plane curve.* One of the 
most ingenious of these is Amsler s planimeter, whose theory affords an interest 
ing application of line integrals. 

Let us consider the areas AI and A* bounded by the curves described by two 
points AI and A 2 of a rigid straight line which moves in a plane in any manner 
arid finally returns to its original position. Let (xi, 2/1) and (x 2 , y 2 ) be the coor 
dinates of the points AI and A, respectively, with respect to a set of rectangu 
lar axes. Let I be the distance AiA 2 , and the angle which A\A Z makes with 

* A description of these instruments is to be found in a work by Abdank- 
Abakanowicx: Les integraphes, la courbe integrate et ses applications (Gauthier- 
Villars, 1886). 



2U2 DEFINITE INTEGRALS [IV, 102 

the positive x axis. In order to define the motion of the line analytically, i, j/i, 
and 6 must be supposed to be periodic functions of a certain variable parameter t 
which resume the same values when t is increased by T. We have x 2 = %i + I cos 6, 
2/2 = V\ + I si n an( l hence 



yidx\ + PdO 
+ I(cos0dyi sintfdxi + 

The areas AI and A 2 of the curves described by the points A\ and A z , under the 
general conventions made above ( 96), have the following values : 

l r 1 r 

AI = - J xidyi - yidxi, A 2 = - J x 2 dy 2 - y z dx 2 . 

Hence, integrating each side of the equation just found, we obtain the equation 
- CdO + - j fcos6dyi - s m0dx l + f (xicosfl + yis\n0)d0\ , 



A 2 = AI -f 



where the limits of each of the integrals correspond to the values t and t Q + T 
of the variable t. It is evident that fd8 = 2Kn, where K is an integer which 
depends upon the way in which the straight line moves. On the other hand, 
integration by parts leads to the formulae 



/ Xi cos 6 d0 Xi sin 01 sin dx\ , 
f yi sin d0 = y\ cos + / cos 9 dyi . 



But Xi sin and y\ cos have the same values for t t Q and t = to + T. Heuce 
the preceding equation may be written in the form 



A 2 = AI + Knl 2 + I CcosOdyi si 



Now let s be the length of the arc described by A i counted positive In a certain 
sense from any fixed point as origin, and let a be the angle which the positive 
direction of the tangent makes with the positive x axis. Then we shall have 

cos dyi sin 6 dxi = (sin a cos sin cos a) ds = sin V da , 

where V is the angle which the positive direction of the tangent makes with the 
positive direction A^A Z of the straight line taken as in Trigonometry. The 
preceding equation, therefore, takes the form 



(38) A 2 = AI + Kxt 2 + ifsinVds. 



Similarly, the area of the curve described by any third point A 3 of the straight 
line is given by the formula 



(39) A 3 = A! + Kl * + I fsiuVds, 



where I is the distance AiA & . Eliminating the unknown quantity fsinVds 
between these two equations, we find the formula 

1 Aj - IA = (* - 1) AI + Kxll\l - Hi 



IV, 102] APPROXIMATE EVALUATION 203 

which may be written in the form 

(40) A! (23) + Ao (31 ) + A 3 (12) + K* (12) (23) (31) = , 

where (ik) denotes the distance between the points Ai and Ak (i, k = 1, 2, 8) 
taken with its proper sign. As an application of this formula, let us consider 
a straight line A\A<L of length (a + 6), whose extremities A\ and A 2 describe the 
same closed convex curve C. The point A 3 , which divides the line into seg 
ments of length a and 6, describes a closed curve C" which lies wholly inside C. 
In this case we have 

A 2 = Ai, (12) = a + b, (23) = - 6, (31) = - a, K = I 

whence, dividing by a + 6, 

AI A 3 = rtab. 

But AI AS is the area between the two curves C and C . Hence this area is 
independent of the form of the curve C. This theorem is due to Holditch. 

If, Instead of eliminating JsinFds between the equations (38) and (39), we 
eliminate AI, we find the formula 

(41) A 3 = A 2 + Kx(V* - I 2 ) + (I - l)CsinVds. 

Amsler s planimeter affords an application of this formula. Let AiA 2 A s be a 
rigid rod joined at A 2 with another rod (L4 2 . The point being fixed, the point 
A 3 , to which is attached a sharp pointer, is made to describe the curve whose area 
is sought. The point A% then 
describes an arc of a circle or 
an entire circumference, accord 
ing to the nature of the motion. 
In any case the quantities A 2 , -K", 
I, I are all known, and the area 
AS can be calculated if the in 
tegral Jsin Vds, which is to be 
taken over the curve C\ described 
by the point A\, can be evaluated. 

f\ ** 

This end A\ carries a graduated V 

, FIG. 20 

circular cylinder whose axis coin 
cides with the axis of the rod AiA 3 , and which can turn about this axis. 

Let us consider a small displacement of the rod which carries AiA 2 A 3 into 
the position AiA ^Az. Let Q be the intersection of these straight lines. About 
Q as center draw the circular arc Ai a and drop the perpendicular A{P from 
AI upon AiA 2 . We may imagine the motion of the rod to consist of a sliding 
along its own direction until AI comes to or, followed by a rotation about Q which 
brings a to A{. In the first part of this process the cylinder would slide, with 
out turning, along one of its generators. In the second part the rotation of 
the cylinder is measured by the arc aA{. The two ratios aA\/A{P and 
A{P/axcAiA{ approach 1 and sinF, respectively, as the arc A[A\ approaches 
zero. Hence ccA{ = As (sinF + e), where e approaches zero with As. It follows 
that the total rotation of the cylinder is proportional to the limit of the sum 
SAs(sinF + e), that is, to the integral JsinFds. Hence the measurement of 
this rotation is sufficient for the determination of the given area. 




204 DEFINITE INTEGRALS [IV, Exs. 



EXERCISES 

1. Show that the sum 1/n + l/(n + !) + + l/2n approaches log 2 as n 
increases indefinitely. 

[Show that this sum approaches the definite integral f Q l [1/(1 + x)]dx as its 
limit.] 

2. As in the preceding exercise, find the limits of each of the sums 

n + -5- + ..- + 



n 2 + 1 n 2 + 2 2 n 2 + (n - I) 2 

1 1 1 



Vn 2 - 1 Vn 2 - 2 2 Vn* - (n - I) 2 

by connecting them with certain definite integrals. In general, the limit of 
the sum 



as n becomes infinite, is equal to a certain definite integral whenever <f>(i, n) is 
a homogeneous function of degree 1 in i and n. 

3. Show that the value of the definite integral f " /2 log sin x dx is 
-(jr/2)log2. 

[This may be proved by starting with the known trigonometric formula 

. it . 2* . (n \)it n 

sm sin sm - ^ , 

n n n 2"- 

or else by use of the following almost self-evident equalities : 

JT 

/I, fy. , 1 C 2 . /sin2x\ _. 

I log sin x dx = I log cos x dx = - log ( I dx. ] 

/o O 2 / u \ 2 / 

4. By the aid of the preceding example evaluate the definite integral 

1 tan x dx . 

2/ 

5. Show that the value of the definite integral 

,1 



/ 

Jo 



1 + x 2 
is (jr/8)log2. 

[Set x = tan <p and break up the transformed integral into three parts.] 

6*. Evaluate the definite integral 



/ITT 

I log (1 2a cos x + a 2 ) dx . 
Jo [POISSOK.] 



IV, Exs.] EXERCISES 205 

[Dividing the interval from to it into n equal parts and applying a well-known 
formula of trigonometry, we are led to seek the limit of the expression 



?r ra 1 

-log - 2n_! 

n La + 1 



as n becomes infinite. If a lies between 1 and + 1, this limit is zero. If 
a 2 > 1, it is 7t log a 2 . Compare 140.] 

1. Show that the value of the definite integral 

sinxdx 



/ 

Jo 



/o Vl 2or cos x + a 2 
where a is positive, is 2 if a < 1, and is 2/a if a > 1. 

15*. Show that a necessary and sufficient condition that /(x) should be inte- 
grable in an interval (a, b) is that, corresponding to any preassigned number e, 
a subdivision of the interval can be found such that the difference S s of the 
corresponding sums S and s is less than e. 

9. Let/(x) and <j>(x) be two functions which are continuous in the interval (a, 6), 
and let (a, Xi, Xj, , b) be a method of subdivision of that interval. If ,, 77. 
are any two values of x in the interval (x,-_i, x,-), the sum 2/(,-) <f> (?;,) (x, x,-_i) 
approaches the definite integral f^f(x) <t>(x) dx as its limit. 

10. Let/(x) be a function which is continuous and positive in the interval (a, b). 
Show that the product of the two definite integrals 



/>* f/l> 



is a minimum when the function is a constant. 

11. Let the symbol I* 1 denote the index of a function ( 77) between 
and Xi. Show that the following formula holds: 



where e = + 1 if /(x ) > and /(Xi) < 0, e = - 1 if f(x ) < and f(xi) > 0, and 
c = if /(x ) and /(Xi) have the same sign. 

[Apply the last formula in the second paragraph of 77 to each of the func 
tions /(x) and l//(x).] 

12*. Let U and V be two polynomials of degree n and n 1, respectively, 
which are prime to each other. Show that the index of the rational fraction 
V/U between the limits oo and -f oo is equal to the difference between the 
number of imaginary roots of the equation U + iV = in which the coefficient 
of i is positive and the number in which the coefficient of i is negative. 

[HERMITE, Bulletin de la Socidte matMmatique, Vol. VII, p. 128.] 

13*. Derive the second theorem of the mean for integrals by integration by 
parts. 



206 DEFINITE INTEGRALS [IV, Exs. 

[Let/(x) and <f>(x) be two functions each of which is continuous in the inter 
val (a, b) and the first of which, /(x), constantly increases (or decreases) and 
has a continuous derivative. Introducing the auxiliary function 

*(z) = f*<t>(x)dx 

J a 

and integrating by parts, we find the equation 

f /(x) </>(x) dx = f(b) *(&) - f / (x) *(x) dx . 

va v a 

Since / (x) always has the same sign, it only remains to apply the first theorem 
of the mean for integrals to the new integral.] 

14. Show directly that the definite integral fxdy ydx extended over a 
closed contour goes over into an integral of the same form when the axes are 
replaced by any other set of rectangular axes which have the same aspect. 

15. Given the formula 

/** 1 

I cos Xx dx = - (sin \b sin Xa) , 

J a X 

evaluate the integrals 

/^fc /ift 

i x 2 P + 1 sinXxdx, I x 2 ^cosXxdx. 

<J a J n 

16. Let us associate the points (x, y) and (x , y ) upon any two given curves 
C and C", respectively, at which the tangents are parallel. The point whose 
coordinates are x\ = px + qx , yi = py + qy , where p and q are given constants, 
describes a new curve C\. Show that the following relation holds between the 
corresponding arcs of the three curves : 

Si = ps qs . 

17. Show that corresponding arcs of the two curves 

c ( x = tf(t) - f(t) +t (t), C \ x = V (t) - f(t) - * () , 

have the same length whatever be the functions /(f) and <f>(t). 

18. From a point M of a plane let us draw the normals -MPi, , MP n to 
n given curves Ci, C*2, , C n which lie in the same plane, and let k be the 
distance MP ( . The locus of the points M, for which a relation of the form 
F(li, k) ) In) = holds between the n distances Z,-, is a curve T. If lengths 
proportional to cF/dli be laid off upon the lines MP^ respectively, according to 
a definite convention as to sign, show that the resultant of these n vectors gives 
the direction of the normal to F at the point M. Generalize the theorem for 
surfaces in space. 

19. Let C be any closed curve, and let us select two points p and p upon the 
tangent to C at a point TO, on either side of TO, making mp = mp . Supposing 
that the distance mp varies according to any arbitrary law as TO describes the 
curve (7, show that the points p and p describe curves of equal area. Discuss 
the special case where mp is constant. 



IV, Exs.] EXERCISES 207 

20. Given any closed convex curve, let us draw a parallel curve by laying oft 
a constant length I upon the normals to the given curve. Show that the area 
between the two curves is equal to it I 2 + s, where s is the length of the given 
curve. 

21. Let C be any closed curve. Show that the locus of the points A, for 
which the corresponding pedal has a constant area, is a circle whose center is 
fixed. 

[Take the equation of the curve C in the tangential form 

x cos t + y sin t = /().] 

22. Let C be any closed curve, C\ its pedal with respect to a point A, and C 2 
the locus of the foot of a perpendicular let fall from A upon a normal to C. 
Show that the areas of these three curves satisfy the relation A = AI A 2 . 

[By a property of the pedal ( 36), if p and u are the polar coordinates of a point 
on d, the coordinates of the corresponding point of C 2 are p and u + n/2, and 
those of the corresponding point of C are r = Vp 2 + p" 2 and <p = w + arc tan p /p.] 

23. If a curve C rolls without slipping on a straight line, every point A which 
is rigidly connected to the curve C describes a curve which is called a roulette. 
Show that the area between an arc of the roulette and its base is twice the area 
of the corresponding portion of the pedal of the point A with respect to C. Also 
show that the length of an arc of the roulette is equal to the length of the corre 
sponding arc of the pedal. ro 

[SXEINEK.] 

[In order to prove these theorems analytically, let X and Y be the coordi 
nates of the point A with respect to a moving system of axes formed of the 
tangent and normal at a point M on C. Let s be the length of the arc OM 
counted from a fixed point on C, and let w be the angle between the tangents 
at and M. First establish the formulae 



and then deduce the theorems from them.] 

24*. The error made in Gauss method of quadrature may be expressed in 
the form 

/ (2n >() x 2 r 1.2.. 3... 71 -ft 
1 . 2 2n 2n + 1 Ll . 2 (2n - 1)J 

where lies between 1 and +1. r ,, .-, 100 ., , 

[MANSION, Comptes renews, 1886.] 



CHAPTER V 
INDEFINITE INTEGRALS 

We shall review in this chapter the general classes of elemen 
tary functions whose integrals can be expressed in terms of ele 
mentary functions. Under the term elementary functions we shall 
include the rational and irrational algebraic functions, the exponen 
tial function and the logarithm, the trigonometric functions and 
their inverses, and all those functions which can be formed by a 
finite number of combinations of those already named. When the 
indefinite integral of a function f(x) cannot be expressed in terms 
of these functions, it constitutes a new transcendental function. 
The study of these transcendental functions and their classification 
is one of the most important problems of the Integral Calculus. 

I. INTEGRATION OF RATIONAL FUNCTIONS 

103. General method. Every rational function /(a:) is the sum of 
an integral function E(x) and a rational fraction P(x~)/Q(x"), where 
P(x) is prime to and of less degree than Q(#). If the real and 
imaginary roots of the equation Q(x~) be known, the rational frac 
tion may be decomposed into a sum of simple fractions of one or the 
other of the two types 

A MX + N 

(cc-a)" 1 [(x - a) 2 + /3 2 ] 

The fractions of the first type correspond to the real roots, those 
of the second type to pairs of imaginary roots. The integral of 
the integral function E(x) can be written down at once. The inte 
grals of the fractions of the first type are given by the formulae 

Adx A 

(x-a) (m-l)(a;-a) > 

A dx 

= A log (x a), if m = 1 . 

x a 

For the sake of simplicity we have omitted the arbitrary constant C, 
which belongs on the right-hand side. It merely remains to examine 

208 



V, 103] RATIONAL FLECTIONS 209 

the simple fractions which arise from pairs of imaginary roots. 
In order to simplify the corresponding integrals, let us make the 

substitution 

x = a -\- fit, dx ftdt. 

The integral in question then becomes 



r MX + N 1 CMa + N+MQt 

J [(x - a)* + W d - 0*- J (1+ * 2 ) 

L- 
and there remain two kinds of integrals : 

c tat r dt 

J (i + t 2 f J (i + a )" 

Since tdt is half the differential of 1 + t 2 , the first of these inte 
grals is given, if n > 1, by the formula 

r 

J 



tdt i _ __ /? 2 



or, if n 1, by the formula 
tdt 1 



The only integrals which remain are those of the type 

dt 



r 
J 



If n = 1, the value of this integral is 



dt x a 

= arc tan t = arc tan - 



1 + t* 

If n is greater than unity, the calculation of the integral may be 
reduced to the calculation of an integral of the same form, in which 
the exponent of (1 -f- 2 ) is decreased by unity. Denoting the inte 
gral in question by /, we may write 



/i -j- 1 2 t z r 
~7T 5T~ dt = I 
(1 + t )" J < 



From the last of these integrals, taking 
tdt 



210 INDEFINITE INTEGRALS [v, 103 

and integrating by parts, we find the formula 

C t*dt __ t __ 1 C 

J (i+O"~ 2(n-l)(l+<V- 1 2(-l)J 



dt 



Substituting this value in the equation for 7 n , that equation becomes 
_ 2n - 3 t 

n o O *n-l I 



2n - 2 2 (n -!)(! + * 2 )"~ 
Repeated applications of this formula finally lead to the integral 
= arc tan t. Retracing our steps, we find the formula 

(2n - 3) (2n - 5) - 3 . 1 





where 72 () is a rational function of which is easily calculated. 
We will merely observe that the denominator is (1 + 2 )"~ 1 , and that 
the numerator is of degree less than 2n 2 (see 97, p. 192). 

It follows that the integral of a rational function consists of 
terms which are themselves rational, and transcendental terms of 
one of the following forms : 

y _ f-f 

log (SB a), log [(x - a) 2 + /8 s ], arc tan - 

P 

Let us consider, for example, the integral /[l/(# 4 1)] dx. The 
denominator has two real roots -f 1 and 1, and two imaginary 
roots + i and i. We may therefore write 

1 A B Cx + 7) 





x* - 1 ~ x-l Z + l 

In order to determine A, multiply both sides by x 1 and then set 
x = 1. This gives A = 1/4, and similarly B = 1/4. The iden 
tity assumed may therefore be written in the form 

r.r + D 



or, simplifying the left-hand side, 

-1 _ Cx + D 

2(1 + x 3 ) ~ 1 + x 2 

It follows that C = and 7) = 1/2, and we have, finally, 
1 1 1 1 

T" A / ~t "I \ .^ / V 1-.- 11 */ ( nr* \ 1 ^ 

*^ -L V t !/ -Ly Trl*^ ^^ / \ 1^ / 

which gives 

rfa; 1 , (x - 1\ 1 



V, 104] RATIONAL FUNCTIONS 211 

Note. The preceding method, though absolutely general, is not 
always the simplest. The work may often be shortened by using 
a suitable device. Let us consider, for example, the integral 

dx 



2 - 1)" 

If n > 1, we may either break up the integrand into partial frac 
tions byliieans of the roots + 1 and 1, or we may use a reduction 
formula similar to that for /. But the most elegant method is to 
make the substitution x = (1 -f )/(! 2), which gives 

4* 2dz 

dx = 



(!-*) (I-*) 

/7 O /"* /1 ~\2 2 

(* - 1)- = 4"- J " * rf * 



Developing (1 z) 2n ~ 2 by the binomial theorem, it only remains 
to integrate terms of the form Az* t where \L may be positive or 
negative. 

104. Hermite s method. We have heretofore supposed that the 
fraction to be integrated was broken up into partial fractions, which 
presumes a knowledge of the roots of the denominator. The fol 
lowing method, due to Hermite, enables us to find the algebraic 
part of the integral without knowing these roots, and it involves 
only elementary operations, that is to say, additions, multiplications, 
and divisions of polynomials. 

Let f(x)/F(x) be the rational fraction which is to be integrated. 
We may assume that f(x) and F(x) are prime to each other, and 
we may suppose, according to the theory of equal roots, that the 
polynomial F(x) is written in the form 



where X l} A 2 , , A ^ are polynomials none of which have multiple 
roots and no two of which have any common factor. We may now 
break up the given fraction into partial fractions whose denomina 
tors are X lt X\, , X p p : 



X\ X* 



where ^4, is a polynomial prime to X t . For, by the theory of high 
est common divisor, if X and Y are any two polynomials which are 



212 INDEFINITE INTEGRALS [v, 104 

prime to each other, and Z any third polynomial, two other poly 
nomials A and B may always be found such that 

BX + AY= Z. 

Let us set X = X lt Y = X\ X p p , and Z =/(*). Then this identity 
becomes 

BXi + AX\--.Xl=f(x), 
or, dividing by F(x), 



It also follows from the preceding identity that if f(x) is prime to 
F(x~), A is prime to X 1 and B is prime to X\ X*. Kepeating the 
process upon the fraction 

B 



and so on, we finally reach the form given above. 

It is therefore sufficient to show how to obtain the rational part 
of an integral of the form 

/A dx 
~" 

where <(.x) is a polynomial which is prime to its derivative. Then, 
by the theorem mentioned above, we can find two polynomials B 
and C such that 



and hence the preceding integral may be written in the form 

f A_dx_ C B<}>+ T(V fBdx C $dx 

J 4>* ~J V ~~J #>"- 1+ J r f 

If n is greater than unity, taking 



u=C, v = 



and integrating by parts, we get 

C c 4Sdx = C 1 C C 

J 4? (n-\}r~ l n-lj fi-" 

whence, substituting in the preceding equation, we find the formula 

C A dx C C A^dx 

J p ~( W -1)^"- 1+ J -&=* 



V, 104] RATIONAL FUNCTIONS 213 

where A! is a new polynomial. If n > 2, we may apply the same 
process to the new integral, and so on : the process may always be 
continued until the exponent of < in the denominator is equal to 
one, and we shall then have an expression of the form 






A dx C \Ldx 

*(*> + J V 

where R(x) is a rational function of x, and ^ is a polynomial whose 
degree we may always suppose to be less than that of <f>, but which 
is not necessarily prime to <. To integrate the latter form we must 
know the roots of <, but the evaluation of this integral will intro 
duce no new rational terms, for the decomposition of the fraction 
\[//<f> leads only to terms of the two types 

A Mx + N 

x-a (z-a) 2 + /3 2 

each of which has an integral which is a transcendental function. 

This method enables us, in particular, to determine whether the 
integral of a given rational function is itself a rational function. 
The necessary and sufficient condition that this should be true is 
that each of the polynomials like ^ should vanish when the process 
has been carried out as far as possible. 

It will be noticed that the method used in obtaining the reduction formula 
for / is essentially only a special case of the preceding method. Let us now 
consider the more general integral 



(Ax* + 2Bx + C) 
From the identity 

A(Ax* + 2Bx + C)- (Ax + B) 2 = AC- B* 
it is evident that we may write 

dx 



C ** = A C 

J (Ax* + 2Bx + C) A C - B* J (Ax* 
-SJ 



(Ax + B) 



2Bx + C)"- 1 

(Ax + B)dx 



AC-B*J ^ (Ax* + 2Bx + C) 

Integrating the last integral by parts, we find 

Ax + B Ax + B 

(Ax -f B) dx = 

\ / A O i C\ Tt . y^v .. 



(Ax* + 2Bx + C) n 2(n- 



2n-2J (Ax* + 2Bx + C) n ~ 



214 INDEFINITE INTEGRALS [V, 104 

whence the preceding relation becomes 



Ax + B 



(Ax* + 2Bx + C) 2(n- \)(AC - B*)(Ax* + 2Bx 
2n-3 



^ /* 

- 2 J (4z* + 2 



2n - 2 AC - 2 J ( Ax i + 2 X + C] 
Continuing the same process, we are led eventually to the integral 

dx 
^Ix 2 + 2Bx + C 

which is a logarithm if B 2 - AC>0, and an arctangent if B 2 - AC<0. 
As another example, consider the integral 

C 5x 3 + 3x - 1 . 
dx. 



J (x + 3x+ , 
From the identity 

5x 3 + 3x - 1 = 6x(x 2 + 1) - (x 3 + 3x + 1) 
it is evident that we may write 

/5x 8 + 3x 1 C 6x(x 2 + l) C 

(x + 3x + l)3 dx =J ^^3x + l)* dX -J (xM 



dx 



+ OX + 

Integrating the first integral on the right by parts, we find 

dx 



C x 6 ( g8 + i)tfg = -x r 

J (X 3 + 3X + I) 3 ~ (X 3 + 3X + 1)2 + J ( 



(x 8 + 3x + 1)2 
whence the value of the given integral is seen to be 

5x 8 + 3x - 1 . - z 

-dx = 



Note. In applying Hermite s method it becomes necessary to solve the fol 
lowing problem : given three polynomials A, B, C, of degrees m, n, p, respectively, 
two of which, A and B, are prime to each other, find two other polynomials u and v 
such that the relation Au + Bv = C is identically satisfied. 

In order to determine two polynomials u and v of the least possible degree 
which solve the problem, let us first suppose that p is at most equal to m + n- 1. 
Then we may take for u and v two polynomials of degrees n - I and m - 1, 
respectively. The m + n unknown coefficients are then given by the system of 
m + n linear non-homogeneous equations found by equating the coefficients. 
For the determinant of these equations cannot vanish, since, if it did, we could 
find two polynomials u and v of degrees n - 1 and m - 1 or less which satisfy 
the identity Au + Bv = 0, and this can be true only when A and B have a 
common factor. 

If the degree of C is equal to or greater than m + n, we may divide C by AB 
and obtain a remainder C" whose degree is less than m + n. Then C = A BQ + C , 
and, making the substitution u - BQ = MI, the relation Au + Bv = C reduces to 
Aui + Bv C . This is a problem under the first case. 



V, 105] RATIONAL FUNCTIONS 215 



105. Integrals of the type /R(x, ^Ax? + 2Bx + c) dx. After the 
integrals of rational functions it is natural to consider the inte 
grals of irrational functions. We shall commence with the case in 
which the integrand is a rational function of x and the square root 
of a polynomial of the second degree. In this case a simple substitu 
tion eliminates the radical and reduces the integral to the preceding 
case. This substitution is self-evident in case the expression under 
the radical is of the first degree, say ax + b. If we set ax + b = t*, 
the integral becomes 

Cll(x, ^ax^b]dx = /V , t] > 
J J \ a J a 

and the integrand of the transformed integral is a rational function. 
If the expression under the radical is of the second degree and 
has two real roots a and b, we may write 



A(x-a)(x-b) = (x-b) 
and the substitution 



Aa - bt 2 

or x = - > 
A - t* 



actually removes the radical. 

If the expression under the radical sign has imaginary roots, the 
above process would introduce imaginaries. In order to get to the 



bottom of the matter, let y denote the radical ^Ax 2 -f 2Bx + C. 
Then x and y are the coordinates of a point of the curve whose 
equation is 

(1) y 2 = Ax 2 + 2Bx + C, 

and it is evident that the whole problem amounts to expressing the 
coordinates of a point upon a conic by means of rational functions 
of a parameter. It can be seen geometrically that this is possible. 
For, if a secant 

y - J3 = t(x - a) 

be drawn through any point (a, /3) on the conic, the coordinates of 
the second point of intersection of the secant with the conic are 
given by equations of the first degree, and are therefore rational 
functions of t. 

If the trinomial Ax 2 + 2Bx + C has imaginary roots, the coeffi 
cient A must be positive, for if it is not, the trinomial will be 
negative for all real values of x. In this case the conic (1) is an 



216 INDEFINITE INTEGRALS [V, 105 

hyperbola. A straight line parallel to one of the asymptotes of 
this hyperbola, 

y = x Vyi + t, 

cuts the hyperbola in a point whose coordinates are 

C - t* r~ C - t* 



2B 2t A - 2B 

If A < 0, the conic is an ellipse, and the trinomial A x 2 + 2Bx + C 
must have two real roots a and b, or else the trinomial is negative 
for all real values of x. The change of variable given above is pre 
cisely that which we should obtain by cutting this conic by the 
moving secant 

y = t(x a) . 

As an example let us take the integral 



(x 2 + k) Vz 2 + k 

The auxiliary conic if = x 2 + k is an hyperbola, and the straight line 
x + y = t, which is parallel to one of the asymptotes, cuts the hyper 
bola in a point whose coordinates are 



Making the substitution indicated by these equations, we find 

_ dt ft* + k\ C dx 4tdt 2 

= ~ 



C dx _ 
J 7~ 



* 

or, returning to the variable x, 

dx _ x Vcc 2 + k _ x 1 

where the right-hand side is determined save for a constant term 
In general, if A C B 2 is not zero, we have the formula 

Ix 1 Ax + B 



(Ax 2 + 2Bx + Cy AC B 2 VAx 2 + 2Bx + C 

In some cases it is easier to evaluate the integral directly without 
removing the radical. Consider, for example, the integral 

dx 



+ 2Bx + C 



V, 105] RATIONAL FUNCTIONS 217 

ff the coefficient A is positive, the integral may be written 
^/Adx C VJdx 



r ^/Adx _ C 

J ^A*x 2 + 2ABx + AC J 



x + ) 2 + AC - B* 
or setting Ax + B = t, 

dt 



i r 

^AJ - 



t-AC B* -VA 

Returning to the variable x, we have the formula 

dx 1 

f 2Bx + C 



B +A Ax* + 2Bx 



If the coefficient of x 2 is negative, the integral may be written in 
the form 



/7 /* 

doc I 

V- Ax 2 + 2Bx + C J VJ 



B 2 - (Ax - BY 

The quantity A C + B 2 is necessarily positive. Hence, making the 
substitution 

Ax - B = t ^/A C + B 2 , 

the given integral becomes 

dt I 



_ r 

I J 



V.4 J Vl - t 2 V.4 
Hence the formula in this case is 



dx 1 Ax B 

= -^= arc sin 



V- Ax 2 + 2Bx + C 

It is easy to show that the argument of the arcsine varies from 1 
to + 1 as x varies between the two roots of the trinomial. 

In the intermediate case when ^4=0 and B = 0, the integral is 
algebraic : 



f 

J 



Integrals of the type 

dx 



(x - a) V^x 2 + 2Bx + C 



218 



INDEFINITE INTEGRALS 



[V, 100 



reduce to the preceding type by means of the substitution x = a -f 1/y. 
We find, in fact, the formula 



r dx r 

J (x a) -^/Ax* + 2Bx + C J 



dy 



where 



A ! = Aa 2 + 2Ba + C, B l = Aa + B, 



It should be noticed that this integral is algebraic if and only if 
the quantity a is a root of the trinomial under the radical. 

Let us now consider the integrals of the type f Va: 2 + A dx. Inte 
grating by parts, we find 



rVa;" + A 



+ A dx = x Va: 2 + A 

On the other hand we have 

fx 2 dx C I. r Adx 

. = I Va; 2 -f A dx I - 

Va: 2 + A J J Va; 2 + J 

= / Va: 2 -\-Adx-A log (x + Va: 2 + A ) . 
From these two relations it is easy to obtain the formulae 

c 2 + A + - log (or + Va: 2 + ^), 



(2) 



C 

/ 
J 



7 2 

+ A * 

The following formulae may be derived in like manner: 



(5) 



x*dx 



arc sin 
z a 



106. Area of the hyperbola. The preceding integrals occur in the evaluation 
of the area of a sector of an ellipse or an hyperbola. Let us consider, for 
example, the hyperbola 



V, 106] 



RATIONAL FUNCTIONS 



219 



and let us try to find the area of a segment AMP bounded by the arc AM, the 
x axis, and the ordinate MP. This area is equal to the definite integral 



6 



that is, by the formula (2), 



- Vx 2 a 2 dx , 
a 



a 2 - a 2 log (?- 



But MP = y = (6/a) Vx 2 a 2 , and the term (b/2a) x Vx 2 a 2 is precisely the 
area of the triangle OMP. Hence the area S of the sector 0AM, bounded by 
the arc AM and the radii vectores OA 
and OM, is 



S = 1 e* loj 

2 



/x + Vx 2 - a 2 \ 

I I 

V a ) 



1 . , x 
= - oft log ( - 

2 \a 



This formula enables us to express 
the coordinates x and y of a, point M 
of the hyperbola in terms of the area S. 
In fact, from the above and from the 
equation of the hyperbola, it is easy to 
show that 




FIG. 21 



h 

y = - (e" b e 



The functions which occur on the right-hand side are called the hyperbolic 
cosine and sine : 

e* + e~ x 



cosh x = 



sinh x = 



2 2 

The above equations may therefore be written in the form 

2S 



x = a cosh 



ab 



, . . 

y = b sinh 
ab 



These hyperbolic functions possess properties analogous to those of the trigo 
nometric functions.* It is easy to deduce, for instance, the following formulae : 

cosh 2 x sinh 2 x = 1, 

cosh (x + y) = cosh x cosh y + sinh x sinh y, 
sinh (x + y) = sinh x cosh y + sinh y cosh x. 

* A table of the logarithms of these functions for positive values of the argument 
is to be found in HoueTs Recueil desformules numeriques. 



220 



INDEFINITE INTEGRALS 



[V, 107 



It may be shown in like manner that the coordinates of a point on an ellipse 
may be expressed in terms of the area of the corresponding sector, as follows : 



2S 

x = a cos , 
ab 



, . 2S 
y = b sin 
ao 



In the case of a circle of unit radius, and in the case of an equilateral hyperbola 
whose semiaxis is one, these formulae become, respectively, 

x = cos2S, 7/ = sin2-S; 

z = cosh2S, y = sinh2/S. 

It is evident that the hyperbolic functions bear the same relations to the equi 
lateral hyperbola as do the trigonometric functions to the circle. 

107. Rectification of the parabola. Let us try to find the length of the arc of 
a parabola 2py = x 2 between the vertex and any point M. The general 
formula gives 



. pj7w^y d z = r^sv 

Jo * W J P 



or, applying the formula (2), 



2p 2 - \ p 

The algebraic term in this result is precisely the length M T of the tangent, 
for we know that OT = x/2, and hence 

x 2 x* x 2 x 2 (x 2 -4- v^) 
4 4p* 4 4p 2 

If we draw the straight line connecting T to the focus F, the angle MTF will 

be a right angle. Hence we 
have 

FT 




V n 2 a-2 1 
H.J = I 



/ whence we may deduce a curi 

ous property of the parabola. 

Suppose that the parabola 
rolls without slipping on the x 
axis, and let us try to find the 
i , " 7"" locus of the focus, which is sup 
posed rigidly connected to the 
parabola. When the parabola 
is tangent at M to the x axis, OM = arc OM. The point T has come into a 
position T such that M T = 3/T, and the focus F is at a point F which is 
found by laying off T F = TF on a line parallel to the y axis. The coordi 
nates X and Y of the point F are then 



T T 
FIG. 22 



V,108] RATIONAL FUNCTIONS 221 

and the equation of the locus is given by eliminating x between these two equa 
tions. From the first we find 

! 

x + Vx 2 + p 2 = pe P , 

to which we may add the equation 



_ _ 

x Vx 2 + j> 2 = pe P 

since the product of the two left-hand sides is equal to p*. Subtracting these 
two equations, we find 



and the desired equation of the locus is 



2 p 

This curve, which is called the catenary, is quite easy to construct. Its form 
is somewhat similar to that of the parabola. 

108. Unicursal curves. Let us now consider, in general, the inte 
grals of algebraic functions. Let 

(6) F(x, y) = 

be the equation of an algebraic curve, and let R(x, y) be a rational 
function of x and y. If we suppose y replaced by one of the roots 
of the equation (6) in R(x, y), the result is a function of the single 
variable x, and the integral 



is called an Abelian integral with respect to the curve (6). When 
the given curve and the function R(x, y) are arbitrary these inte 
grals are transcendental functions. But in the particular case where 
the curve is unicursal, i.e. when the coordinates of a point on the 
curve can be expressed as rational functions of a variable param 
eter t, the Abelian integrals attached to the curve can be reduced at 
once to integrals of rational functions. For, let 



be the equations of the curve in terms of the parameter t. Taking 
t as the new independent variable, the integral becomes 



j 



R(x, y}dx = 
and the new integrand is evidently rational. 



222 INDEFINITE INTEGRALS [v, 108 

It is shown in treatises on Analytic Geometry* that every uni- 
cursal curve of degree n has (n l)(n 2)/2 double points, and, 
conversely, that every curve of degree n which has this number of 
double points is unicursal. I shall merely recall the process for 
obtaining the expressions for the coordinates in terms of the param 
eter. Given a curve C B of degree n, which has 8 = (n !)(. 2)/2 
double points, let us pass a one-parameter family of curves of degree 
n 2 through these 8 double points and through n 3 ordinary points 
on C a . These points actually determine such a family, for 






- 1, 



whereas (n 2)(n -(-l)/2 points are necessary to determine uniquely 
a curve of order n 2. Let P(x, y) + tQ(x, ?/) = be the equation 
of this family, where t is an arbitrary parameter. Each curve of the 
family meets the curve C n in n(n 2) points, of which a certain num 
ber are independent of t, namely the n 3 ordinary points chosen 
above and the 8 double points, each of which counts as two points of 
intersection. But we have 

- 3 + 28 = n - 3 + (ft -l)(n - 2) = n(n - 2) -1, 

and there remains just one point of intersection which varies with t. 
The coordinates of this point are the solutions of certain linear equa 
tions whose coefficients are integral polynomials in t, and hence they 
are themselves rational functions of t. Instead of the preceding we 
might have employed a family of curves of degree n 1 through the 
(n l)(w 2)/2 double points and 2n 3 ordinary points chosen at 
pleasure on C n . 

If n = 2, (n l)(w 2)/2 = 0, every curve of the second 
degree is therefore unicursal, as we have seen above. If n = 3, 
(n l)(w 2)/2 = 1, the unicursal curves of the third degree 
are those which have one double point. Taking the double point 
as origin, the equation of the cubic is of the form 

4>s (x, y) + fa (x, y) = , 

where < 3 and < 2 are homogeneous polynomials of the degree of their 
indices. A secant y = tx through the double point meets the cubic 
in a single variable point whose coordinates are 

(!, Q <&(!, Q 

<MM) " *i(M) 

*See, e.g., Niewenglowski, Cours de Geometric analytique, Vol. II, pp. 99-114. 



V, 108] .RATIONAL FUNCTIONS 223 

A unicursal curve of the fourth degree has three double points. 
In order to find the coordinates of a point on it, we should pass a 
family of conies through the three double points and through another 
point chosen at pleasure on the curve. Every conic of this family 
would meet the quartic in just one point which varies with the 
parameter. The equation which gives the abscissae of the points of 
intersection, for instance, would reduce to an equation of the first 
degree when the factors corresponding to the double points had 
been removed, and would give x as a rational function of the 
parameter. We should proceed to find y in a similar manner. 

As an example let us consider the lemniscate 



which has a double point at the origin and two others at the imagi 
nary circular points. A circle through the origin tangent to one of 
the branches of the lemniscate, 

x* + y 2 = t(x - y} , 

meets the curve in a single variable point. Combining these two 
equations, we find 



or, dividing by x y, 



This last equation represents a straight line through the origin which 
cuts the circle in a point not the origin, whose coordinates are 

_ o a *(< 3 + a 2 ) _ a 2 t(t 2 - a 2 ) 

t* + a* t* + a< 

These results may be obtained more easily by the following 
process, which is at once applicable to any unicursal curve of the 
fourth degree one of whose double points is known. The secant 
y = \x cuts the lemniscate in two points whose coordinates are 



The expression under the radical is of the second degree. Hence, 
by 105, the substitution (1 - X)/(l + A) = (a/t) 2 removes the radi 
cal. It is easy to show that this substitution leads to the expressions 
just found. 



224 INDEFINITE INTEGRALS [V, 109 

Note. When a plane curve has singular points of higher order, it 
can be shown that each of them is equivalent to a certain number of 
isolated double points. In order that a curve be unicursal, it is suffi 
cient that its singular points should be equivalent to (n l)(n 2)/2 
isolated double points. For example, a curve of order n which has 
a multiple point of order n 1 is unicursal, for a secant through 
the multiple point meets the curve in only one variable point. 

109. Integrals of binomial differentials. Among the other integrals 
in which the radicals can be removed may be mentioned the follow 
ing types : 

/ R\_x, (ax + b) 1 \dx , I R(X, ~vax -f- b, ~V ex + d)dx, 

R(x a , x a \ x a ", -)dx, 



where R denotes a rational function and where the exponents 
a, a , a", are commensurable numbers. For the first type it is 
sufficient to set ax -f b = t q . In the second type the substitution 
ax + b = t* leaves merely a square root of an expression of the 
second degree, which can then be removed by a second substitution. 
Finally, in the third type we may set x = t D , where D is a common 
denominator of the fractions n-, a , a", 

In connection with the third type we may consider a class of 
differentials of the form 



which are called binomial differentials. Let us suppose that the 
three exponents m, n, p are commensurable. If p is an integer, the 
expression may be made rational by means of the substitution 
x = t D , as we have just seen. In order to discover further cases 
of integrability, let us try the substitution ax n + b t. This gives 



\ x (ax n + by dx = |W^ -H 

J naj \ a / 



dx 

\ a / 

" dt. 



The transformed integral is of the same form as the original, and 
the exponent which takes the place of p is (m + 1) /n 1. Hence 
the integration can be performed if (m + l)/w is an integer. 



V,109] RATIONAL FUNCTIONS 225 

On the other hand, the integral may be written in the form 



whence it is clear that another case of integrability is that in which 
(m + np 4- l)/w = (m + l)/w + p is an integer. To sum up, the 
integration can be performed whenever one of the three numbers 
2^ (in +l)/n, (m+V)/n +p is an integer. In no other case can the 
integral be expressed by means of a finite number of elementary 
functional symbols when m, n, and p are rational. 

In these cases it is convenient to reduce the integral to a simpler 
form in which only two exponents occur. Setting ax n = bt, we find 

/As l 1/Asl-, 

x = (- } t n , dx = - - ) t n dt, 

a n a 



IP /*\!ii r " 1 + 1 i 

x m (ax n + bydx = - - " lt n 
n \a/ J 



Neglecting the constant factor and setting q = (m + l)/n 1, we 

are led to the integral 

/ 

tydt. 

The cases of integrability are those in which one of the three num 
bers p, q, p + q is an integer. If p is an integer and q = r/s, we 
should set t u s . If q is an integer and p r/s, we should set 
l+t = u*. Finally, if p + q is an integer, the integral may be 
written in the form 



and the substitution 1 + t = tu s , where p r/s, removes the radical. 
As an example consider the integral 



x Vl + x 3 dx . 



Here m = 1, n = 3, p = 1/3, and (m + l)/w + p = 1. Hence this 
is an integrable case. Setting x 3 = t, the integral becomes 



dt, 
and a second substitution 1 + t = tu 8 removes the radical. 



226 INDEFINITE INTEGRALS [V, 110 



II. ELLIPTIC AND HYPERELLIPTIC INTEGRALS 

110. Reduction of integrals. Let P(x) be an integral polynomial 
of degree/? which is prime to its derivative. The integral 



where R denotes a rational function of x and the radical y Vp(ce), 
cannot be expressed in terms of elementary functions, in general, 
when p is greater than 2. Such integrals, which are particular 
cases of general Abelian integrals, can be split up into portions which 
result in algebraic and logarithmic functions and a certain number 
of other integrals which give rise to new transcendental functions 
which cannot be expressed by means of a finite number of elemen 
tary functional symbols. We proceed to consider this reduction. 

The rational function R(x, y) is the quotient of two integral 
polynomials in x and y. Replacing any even power of y, such as 
y 2q > by [P(V)] 9 > and any odd power, such as y* g + l , by y [_P(x)~] q , we 
may evidently suppose the numerator and denominator of this frac 
tion to be of the first degree in y, 

A + By 

R(x, in > 

v J) C + Dy 

where A, B, C, D are integral polynomials in x. Multiplying the 
numerator and the denominator each by C Dy, and replacing y 2 
by P(x), we may write this in the form 

R(x, y} -- 



K 



where F, G, and K are polynomials. The integral is now broken 
up into two parts, of which the first JF/K dx is the integral of a 
rational function. For this reason we shall consider only the second 
integral fOy/K dx, which may also be written in the form 



fpft 

where M and N are integral polynomials in x. The rational frac 
tion M/N may be decomposed into an integral part E(x) and a 
sum of partial fractions 



V,lio] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 227 

where each of the polynomials A ,- is prime to its derivative. We 
shall therefore have to consider two types of integrals, 



/x 
V 



l dx C Adx 




If the degree of P(x) is p, all the integrals Y m may be expressed 
in terms of the first p 1 of them, F , Y i} -, Y p _ 2 , and certain 
algebraic expressions. 

For, let us write 

P(X) = a x p + a^x* + 
It follows that 

~JZ( A 



2 VP(Z) 
_ 2mx m - 1 P(x) + x m P (x) 
2 



The numerator of this expression is of degree m + p 1, and its 
highest term is (2m + p~)a x m+p ~ l . Integrating both sides of the 
above equation, we find 

2x Vp(xj=(2m+p}a Y m+p _ l + .--, 

where the terms not written down contain integrals of the type 
Y whose indices are less than m + p 1. Setting m = 0, 1, 2, , 
successively, we can calculate the integrals Y p _ l} Y p , succes 
sively in terms of algebraic expressions and the p 1 integrals 
YO> YI, , Y p _ 2 . 

With respect to the integrals of the second type we shall distin 
guish the two cases where X is or is not prime to P(x)> 

1) If X is prime to P(x), the integral Z n reduces to the sum of 
an algebraic term, a number of integrals of the type Y k , and a new 
integral 

B dx 



f- 



X V P(x) 
where B is a polynomial whose degree is less than that of X. 

Since X is prime to its derivative X and also to P(ar), X n is prime 
to PA". Hence two polynomials A and fi can be found such that 
XX" + p.X P = A, and the integral in question breaks up into 
two parts: 



f ii== f *^_ i c 

J X n ^P(x) J VP(X) J 



X n 



228 INDEFINITE INTEGRALS [V, 110 

The first part is a sum of integrals of the type Y. In the second 
integral, when n > 1, let us integrate by parts, taking 
1 

f I ~\./ T-) *3/ tt TT~" 

** (n -I)*- 1 

which gives 

r^px dx = -pV7> l i rwp + pp* ^ 

J X n (n-l)X n ~ l n-lJ 2J"- 1 VP(a:) 

The new integral obtained is of the same form as the first, except 
that the exponent of X is diminished by one. Repeating this 
process as often as possible, i.e. as long as the exponent of X is 
greater than unity, we finally obtain a result of the form 




P(x) J .YVP J VP X n ~ 

where B, C, D are all polynomials, and where the degree of B may 
always be supposed to be less than that of X. 

2) If X and P have a common divisor D, we shall have X = YD, 
P = SD, where the polynomials D, S, and Y are all prime to each 
other. Hence two polynomials X and /u. may be found such that 
A = XD n + p.Y", and the integral may be written in the form 

/Adx C \dx C fidx 
jpVp J r n Vp J D n 



The first of the new integrals is of the type just considered. The 
second integral, 



where D is a factor of P, reduces to the sum of an algebraic term 
and a number of integrals of the type Y. 

For, since D n is prime to the product D S, we can find two poly 
nomials A! and //,! such that X^" -f piD S = /x. Hence we may write 



C dx_ = C\dx + Cr^ 
J > n VP J VP J D" 



VP 

Replacing P by DS, let us write the second of these integrals in the 
form 

D 



V,110] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 229 

and then integrate it by parts, taking 



-11 

V - - 



which gives 

Cjidx^ r, | 

J D"VP J 



This is again a reduction formula ; but in this case, since the expo 
nent n 1/2 is fractional, the reduction may be performed even 
when D occurs only to the first power in the denominator, and we 
finally obtain an expression of the form 



C p.dx _ KVP C 
J D n VP "fP V 



Hdx 



where H and K are polynomials. 

To sum up our results, we see that the integral 

M dx 



can always be reduced to a sum of algebraic terms and a number of 
integrals of the two types 



/rpftl ///) / "V /7 Tf 

JO \JiJCi -**-\ UJU 

VP J xVp 



where ra is less than or equal to p 2, where X is prime to its 
derivative X and also to P, and where the degree of ^ is less than 
that of X. This reduction involves only the operations of addition, 
multiplication, and division of polynomials. 

If the roots of the equation X = are known, each of the rational 
fractions X l /X can be broken up into a sum of partial fractions of 

the two forms 

A Bx + C 

x-a (x-af + p* 

where A, B, and C are constants. This leads to the two new types 



r dx C (Bx 

J x - a VP(V> J x - a) 2 



(x - a) VP(V> J [_(x - a) 2 + /3 2 

which reduce to a single type, namely the first of these, if we agree 
to allow a to have imaginary values. Integrals of this sort are 



230 INDEFINITE INTEGRALS [V, 110 

called integrals of the third kind. Integrals of the type Y m are 
called integrals of the first kind when ra is less than p/2 1, and 
are called integrals of the second kind when m is equal to or greater 
than p/2, 1. Integrals of the first kind have a characteristic 
property, they remain finite when the upper limit increases 
indefinitely, and also when the upper limit is a root of P(x) 
( 89, 90); but the essential distinction between the integrals of 
the second and third kinds must be accepted provisionally at this 
time without proof. The real distinction between them will be 
pointed out later. 

Note. Up to the present we have made no assumption about the 
degree p of the polynomial P(x). If p is an odd number, it may 
always be increased by unity. For, suppose that P(x) is a poly 
nomial of degree 2q 1 : 

P(x) = A x*- 1 + A.x^-* + .-. + A. 2q _ v . 

Then let us set x = a -f- 1/y, where a is not a root of P(z). This 
gives 

I p(2<,-l)( fl \ 1 P,((/\ 

P(x)=P(a) + P (a) i + ... + I__Lii -i-j = fJ-134, 
y (2q - 1)! y- < 1 i/ 2 " 

where P^ (y) is a polynomial of degree 2q. Hence we have 



and any integral of a rational function of x and Vp(a-) is trans 
formed into an integral of a rational function of y and \/P l (y). 

Conversely, if the degree of the polynomial P(x) under the radi 
cal is an even number 2q, it may be reduced by unity provided a 
root of P(x) is known. For, if a is a root of P(x), let us set 
x = a -f- 1/y. This gives 



y (2?)! y y 

where Pi(y) is of degree 2y 1, and we shall have 




Hence the integrand of the transformed integral will contain no 
other radical than 



V, lllj ELLIPTIC AND IIYPERELLIPTIC INTEGRALS 231 

111. Case of integration in algebraic terms. We have just seen that an integral 
of the form 

C R[X, VP(x)]dx 

can always be reduced by means of elementary operations to the sum of an inte 
gral of a rational fraction, an algebraic expression of the form G VP(x)/L, and 
a number of integrals of the first, second, and third kinds. Since we can also 
find by elementary operations the rational part of the integral of a rational 
fraction, it is evident that the given integral can always be reduced to the form 



]dx = F[x, VP(xj] + T, 

where F is a rational function of x and VP(x), and where T is a sum of inte 
grals of the three kinds and an integral fXi /Xdx, X being prime to its deriva 
tive and of higher degree than X\ . Liouville showed that if the given integral 
is integrable in algebraic terms, it is equal to F[x, VP(x)]. We should there 
fore have, identically, 



R[x, VP(x)~\ = ~ 



and hence T = 0. 



Hence we can discover by means of multiplications and divisions of polynomials 
whether a given integral is integrable in algebraic terms or not, and in case it is, 
the same process gives the value of the integral. 

112. Elliptic integrals. If the polynomial P(x) is of the second 
degree, the integration of a rational function of x and P(x) can be 
reduced, by the general process just studied, to the calculation of the 
integrals 

/dx C dx 

VP(z) J (x - a)VP(z) 

which we know how to evaluate directly ( 105). 

The next simplest case is that of elliptic integrals, for which P(x) 
is of the third or fourth degree. Either of these cases can be 
reduced to the other, as we have seen just above. Let P(x) be a 
polynomial of the fourth degree whose coefficients are all real and 
whose linear factors are all distinct. We proceed to show that 
a real substitution can always be found which carries P(x~) into a 
polynomial each of whose terms is of even degree. 

Let a, b, c, d be the four roots of P(x). Then there exists an 
involutory relation of the form 

(7) Lx x" + M(x + x") + N = 0, 



232 INDEFINITE INTEGRALS [V, 112 

which is satisfied by x = a, x" b, and by x = c, x" = d. For the 
coefficients L, M, N need merely satisfy the two relations 

Lab + M(a + b) -f N 0, 
Led + M(c + d) + N = 0, 

which are evidently satisfied if we take 

L = a -+- b c d, M = cd ab , N = ab (c + d) cd (a + b). 

Let a and ft be the two double points of this involution, i.e. the 
roots of the equation 

Zt* 2 + 2ATw + N = 0. 

These roots will both be real if 

(cd - ab)*-(a + b - c - d) [ai (c + d) - cd(a + J)] > 0, 
that is, if 

(8) (a-c)(a-d)(b-c)(b-d)>Q. 

The roots of P(x) can always be arranged in such a way that this 
condition is satisfied. If all four roots are real, we need merely 
choose a and b as the two largest. Then each factor in (8) is positive. 
If only two of the roots are real, we should choose a and b as the real 
roots, and c and d as the two conjugate imaginary roots. Then the 
two factors a c and a d are conjugate imaginary, and so are the 
other two, b c and b d. Finally, if all four roots are imaginary, 
we may take a and b as one pair and c and d as the other pair of 
conjugate imaginary roots. In this case also the factors in (8) are 
conjugate imaginary by pairs. It should also be noticed that these 
methods of selection make the corresponding values of L, M, N real. 
The equation (7) may now be written in the form 

(9} ~- 4- x "~ a = 

x > _ ft + X " - (3 

If we set (x a)/(x ft~) = y, or x = (ft// <*)/(y 1), we find 



where P\(y) is a new polynomial of the fourth degree with real 
coefficients whose roots are 

a a b a c a d a 

a-p b- ft c- ft d- ft 

It is evident from (9) that these four roots satisfy the equation 



V, 112] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 233 

y -f- y" = by pairs ; hence the polynomial / i(y) contains no term 
of odd degree. 

If the four roots , b, c, d satisfy the equation a + b = c + d, we 
shall have L = 0, and one of the double points of the involution lies 
at infinity. Setting a N/2M, the equation (7) takes the form 

x a + x" a 0, 

and we need merely set x = a + y in order to obtain a polynomial 
which contains no term of odd degree. 

We may therefore suppose P(x) reduced to the canonical form 



It follows that any elliptic integral, neglecting an algebraic term 
and an integral of a rational function, may be reduced to the sum 
of integrals of the forms 



dx C xdx C 

^AtX*+AiX*+Ai J ^/A^+AiXt+At J 



x*dx 

^/A^+AiXt+At J ^A 9 x*+A l 
and integrals of the form 



/; 



dx 



(x a 
The integral 

dx 



is the elliptic integral of the first kind. If we consider x, on the 
other hand, as a function of u, this inverse function is called an 
elliptic function. The second of the above integrals reduces to an 
elementary integral by means of the substitution x 2 = u. The third 

integral 

x 2 dx 



is Legendre s integral of the second kind. Finally, we have the 
identity 

/dx _ C xdx C dx 

(x - a)Vp(x) ~J (x 2 - a 2 )VP(x) V (x 2 - a 2 ) VP(^) 



The integral 

dx 



(x 2 + /i) VJ x 4 + /Ijx 2 + A. 4 
is Legendre s integral of the third kind. 



234 INDEFINITE INTEGRALS [V, 113 

These elliptic integrals were so named because they were first 
met with in the problem of rectifying the ellipse. Let 

x = a cos <f> , y = b sin <f> 
be the coordinates of a point of an ellipse. Then we shall have 

ds 2 = dx 2 + dy* = (a 2 sin 2 < + t>* cos 2 <) d<j> 2 , 
or, setting a 2 b 3 = e 2 a 2 , 

ds = a Vl e 2 cos 2 < d<f> . 

Hence the integral which gives an arc of the ellipse, after the sub 
stitution cos < = t, takes the form 




It follows that the arc of an ellipse is equal to the sum of an inte 
gral of the first kind and an integral of the second kind. 
Again, consider the lemniscate defined by the equations 



y = a" 



. 
t* + a 4 t 4 + a 4 

An easy calculation gives the element of length in the form 



a 
ds 2 = dx 2 + dif = - - - 

t ~p Ct 



dt*. 



Hence the arc of the lemniscate is given by an elliptic integral of 
the first kind.* 

113. Pseudo-elliptic integrals. It sometimes happens that an integral of the 
form f F[x, VP(x)] dx, where P(x) is a polynomial of the third or fourth 
degree, can be expressed in terms of algebraic functions and a sum of a finite 
number of logarithms of algebraic functions. Such integrals are called pseudo- 
elliptic. This happens in the following general case. Let 

(10) Lx x" + M(x + x") + N=0 

be an involutory relation which establishes a correspondence between two pairs of 
the four roots of the quartic equation P(x) = 0. If the function f(x) be such that 
the relation 

(ID 

Lx 
is identically satisfied, the integral /[/(x)/VP(x)] dx is pseudo-elliptic. 

* This is a common property of a whole class of curves discovered by Serret 
(Cours de Calcul differentiel et integral, Vol. II, p. 264). 



V, 113] ELLIPTIC AND HYPERELLIPTIC INTEGRALS 235 

Let a and p be the double points of the involution. As we have already 
seen, the equation (10) may be written in the form 



(12) 



x - /3 x" - p 
Let us now make the substitution (z a)/(x P) = y. This gives 

(a-^, P(2) = ^L, 

(l-y) a (1-y)* 

and consequently 

dx _ ( p) dy 



where PI (y) is a polynomial of the fourth degree which contains no odd powers 
of y ( 112). On the other hand, the rational fraction f(x) goes over into a 
rational fraction 4>(y), which satisfies the identity <p(y) + <( y) = 0. For if 
two values of x correspond by means of (12), they are transformed into two 
values of y, say y and y" , which satisfy the equation y + y" = 0. It is evident 
that 4>(y) is of the form y^(y z ), where ^ is a rational function of y 2 . Hence 
the integral under discussion takes the form 



and we need merely set y 2 = z in order to reduce it to an elementary integral. 
Thus the proposition is proved, and it merely remains actually to carry out 
the reduction. 

The theorem remains true when the polynomial P(x) is of the third degree, 
provided that we think of one of its roots as infinite. The demonstration is 
exactly similar to the preceding. 

If, for example, the equation P(x) = is a reciprocal equation, one of the 
involutory relations which interchanges the roots by pairs is x x" = 1. Hence, 
if f(x) be a rational function which satisfies the relation /(x) + /(1/x) = 0, 
the integral /[/()/ VP(x)] dx is pseudo-elliptic, and the two substitutions 
(x l)/(z + 1) = y, y 2 = z, performed in order, transform it into an elementary 
integral. 

Again, suppose that P(x) is a polynomial of the third degree, 



Let us set a = o>, b = 0, c = 1, d = I/A; 2 . There exist three involutory rela 
tions which interchange these roots by pairs : 



l-Jk 2 x" 
Hence, if /(x) be a rational function which satisfies one of the identities 



236 INDEFINITE INTEGRALS [V, 114 

the integral 

f(z)dx 



Vx(l- )(! 

is pseudo-elliptic. From this others may be derived. For instance, if we set 
z = 2 2 , the preceding integral becomes 



whence it follows that this new integral is also pseudo-elliptic if /(z 2 ) satisfies 
one of the identities 



The first of these cases was noticed by Euler.* 



III. INTEGRATION OF TRANSCENDENTAL FUNCTIONS 

114. Integration of rational functions of sin x and cos x. It is well 
known that since and cos a? may be expressed rationally in terms 
of tan ce/2 = t. Hence this change of variable reduces an integral 
of the form 



sinx, cosxjdx 

to the integral of a rational function of t. For we have 

2dt 2t l-t 2 

x = 2 arc tan t , ax = - - -^ > sin x = 2 > cos x = > 

and the given integral becomes 



where &(t) is a rational function. For example, 

C dx Cdt 
\ - - = I = log t ; 
J sin a; J t 



hence 



/dx 
sin a; 



dx x 

= log tan-- 

sin a; 2 



* See Hermite 3 lithographed Cours, 4th ed., pp. 25-28. 



V, 114] TRANSCENDENTAL FUNCTIONS 237 

The integral f [I/cos x~]dx reduces to the preceding by means of the 
substitution x = 7r/2 y, which gives 



/dx 
cos x 



dx ITT x\ ITT x 

- = - log tan T - I = log tan - + - 
coscc \4 2/ \4 2 

The preceding method has the advantage of generality, but it is 
often possible to find a simpler substitution which is equally suc 
cessful. Thus, if the function /(sin x, cos x~) has the period IT, it is 
a rational function of tana;, F(tan x). The substitution t&nx = t 
therefore reduces the integral to the form 



f 



F(tan x~) dx = 

As an example let us consider the integral 

dx 



A cos 2 x -f- B sin x cos x + C sin 2 x + D 

where A, B, C, D are any constants. The integrand evidently has the 
period IT ; and, setting tan x = t, we find 

1 



cos" 1 * x = ) sin x cos x > sin 2 x = 

-L ~~T~ t 1 ~T~ * 

Hence the given integral becomes 



r 

J 



The form of the result will depend upon the nature of the roots 
of the denominator. Taking certain three of the coefficients zero, 
we find the formulae 

/dx r dx 

= tan:r, / -; - = log tan x, 

cos 2 x J sin x cos x 



/; 



dx 

r-r- = COt X. 



When the integrand is of the form R(sm x) cos x, or of the form 
R (cos x) since, the proper change of variable is apparent. In the 
first case we should set sin x = t ; in the second case, cos x = t. 

It is sometimes advantageous to make a first substitution in order 
to simplify the integral before proceeding with the general method. 
For example, let us consider the integral 

dx 



a cos x + b sin x -f 



238 INDEFINITE INTEGRALS [V, 114 

where a, b, c are any three constants. If p is a positive number 
and < an angle determined by the equations 

a = p cos </>, b = p sin <f>, 
we shall have 

/ a , n a b 

p = V a -f- ) cos o> = 7== > sin d> = p== > 

Va 2 + b 2 Va 2 + 6 2 

and the given integral may be written in the form 

/dx _ C dy 

p cos (x -</>)+ c J p cos y + c 

where x <f> = y. Let us now apply the general method, setting 
tan y/2 t. Then the integral becomes 

2dt 



and the rest of the calculation presents no difficulty. Two different 
forms will be found for the result, according as p 2 c 2 = a 2 + b 2 c 2 
is positive or negative. 
The integral 

" m cos x -f n sin x + p . 

; : *- dx 

a cos x + o sin x + c 



f 



may be reduced to the preceding. For, let u = a cos x + b sin x + c, 
and let us determine three constants X, p., and v such that the equation 

du 

m cos x -\- n sin x + 7> = MI + u. \- v 

ax 

is identically satisfied. The equations which determine these num 
bers are 

m = \a + p.b, n = Xb p,a, p = \c + v, 

the first two of which determine X and /A. The three constants hav 
ing been selected in this way, the given integral may be written in 
the form 

/du 
Xt< + fi + v 
. dx = \x -\- u. log u 4- v / 
u J a 



cos x + b sin x + c 
Example. Let us try to evaluate the definite integral 
dx 



1 + ecosx 



where |e|<l. 



y, 115] TRANSCENDENTAL FUNCTIONS 239 

Considering it first as an indefinite integral, we find successively 

C dx _ 2 C dt 2 C du 

J 1 + ecosx J l + e + (l- e)V ~ vT^J 1 + w 2 



by means of the successive substitutions tanx/2 = f, t = u V(l + e)/(l e). 
Hence the indefinite integral is equal to 



2 / ll-e. x\ 
. arc tan I \l tan - J 

TIT^ \ \ i + c 2/ 



As x varies from to x, V(l e)/(l + e) tan x/2 increases from to + oo, and 
the arctangent varies from to it/2. Hence the given definite integral is equal 
to ;r/V(l - e 2 ). 

115. Reduction formulae. There are also certain classes of integrals 
for which reduction formulae exist. For instance, the formula for 
the derivative of tan n ~ l may be written 

j- (tan"- 1 a;) = (?i -l)tan"- 2 x(l+ tan 2 a;), 
whence we find 



/tan" ~ l x C 
ka,n"xdx = -- / tan"~ 
n-l J 



The exponent of tan x in the integrand is diminished by two units. 
Repeated applications of this formula lead to one or the other of 
the two integrals 

I dx = x , I tan x dx = log cos x . 

The analogous formula for integrals of the type /cot" x dx is 

C cot"- 1 ^ r 

I c*r\i~ ** T* fiw - - ____ __ I r/-k i r * * * fit* 

I L/UU <C W-i-C ^^ I CUu X (L3C , 

J n-l J 

In general, consider the integral 

sin m x cos n xdx, 

where m and n are any positive or negative integers. When one of 
these integers is odd it is best to use the change of variable given 
above. If, for instance, n = 2p + 1, we should set sin x = t, which 
reduces the integral to the form />(! t 2 ) p dt. 

Let us, therefore, restrict ourselves to the case where m and n are 
both even, that is, to integrals of the type 



= f si 
J 



sin 2 " a; cos j "ccc?x, 



240 INDEFINITE INTEGRALS [V, 116 

which may be written in the form 

T = I sin 2m ~ l xcos 2n xsmxdx. 

*m,m J 

Taking cos 2n z since dx as the differential of [ l/(2n -f I)]cos 2n + 1 a;, 
an integration by parts gives 

pQgZn-fl^ 2m 1 C 

T = -sin 2m - l x- r + - r / sin sm - 8 ajcos*"aj(l-sin a x)cte, 
- Lm n 2ft + 1 2n+lJ 

which may be written in the form 

7" sin 2 "*" 1 ^ cos 2 " +1 a; 2m 1 j 

-** ~ 2 (m + n) 2 (m + ri) -*-!, 

This formula enables us to diminish the exponent m without alter 
ing the second exponent. If m is negative, an analogous formula 
may be obtained by solving the equation (A) with respect to / m _ lin 
and replacing m by 1 m : 

j sin 1 ~ 2 ? a; cos 2n+1 a; 2 (n m + 1) ,- 

**.. 1-2OT 1-2WI ^l-m,n 

The following analogous formulae, which are easily derived, enable 
us to reduce the exponent of cos a; : 

r sin 2m+1 a;cos 2n ~ 1 a; 2n 1 T 

M : 2 (m + w) 2(m + w) ^ m ,-i 

^- _ sin m+1 a;cos 1 ~ 8 *a; 2 (m + 1 n) j 

( t **,- l2n l2n -*,-+!* 

Eepeated applications of these formulae reduce each of the num 
bers m and n to zero. The only case in which we should be unable to 
proceed is that in which we obtain an integral / TOiB , where m + n = 0. 
But such an integral is of one of the types for which reduction for 
mulae were derived at the beginning of this article. 

116. Wallis formulae. There exist reduction formulae whether the exponents 
m and n are even or odd. 

As an example let us try to evaluate the definite integral 

n 

I m = I s m m xdx, 
Jo 

where m is a positive integer. An integration by parts gives 

7T 7T E 

| 2 siu m - 1 xsinxtZx = [cosxsin-^]^ + (m 1) | 2 sin m - 2 cos 2 xdx, 
Jo o Jo 



V, 117] TRANSCENDENTAL FUNCTIONS 241 

whence, noting that cosz s\n m ~ l x vanishes at both limits, we find the formula 

7T 

I m = (m - 1) f 2 sin- 2 x(l - sin 2 x)<Jx = (m - l)(/ m _ 2 - /m), 
Jo 

ivhich leads to the recurrent formula 

m 1 T 

(13) I m = - - Im-2- 

m 

Repeated applications of this formula reduce the given integral to IQ = it/2 
if m is even, or to Ii = 1 if m is odd. In the former case, taking m = 2p and 
replacing m successively by 2, 4, 6, , 2j>, we find 

1 T T 3 T T - 2 P- 1 T 

-*a = -- 05 -M = 7 -* 2 > * J 2p 2n~ 

or, multiplying these equations together, 

_ 1 . 3 . 6 (2p - 1) 5 
2 . 4 . 6 2p 2 
Similarly, we find the formula 

2 . 4 . 6 2p 



12.P + 1 = 



1 . 3 . 5 (2j> + 1) 



A curious result due to Wallis may be deduced from these formulae. It is 
evident that the value of I m diminishes as m increases, for sin m + 1 x is less than 
sin m z. Hence 



and if we replace I 2 p + i> ^2 P , IS P -I by their values from the formulae above, we 
find the new inequalities 



where we have set, for brevity, 

2244 2p - 2 2p 



1335 2p - 1 2p - 1 

It is evident that the ratio n/2H p approaches the limit one as p increases indefi 
nitely. It follows that 7T/2 is the limit of the product H p as the number of 
factors increases indefinitely. The law of formation of the successive factors is 
apparent. 

117. The integral /cos (ax + b) cos (a x + b ) dx. Let us consider 
a product of any number of factors of the form cos (ax + b), where 
a and b are constants, and where the same factor may occur several 
times. The formula 

cos (u + ? ) cos (u v) 
cos u cos v = ^ H ^ 



242 INDEFINITE INTEGRALS [V, 117 

enables us to replace the product of two factors of this sort by the 
sum of two cosines of linear functions of x ; hence also the product 
of n factors by the sum of two products of n 1 factors each. 
Kepeated applications of this formula finally reduce the given inte 
gral to a sum of the form 2 H cos (Ax + B), each term of which is 
immediately integrable. If A is not zero, we have 

/ , sin (Ax 4- B} 
cos (Ax -f B}dx = I + C, 

J\. 

while, in the particular case when A = 0, /cos B dx = x cos B + C. 
This transformation applies in the special case of products of 
the form 

cos m a; sin n ic, 

where m and n are both positive integers. For this product may 
be written 



and, applying the preceding process, we are led to a sum of sines and 
cosines of multiples of the angle, each term of which is immediately 
integrable. 

As an example let us try to calculate the area of the curve 



which we may suppose given in the parametric form x = acos 0, 
y = b sin 0, where 6 varies from to 2-rr for the whole curve. The 
formula for the area of a closed curve, 

A = - / xdy ydx, 

i/(C) 

gives 



I 

Jo 



But we have the formula 

(sin 6 cos 0) 2 = - sin 2 20 = - (1 - cos 

Hence the area of the given curve is 

Sab [~ sin40~l 27r 3-rrab 



V, 117] 



TRANSCENDENTAL FUNCTIONS 



243 



It is now easy to deduce the following formulae . 





/I cos 2x 


a; sin 2x 


hr< 


sin 2 x dx 


2 
/3 sin x sin 3x ^ 


2 4 
3 cos cc 


c > 
cos 3cc j 


1 sin 3 a? dx 

r . 


4 
/3 4 cos 2x 4- cos 4x ^ 


3ic sin 2x 


12 
, sin 4x 


| sin 4 cc dx 


8 


S 4 


32 


r 


T1+OOS2X 


a; sin 2x 




1 cos x dx 


J 2 

/3 cos cc + cos 3x j 


2 4 

3 sin x t si 


T t7> 

n3x 


1 cos 3 a; dx 

r 4 


4 
/3 + 4 cos 2x 4- cos 4x ^ 


4 

3x t sin2x 


12 C 


J 




8 4 


32 



C, 



A general law may be noticed in these formulae. The integrals 
F(x) f*sin. n xdx and <J>(cc) = f* cos"x dx have the period 2?r 
when n is odd. On the other hand, when n is even, these integrals 
increase by a positive constant when x increases by 2?r. It is evi 
dent a priori that these statements hold in general. For we have 



F(x + 2-Tr) 



/-2Jr r27r + a: 

= I sin";rcfa; + I sin" 
Jo Jz-rr 



xdx, 



or 



/o2ir r>x /^2ir 

x + 27r) = I sin n a;c?ic+ I sin n a;rfa; = F(x~) + I sin n co?a;, 

i/O i/O i/O 



F( 



since sin x has the period 2?r. If n is even, it is evident that the 
integral f 2v sin n x dx is a positive quantity. If n is odd, the same 
integral vanishes, since sin (x + TT) = sin x. 

Note. On account of the great variety of transformations appli 
cable to trigonometric functions it is often convenient to introduce 
them in the calculation of other integrals. Consider, for example, 
the integral /[!/(!+ x*)*~\dx. Setting x = ts,n<f>, this integral 
becomes f cos </> d<j> = sin <j> + C. Hence, returning to the variable x, 

dx 






which is the result already found in 105. 



244 INDEFINITE INTEGRALS [V, 118 

118. The integral /R(x)e wx dx. Let us now consider an integral 
of the form /R(x)e" x dx, where R (x) is a rational function of x. 
Let us suppose the function R(x) broken up, as we have done 
several times, into a sum of the form 



where E(x), A l , A 2 , , A p , X 1} , X p are polynomials, and X t is 
prime to its derivative. The given integral is then equal to the 
sum of the integral / E(x)e ax dx, which we learned to integrate in 
85 by a suite of integrations by parts, and a number of integrals 
of the form 



There exists a reduction formula for the case when n is greater 
than unity. For, since X is prime to its derivative, we can determine 
two polynomials A. and /* which satisfy the identity A = \X + p.X . 
Hence we have 



and an integration by parts gives the formula 

; dx. 



f 

J 



-1 X" 



Uniting these two formulas, the integral under consideration is 
reduced to an integral of the same type, where the exponent n is 
reduced by unity. Eepeated applications of this process lead to 
the integral 

dx, 



X 

where the polynomial B may always be supposed to be prime to 
and of less degree than X. The reduction formula cannot be applied 
to this integral, but if the roots of X be known, it can always be 
reduced to a single new type of transcendental function. For 
defmiteness suppose that all the roots are real. Then the integral 
in question can be broken up into several integrals of the form 



dx. 
x a 



v, 119] TRANSCENDENTAL FUNCTIONS 245 

Neglecting a constant factor, the substitutions x = a + y/u>, u = e" 
enable us to write this integral in either of the following forms : 



/e" dy C du 

y J log it 



The latter integral f [I/log u~\du is a transcendental function which 
is called the integral logarithm. 

119. Miscellaneous integrals. Let us consider an integral of the form 

inx, cos a;) eta, 



where / is an integral function of sin x and cos x. Any term of 
this integral is of the form 



where m and n are positive integers. We have seen above that the 
product sin m x cos"x may be replaced by a sum of sines and cosines 
of multiples of x. Hence it only remains to study the following 
two types : 

I e ax cosbxdx, I e ax sinbxdx. 

Integrating each of these by parts, we find the formulae 

C e ax sinbx a T 

I e"* cos bx dx = ; I e"* sin bx dx , 

J b b J 

/pQZC QQg Jjnf* ft, I 

e ax sin bx dx = h T I e " x cos & x dx. 
b b J 

Hence the values of the integrals under consideration are 

e" x (a cos bx -f b sin bx) 

e ax cos bx dx = * r: L > 

a* + 6* 

e ax (a sin bx b cos bx) 

e"* sm bx dx = 5 - 

a 2 + i 2 

Among the integrals which may be reduced to the preceding 
types we may mention the following cases : 

I /(log x) x dx , I /(arc sin x) dx , 

I /(x) arc sin x dx , I /(x) arc tan x dx , 



246 INDEFINITE INTEGRALS [V, EM. 

where / denotes any integral function. In the first two cases we 
should take log x or arc sin x as the new variable. In the last 
two we should integrate by parts, taking /(#) dx as the differential 
of another polynomial F(x), which would lead to types of integrals 
already considered. 

EXERCISES 

1. Evaluate the indefinite integrals of each of the following functions : 



z* - x 8 - 3x 2 - x I + Vl + x 



(x* + I) 2 x (x 3 + I) 3 (x 2 + I) 8 i _ Vx 

3 . 

1 1 + VI + x 1 x 

1 + x + Vl + x 2 i_vT^~x Vx + Vx + 1 + Vx(x + 1) cos 2 x 

x 2 P 

XC*COSX, , TV tan T 

/ t* * L til 1 X 

V a -f x" + 2 

2. Find the area of the loop of the folium of Descartes : 

x 3 + y 3 Saxy 0. 

3. Evaluate the integral fy dx, where x and y satisfy one of the following 
identities : 

(x 2 - a 2 ) 2 - ay 2 (2y + 3a) = , y 2 (a - x) = x 3 , y (x 2 + y 2 ) = a (y 2 - x 2 ) . 

4. Derive the formulae 



xcosnx 

+ (7. 

[EDLER.] 



/. , , , sin" x cos nx 

snv -^x cos(n + l)xdx = f- C, 

n 

/sin n xsinnx 
sm n - l x sin (n + l)xdx = \-C, 
n 

/cos"x sinnx 
cos- 1 xcos(n + l)xdx= + C, 

/cos n x cos 
cos"- 1 x sin (n + l)xdx = 

5. Evaluate each of the following pseudo-elliptic integrals : 

/(l + x 2 )dx C (l-x 2 )dx 
,.> . 

6. Reduce the following integrals to elliptic integrals : 

E(x)dx 



Va(l + x 6 ) + 6x(l + x*) + cx 2 (l + x 2 ) + dx* 

R(x)dx 

Va(l + x 8 ) + 6x 2 (1 + x*) + ex* 

where R(x) denotes a rational function. 



V, E.] EXERCISES 247 

7*. Let a, 6, c, d be the roots of an equation of the fourth degree P(x) = 0. 
Then there exist three involutory relations of the form 

Mix" + Ni 

t <= 1, 2, 3, 



which interchange the roots by pairs. If the rational function f(x) satisfies the 
identity 



the integral f[f(x)/ VP (x)] dx is pseudo-elliptic (see Bulletin de la SocMM matM- 
matique, Vol. XV, p. 106). 

8. The rectification of a curve of the type y = Ax* leads to an integral of 
a binomial differential. Discuss the cases of integrability. 



9. If a > 1, show that 



+ 1 , 

dx 



/_, (a - x) Vl - x 2 Va 2 - 1 
Hence deduce the formula 
1 



X 
t 



n dx _ 1. 3. 5-- -(2re - 1) 



l - x 2 ~ 2.4.6..-2n 

10. If ^1C - J5 2 > 0, show that 

X*"* dx _ 1 . 3 . 5 (2ft - 3) 

x (^lx 2 + 2Bz + C) ~ 2 . 4 . 6 (2n - 2) 

[Apply the reduction formula of 104.] 

11. Evaluate the definite integral 

sin 2 xdx 



C 

J o 



1 + 2a cos x + a 2 
12. Derive the following formulae : 

dx 1 



C 
JL 



LI Vl - 2ax + a 2 Vl - 2/3x 
+ 1 






- ax)(l - /3x) dz TT 2 - 



- 2ax + a 2 )(l - 2^x + p?) Vl - x 2 2 1 - 
13*. Derive the formula 

x m ~ l dx it 



f 

I/O 



, * 1 nsin M* 

n 



where m and n are positive integers (m<n). [Break up the integrand into 
partial fractions.] 



248 INDEFINITE INTEGRALS [V, Exs 

14. From the preceding exercise deduce the formula 



/ 

i/O 



x n - } dx it 



1 + x 



15. Setting I p<q = ft q (t + l) p dt, deduce the following reduction formulae : 

(p + q + l)I p , q = * + !( + !)*+!/_,, 

(P ~ !)*-*, = i9 + l (t + I) 1 -* ~ (2 + q-p)I- P + i, g , 
and two analogous formulae for reducing the exponent q. 

16. Derive formulae of reduction for the integrals 

7 _ C xdx z _ C dx 

J V2x 2 + 2Bx + C ~J (x - a)* V^4z 2 + 2Bx + C 



17*. Derive a reduction formula for the integral 

C x n dx 

J vnr^ 



Hence deduce a formula analogous to that of Wallis for the definite integral 
18. Has the definite integral 







1 dx 



o vl-x* 



/ 

Jo 



dx 



^ ^ 1+ x 4 sin 2 x 
a finite value ? 

19. Show that the area of a sector of an ellipse bounded by the focal axis 
and a radius vector through the focus is 



P 2 r 
= ^j 

tSQ 



(1 + e cos w) 2 



where p denotes the parameter W/a and e the eccentricity. Applying the gen 
eral method, make the substitutions tan w/2 = t, t = u V(l + e)/(l e) succes 
sively, and show that the area in question is 



A = ab I arc tan u e ) 

V 1 + M/ 



Also show that this expression may be written in the form 



ab . 
A = (* - e sin 



where <f> is the eccentric anomaly. See p. 406. 

20. Find the curves for which the distance NT, or the area of the triangle 
MNT, is constant (Fig. 3, p. 31). Construct the two branches of the curve. 

[Licence, Paris, 1880; Toulouse, 1882.] 



V, Exs.j EXERCISES 249 

21*. Setting 

X 2n + l /.I 

A n = - / (1 z 2 )"cosxzdz t 
2 . 4 . 6 . 2n Jo ^ 



derive the recurrent formula 



From this deduce the formulae 

AZ P = Ut p sin x + Vzp cos z , 
A.i p + 1 = Uz p + 1 sin x + Vz p + 1 cos x , 

where UZ P , V% p , UV P + I, V^^ + i are polynomials with integral coefficients, and 
where 7 2p and U^ p + \ contain no odd powers of x. It is readily shown that 
these formulae hold when n = 1, and the general case follows from the above 
recurrent formula. 

The formula for Ao p enables us to show that n 2 is incommensurable. For if 
we assume that 7T 2 /4 = b/a, and then replace x by ir/2 in A^ p , we obtain a 
relation of the form 



a 2 . 4 . 6 4p 



f V -*) COB ^<b, 

Jo 2 



where HI is an integer. Such an equation, however, is impossible, for the right* 
nand side approaches zero as p increases indefinitely. 



CHAPTER VI 
DOUBLE INTEGRALS 

I. DOUBLE INTEGRALS METHODS OF EVALUATION 
GREEX S THEOREM 

120. Continuous functions of two variables. Let 2 = f(x, y} be a 
function of the two independent variables x and y which is contin 
uous inside a region A of the plane which is bounded by a closed 
contour C, and also upon the contour itself. A number of proposi 
tions analogous to those proved in 70 for a continuous function 
of a single variable can be shown to hold for this function. For 
instance, given any positive number c, the region A can be divided into 
subregions in such a, way that the difference between the values of z at 
any tivo points (x, y), (x , y ) in the same subregion is less than e. 

We shall always proceed by means of successive subdivisions as 
follows : Suppose the region A divided into subregions by drawing 

parallels to the two axes at equal dis 
tances 8 from each other. The corre 
sponding subdivisions of A are either 
squares of side 8 lying entirely inside C, 
or else portions of squares bounded in 
part by an arc of C. Then, if the prop 
osition were untrue for the whole region 
A, it would also be untrue for at least 
x one of the subdivisions, say A^. Sub 
dividing the subregion A l in the same 
manner and continuing the process indefinitely, we would obtain a 
sequence of squares or portions of squares A, A lf , A n , , for 
which the proposition would be untrue. The region A n lies between 
the two lines x = a n and x = b n , which are parallel to the y axis, 
and the two lines y = c n , y = d n , which are parallel to the x axis. 
As n increases indefinitely a n and b n approach a common limit A, 
and c n and d n approach a common limit /A, for the numbers , 
for example, never decrease and always remain less than a fixed 
number. It follows that all the points of A H approach a limiting 

250 



FIG. 23 



VI, 120] INTRODUCTION GREEN S THEOREM 251 

point (\, //,) which lies within or upon the contour C. The rest of 
the reasoning is similar to that in 70 ; if the theorem stated were 
untrue, the function f(x, y) could be shown to be discontinuous at 
the point (A, /*), which is contrary to hypothesis. 

Corollary. Suppose that the parallel lines have been chosen 
so near together that the difference of any two values of z in any 
one subregion is less than e/2, and let -^ be the distance between 
the successive parallels. Let (x, y*) and (x 1 , y ) be two points inside 
or upon the contour C, the distance between which is less than rj. 
These two points will lie either in the same subregion or else in 
two different subregions which have one vertex in common. In 
either case the absolute value of the difference 

f(x,y}-f(x ,y<} 

cannot exceed 2e/2 = c. Hence, given any positive number e, another 
positive number 17 can be found such that 

\f(x, y}-f(x>, y )\< 

whenever the distance between the two points (x, y*) and (x , y ), which 
lie in A or on the contour C, is less than rj. In other words, any func 
tion which is continuous in A and on its boundary C is uniformly 
continuous. 

From the preceding theorem it can be shown, as in 70, that every 
function which is continuous in A (inclusive of its boundary) is neces 
sarily finite in A. If M be the upper limit and m the lower limit of 
the function in A, the difference M m is called the oscillation. The 
method of successive subdivisions also enables us to show that the 
function actually attains each of the values m and M at least once 
inside or upon the contour C. Let a be a point for which z = m 
and b a point for which z = M, and let us join a and b by a broken 
line which lies entirely inside C. As the point (x, y) describes this 
line, z is a continuous function of the distance of the point (x, y) 
from the point a. Hence z assumes every value p. between m and 
M at least once upon this line ( 70). Since a and b can be joined 
by an infinite number of different broken lines, it follows that the 
f unction f(x, ?/) assumes every value between m and M at an infinite 
number of points which lie inside of C. 

A finite region A of the plane is said to be less than I in all its 
dimensions if a circle of radius I can be found which entirely 
encloses A. A variable region of the plane is said to be infinitesimal 



252 DOUBLE INTEGRALS [VI, 121 

in all its dimensions if a circle whose radius is arbitrarily preas- 
signed can be found which eventually contains the region entirely 
within it. For example, a square whose side approaches zero or au 
ellipse both of whose axes approach zero is infinitesimal in all its 
dimensions. On the other hand, a rectangle of which only one side 
approaches zero or an ellipse only one of whose axes approaches zero 
is not infinitesimal in all its dimensions. 

121. Double integrals. Let the region A of the plane be divided 
into subregions a x , a 2 , -, a n in any manner, and let u>,- be the area of 
the subregion a,-, and M { and m,- the limits of /(a;, y) in a t . Consider 
the two sums 



each of which has a definite value for any particular subdivision 
of A. None of the sums are less than ml* where ft is the area of 
the region A of the plane, and where m is the lower limit of f(x, y) 
in the region A ; hence these sums have a lower limit /. Likewise, 
none of the sums s are greater than 3/ft, where M is the upper limit 
of f(x, y) in the region A ; hence these sums have an upper limit / . 
Moreover it can be shown, as in 71, that any of the sums S is 
greater than or equal to any one of the sums s; hence it follows 

that 

/>/ . 

If the function f(x, y) is continuous, the sums S and s approach 
a common limit as each of the subregions approaches zero in all its 
dimensions. For, suppose that rj is a positive number such that the 
oscillation of the function is less than c in any portion of A which 
is less in all its dimensions than 77. If each of the subregions a if 
a 2 , , a n be less in all its dimensions than rj, each of the differences 
M i nii will be less than e, and hence the difference S s will be 
less than eft, where ft denotes the total area of A. But we have 

S -s = S-I+I- / + / -*, 

where none of the quantities S 7, 7 / ,/ s can be negative. 
Hence, in particular, / 7 <eft; and since e is an arbitrary posi 
tive number, it follows that 7 = / . Moreover each of the numbers 
S / and / s can be made less than any preassigned number by 

*If f(x, y) is a constant k, M = m = Mf = m< = k, and S = s = mft = MQ. 
TRANS. 



VI, 121] INTRODUCTION GREEN S THEOREM 253 

a proper choice of e. Hence the sums >S and s have a common limit 
/, which is called the double integral of the function f(x, y\ extended 
over the region A. It is denoted by the symbol 



-// 

J J(A) 



and the region A is called the field of integration. 

If (|,., 77,) be any point inside or on the boundary of the sub- 
region it is evident that the sum 2/(, 77,-) to, lies between the two 
sums S and s or is equal to one of them. It therefore also 
approaches the double integral as its limit whatever be the method 
of choice of the point (,-, 77,). 

The first theorem of the mean may be extended without difficulty ( 
to double integrals. Let f(x, y) be a function which is continuous 
in A, and let <f>(x, y) be another function which is continuous and 
which has the same sign throughout A. For definiteness we shall 
suppose that <$>(x, y) is positive in A. If M and m are the limits of 
f(x, y] in A, it is evident that* 



Adding all these inequalities and passing to the limit, we find the 
formula 



I I A x > y)<K x > y)dxdy = nl I *(* 

J J(A) J J(A) 



where /x lies between M and m. Since the function f(x, y) assumes 
the value /A at a point (, 77) inside of the contour C, we may write 
this in the form 

(1) ff f(x, y)4>(x, y)dxdy =/(, 77) \\ 4>(x, 

J J(A) J J<A 

which constitutes the law of the mean for double integrals. If 
<(a:, y~) = 1, for example, the integral on the right, ffdx dy, extended 
over the region A, is evidently equal to the area O of that region. 
In this case the formula (1) becomes 

(2) f f f(x, y) dx dy = fl/( 77) . 

J J(A) 

* If f(x, y) is a constant k, we shall have M = m = k, and these inequalities become 
equations. The following formula holds, however, with M= k. TRANS. . 



254 DOUBLE INTEGRALS [VI, 122 

122. Volume. To the analytic notion of a double integral corre 
sponds the important geometric notion of volume. Let f(x, y) be 
a function which is continuous inside and upon a closed contour C. 
We shall further suppose for definiteness that this function is posi 
tive. Let S be the portion of the surface represented by the equa 
tion z =f(x, y) which is bounded by a curve T whose projection 
upon the xy plane is the contour C. We shall denote by E the por 
tion of space bounded by the xy plane, the surface S, and the cylinder 
whose right section is C. The region A of the xy plane which is 
bounded by the contour C being subdivided in any manner, let a,- be 
one of the subregions bounded by a contour c t -, and o> f the area of 
this subregion. The cylinder whose right section is the curve c { cuts 
out of the surface 5 a portion s t - bounded by a curve y { . Let p { and 
Pf be the points of s { whose distances from the xy plane are a mini 
mum and a maximum, respectively. If planes be drawn through 
these two points parallel to the xy plane, two right cylinders are 
obtained which have the same base o> and whose altitudes are the 
limits M { and m t - of the function /(cc, y) inside the contour c,, respec 
tively. The volumes V t and v { of these cylinders are, respectively, 
co, M f and w,-m,-.* The sums S and s considered above therefore repre 
sent, respectively, the sums 2F f and ^v t of these two types of cylin 
ders. We shall call the common limit of these two sums the volume 
of the portion E of space. It may be noted, as was done in the case 
of area ( 78), that this definition agrees with the ordinary concep 
tion of what is meant by volume. 

If the surface S lies partly beneath the xy plane, the double integral 
will still represent a volume if we agree to attach the sign to the 
volumes of portions of space below the xy plane. It appears then that 
every double integral represents an algebraic sum of volumes, just as 
a simple integral represents an algebraic sum of areas. The limits of 
integration in the case of a simple integral are replaced in the case of a 
double integral by the contour which encloses the field of integration. 

123. Evaluation of double integrals. The evaluation of a double 
integral can be reduced to the successive evaluations of two simple 
integrals. Let us first consider the case where the field of integration 

*By the volume of a right cylinder we shall understand the limit approached by 
the volume of a right prism of the same height, whose base is a polygon inscribed in 
a right section of the cylinder, as each of the sides of this polygon approaches zero. 
[This definition is not necessary for the argument, but is useful in showing that the 
definition of volume in general agrees with our ordinary conceptions. TRANS.] 



VI, 123] INTRODUCTION GREEN S THEOREM 



255 



is a rectangle R bounded by the straight lines x = x , x = X, 
y = y , y = Y, where x < X and y < F. Suppose this rectangle 
to be subdivided by parallels to the two axes x = x ( , y = y k 
(i = 1, 2, , n ; k = 1, 2, , TO). The area of the small rectangle 
R ik bounded by the lines x = #,_ a; = or,-, y = y k _ l , y = y k is 



Hence the double integral is the limit of the sum 



where (,*, r) ik ) is any point 
inside or upon one of the 
sides of R ik . 

We shall employ the inde- 
termination of the points 
(it> Vik) i n order to simplify 
the calculation. Let us re 
mark first of all that if /(a-) 
is a continuous function in 

the interval (a, ), and if the interval (a, b~) be subdivided in any 
manner, a value can be found in each subinterval (x i _ l , #,-) such 
that 



y 



















Y 
















^ 


v k 




































t/2 


















2/i 


















2/o 
















B 





a 


o 


c,; 


5 8 2 


i-1 


Bf 


;-! 


X * 



For we need merely apply the law of the mean for integrals to each of 
the subintervals (a, Xj), (a^, a- 2 ), , (#_,, &) to find these values of &. 
Now the portion of the sum S which arises from the row of rec 
tangles between the lines x = a:,-.! and x = x ( is 

^ 2 )G/2 - //i) + 



Let us take ^ a = | /2 = = ^ im = x^ l} and then choose i; n , 77,2, 
in such a way that the sum 

/(*,-! i?n)(yi - 2/0) +/(,-!> i7,-s)(y - yO H 

is equal to the integral f !/ Y f(x i _- [ , y)dy, where the integral is to be 
evaluated under the assumption that x i _ l is a constant. If we pro 
ceed in the same way for each of the rows of rectangles bounded by 
two consecutive parallels to the y axis, we finally find the equation 

(5) S = *(or )(x 1 - 



256 DOUBLE INTEGRALS [VI, 123 

where we have set for brevity 

*00 = / f( x > y) d y- 

JV* 

This function 4>(x), defined by a definite integral, where x is con 
sidered as a parameter, is a continuous function of x. As all the 
intervals x i x i _ l approach zero, the formula (5) shows that S 
approaches the definite integral 

..r 

(#) dx . 



Jx. 



Hence the double integral in question is given by the formula 
( 6 ) f f /(*> !/) dxdy = f dx ff(x, y) dy . 

J J(,R) J*t A 

In other words, in order to evaluate the double integral, the function 
f(x, y) should first be integrated between the limits y and Y, regard 
ing x as a constant and y as a variable ; and then the resulting func 
tion^ which is a function of x alone, should be integrated again between 
the limits x and X. 

If we proceed in the reverse order, i.e. first evaluate the portion 
of S which comes from a row of rectangles which lie between two 
consecutive parallels to the x axis, we find the analogous formula 

/ I /(*> y)dxdy = I dy I f(x, y)dx. 
J J(R) Jy Jx 9 

A comparison of these two formulae gives the new formula 

x;r />r /->Y ^x 

I dx I f(x, y) dy = I dy I f(x, y)dx } 

Jx J V(S Jy Jx 

which is called the formula for integration under the integral sign. 
An essential presupposition in the proof is that the limits x , X, y , Y 
are constants, and that the function f(x, y} is continuous throughout 
the field of integration. 

Example. Let z = xy/a. Then the general formula gives 



cc ^- 

/JU>* 



VI, 123] INTRODUCTION GREEN S THEOREM 257 

In general, if the function f(x, y*) is the product of a function of x 
alone by a function of y alone, we shall have 



/ / <t>(x)$(y)dxdy = I $(x)dx x I 

J J(R) Jx n J .I 



The two integrals on the right are absolutely independent of each 
other. 

Franklin * has deduced from this remark a very simple demonstration of cer 
tain interesting theorems of Tchebycheff. Let <f>(x) and f (x) be two functions 
which are continuous in an interval (a, b), where a < b. Then the double integral 



extended over the square bounded by the lines x = o, x = 6, y = a, y = b is equal 
to the difference 

2(6 - a) C <t>(x)\l/(x)dx 2 C (p(x)dx x C \f/(x)dx. 

Ja Ja */a 

But all the elements of the above double integral have the same sign if the two 
lunctions 0(z) and ^(z) always increase or decrease simultaneously, or if one of 
them always increases when the other decreases. In the first case the two func 
tions (f>(x) (f>(y) and \f/(x) ^(y) always have the same sign, whereas they have 
opposite signs in the second case. Hence we shall have 

(b -a) C <j>(x)t(x)dx > C $(x)dx x C ^(x)dx 

Ja J a Ja 

whenever the two functions <j>(x) and \f/(x) both increase or both decrease through 
out the interval (a, b). On the other hand, we shall have 



(b- a) f <f>(x)^(x)dx < f 0(z)dz x f 

Ja Ja Ja 



whenever one of the functions increases and the other decreases throughout the 
interval. 

The sign of the double integral is also definitely determined in case 0(z) = ^(z), 
for then the integrand becomes a perfect square. In this case we shall have 

(b -a] 

whatever be the function 0(z), where the sign of equality can hold only when 
<p(x) is a constant. 

The solution of an interesting problem of the calculus of variations may be 
deduced from this result. Let P and Q be two fixed points in a plane whose 
coordinates are (a, A) and (6, B), respectively. Let y =/(z) be the equation of 
any curve joining these two points, where /(z), together with its first derivative 

* American Journal of Mathematics, Vol. VII, p. 77. 



258 



DOUBLE INTEGRALS 



[VI, 124 



/ (x), is supposed to be continuous in the interval (a, b). The problem is to 
find that one of the curves y=f(x) for which the integral f^ y 2 dx is a 
minimum. But by the formula just found, replacing <f>(x) by y" and noting 
that /(a) = A and f(b) = B by hypothesis, we have 



(b-a) Cy *dx^(B- 

) a 



A 



B 



The minimum value of the integral is therefore (B A)*/(b a), and that value 
is actually assumed when y is a constant, i.e. when the curve joining the two 
fixed points reduces to the straight line PQ. 

124. Let us now pass to the case where the field of integration is 
bounded by a contour of any form whatever. We shall first suppose 
that this contour is met in at most two points by any parallel to the 
y axis. We may then suppose that it is composed of two straight 

lines x = a and x = b (a < 6) 
and two arcs of curves APB 
and A QB whose equations are 
YI = < : (cc) and F 2 = $2 (#)> re ~ 
spectively, where the functions 
<f>! and < 2 are continuous be 
tween a and b. It may happen 
that the points A and A coin 
cide, or that B and B coin 
cide, or both. This occurs, for 
instance, if the contour is a convex curve like an ellipse. Let us 
again subdivide the field of integration R by means of parallels to 
the axes. Then we shall have two classes of subregions : regular if 
they are rectangles which lie wholly within the contour, irregular 
if they are portions of rectangles bounded in part by arcs of the 
contour. Then it remains to find the limit of the sum 



FIG. 25 



where o> is the area of any one of the subregions and (, rj) is a point 
in that subregion. 

Let us first evaluate the portion of S which arises from the row 
of subregions between the consecutive parallels x = x i _ l , x = x ( . 
These subregions will consist of several regular ones, beginning 
with a vertex whose ordinate is y ^ Y t and going to a vertex whose 
ordinate is y" ^ Y 2 , and several irregular ones. Choosing a suitable 
point (, 77) in each rectangle, it is clear, as above, that the portion 
of S which comes from these regular rectangles may be written in 
the form 



VI, 124] INTRODUCTION GREEN S THEOREM 259 

(,-_ i, y)dy. 



Suppose that the oscillation of each of the functions <i(#) and < 2 (a;) 
in each of the intervals (x { _ l} x t ) is less than 8, and that each of the 
differences y k y k _ l is also less than 8. Then it is easily seen that 
the total area of the irregular subregions between x = x i _ 1 and x = xt 
is less than 48(x t - ;_,), and that the portion of S which arises 
from these regions is less than 4:HB(x { x t _^) in absolute value, 
where H is the upper limit of the absolute value of f(x, y) in the 
whole field of integration. On the other hand, we have 



XV" f*Y t s*Yi ny 

/(*<-i> y)dy = I /(,-i, y}dy+ I + / , 
Jf\ Jy J Y* 

and since \Y l y \ and |F 2 y"\ are each less than 28, we may write 
f/fa-u y)dy= C f(x M) 

Jy J } , 



The portion of S which arises from the row of subregions under 
consideration may therefore be written in the form 



where 0,- lies between 1 and + 1. The sum SH8 2,O i (x i ar _j) is 
less than 87/8(6 a) in absolute value, and approaches zero with 8, 
which may be taken as small as we please. The double integral is 
therefore the limit of the sum 



where 



Hence we have the formula 

( 7 ) f f f(*, V) dxd l/ = fdx f f(x, y) dy. 

J J(R) Ja J Y l 

In the first integration x is to be regarded as a constant, but 
the limits Y l and F 2 are themselves functions of x and not 
constants. 



260 DOUBLE INTEGRALS [VI, 124 

Example. Let us try to evaluate the double integral of the function xy/a 
over the interior of a quarter circle bounded by the axes and the circumference 

X 2 + yl _ #2 _ . 

The limits for x are and R, and if x is constant, y may vary from to VR 2 z 2 
Hence the integral is 



p r^ , _. i(fe = /.(. ^ 

Jo Jo < Jo 2 L Jo J 2 

The value of the latter integral is easily shown to be R*/8a. 

When the field of integration is bounded by a contour of any form 
whatever, it may be divided into several parts in such a way that 
the boundary of each part is met in at most two points by a parallel 
to the y axis. We might also divide it into parts in such a way that 
the boundary of each part would be met in at most two points by 
any line parallel to the x axis, and begin by integrating with respect 
to x. Let us consider, for example, a convex closed curve which lies 
inside the rectangle formed by the lines x = a, x = b, y = c, y = d, 
upon which lie the four points A, B, C, D, respectively, for which x 
or y is a minimum or a maximum.* Let y v = <#>i() and y 2 = < 2 (cc) 
be the equations of the two arcs ACS and ADB, respectively, and 
let o^ = 1/^1 (y) and x 2 = \j/ 2 (y) be the equations of the two arcs CAD 
and CBD, respectively. The functions <f>i(x~) and fa(x") are continu 
ous between a and b, and i/^ (y) and i/^ (y) are continuous between c 
and d. The double integral of a f unction /(x, y), which is continuous 
inside this contour, may be evaluated in two ways. Equating the 
values found, we obtain the formula 

~& ~y 2 ~fi -.T, 

(8) I dx I f(x, y}dy = \ dy I /(or, y)dx. 

Jo, i/i/j t/c lyj-, 

It is clear that the limits are entirely different in the two integrals. 
Every convex closed contour leads to a formula of this sort. For 
example, taking the triangle bounded by the lines y = 0, x = a, 
y = x as the field of integration, we obtain the following formula, 
which is due to Lejeune Dirichlet : 

/ dx I f(x, y)dy=\dy\ f(x, y)dx. 

Jo Jo Jo Jy 



*The reader is advised to draw the figure. 



VI, 125] INTRODUCTION GREEN S THEOREM 261 

125. Analogies to simple integrals. The integral JJf(t)dt, considered as a 
function of x, has the derivative /(x). There exists an analogous theorem for 
double integrals. Let f(x, y) be a function which is continuous inside a rec 
tangle bounded by the straight lines x = a, x A, y b, y = J5,(a < A, b < B). 
The double integral of /(x, y) extended over a rectangle bounded by the lines 
x = a, x = X, y = b, y = F,(a < X < A, b < Y < .B), is a function of the coordi 
nates X and Y of the variable corner, that is, 



F(X,Y)= C dxCf(x,y)dy. 

J a J b 

Setting *(x) = f b /(x, y) dy, a first differentiation with respect to X gives 



= *(X) = f /<-*, V)dy. 
A second differentiation with respect to F leads to the formula 

2 F 

(9) 



The most general function u(X, Y) which satisfies the equation (9) is evi 
dently obtained by adding to F(X, Y) a function z whose second derivative 
d 2 z/dXdY is zero. It is therefore of the form 

(10) u(X, Y) = C A dx C Y f(x, y) dy + <f(X) + f (F) , 

J a Jb 

where <t>(X) and \1<(Y) are two arbitrary functions (see 38). The two arbitrary 
functions may be determined in such a way that u(X, Y) reduces to a given 
function V(Y) when X = a, and to another given function U(X) when Y b. 
Setting X = a and then Y = b in the preceding equation, we obtain the two 
conditions 

V(Y) = 0(a) + *(F) , U(X) - t(X) + f (6) , 

whence we find 



= F(F) - *(a) , *(&) = F(6) - 0(a) , <i>(X) = U(X) - F(6) 
and the formula (10) takes the form 

(11) u(X, Y) = ( X dx ( /(x, y) dy + U(X) + F(F) - F(6) . 

*/ a J b 



Conversely, if, by any means whatever, a function u(X, Y) has been found 
which satisfies the equation (9), it is easy to show by methods similar to tne 
above that the value of the double integral is given by the formula 

(12) f dx f /(x, y)dy = u(X, Y) - u(X, b) - u(o, F) + u(a, b). 

i/a /6 

This formula is analogous to the fundamental formula (6) on page 156. 

The following formula is in a sense analogous to the formula for integration 
by parts. Let A be a finite region of the plane bounded by one or more curves 



262 DOUBLE INTEGRALS [VI, 126 

of any form. A function /(a;, y) which is continuous in A varies between its 
minimum t and its maximum V. Imagine the contour lines /(x, y) = v drawn 
where v lies between v and F, and suppose that we are able to find the area of 
the portion of A for which /(x, y) lies between v and v. This area is a func 
tion F(v) which increases with u, and the area between two neighboring contour 
lines is F(v + A) - F(v) = AvF (v + 0Av). If this area be divided into infinitesi 
mal portions by lines joining the two contour lines, a point (, 77) may be found 
in each of them such that /(, i}) v + 6A.v. Hence the sum of the elements 
of the double integral / ffdxdy which arise from this region is 



(V 

It follows that the double integral is equal to the limit of the sum 

that is to say, to the simple integral 

v r v 

v F (v) dv = VF( F) - I F(v) dv . 



This method is especially convenient when the field of integration is bounded 
by two contour lines 

/(x, y) = v , /(x, y}= V. 

For example, consider the double integral // Vl + x 2 + y 2 dx dy extended over 
the interior of the circle x 2 + j/ 2 = 1. If we set v = Vl + x 2 + y 2 , the field of 
integration is bounded by the two contour lines v = 1 and v = \/2, and the 
function F(v), which is the area of the circle of radius Vv 2 1, is equal to 
7[(v* I). Hence the given double integral has the value 

/v/iT 2ir 

| 2itv 2 dv = (2V5-1). * 

J\ 3 

The preceding formula is readily extended to the double integral 



where F(o) now denotes the double integral ff<f>(x, y)dxdy extended over that 
portion of the field of integration bounded by the contour line v =f(x, y). 

126. Green s theorem. If the function f(x, y} is the partial deriva 
tive of a known function with respect to either x or y, one of the 
integrations may be performed at once, leaving only one indicated 
integration. This very simple remark leads to a very important 
formula which is known as Green s theorem. 



* Numerous applications of this method are to be found in a memoir by Catalan 
(Journal de Liouville, 1st series, Vol. IV, p. 233). 



VI, 126] INTRODUCTION GREEN S THEOREM 263 

Let us consider first a double integral // cP/dy dx dy extended 
over a region of the plane bounded by a contour C, which is met 
in at most two points by any line parallel to the y axis (see Fig. 15, 
p. 188). 

Let A and B be the points of C at which x is a minimum and a 
maximum, respectively. A parallel to the y axis between Aa and 
Bb meets C in two points m x and m z whose ordinates are y-^ and y z , 
respectively. Then the double integral after integration with respect 
to y may be written 



CCcP C b C"^P C 

JJ ~fy dxd> J = J dx j ^y dy= j 



But the two integrals f a P(x, y\)dx and f a P(x, y^)dx are line 
integrals taken along the arcs Am l B and Am 2 B, respectively; hence 
the preceding formula may be written in the form 

(13) 

where the line integral is to be taken along the contour C in the 
direction indicated by the arrows, that is to say in the positive 
sense, if the axes are chosen as in the figure. In order to extend 
the formula to an area bounded by any contour we should proceed 
as above ( 94), dividing the given region into several parts for each 
of which the preceding conditions are satisfied, and applying the for 
mula to each of them. In a similar manner the following analogous 
form is easily derived : 



< u > // 1? 



where the line integral is always taken in the same sense. Sub 
tracting the equations (13) and (14), we find the formula 

(15) 

where the double integral is extended over the region bounded by C. 
This is Green s formula ; its applications are very important. Just 
now we shall merely point out that the substitution Q = x and 
P = y gives the formula obtained above ( 94) for the area of a 
closed curve as a line integral. 



264 DOUBLE INTEGRALS [VI, 127 

II. CHANGE OF VARIABLES AREA OF A SURFACE 

In the evaluation of double integrals we have supposed up to the 
present that the field of integration was subdivided into infinitesimal 
rectangles by parallels to the two coordinate axes. We are now going 
to suppose the field of integration subdivided by any two systems of 
curves whatever. 

127. Preliminary formula. Let u and v be the coordinates of a point 
with respect to a set of rectangular axes in a plane, x and y the coor 
dinates of another point with respect to a similarly chosen set of 
rectangular axes in that or in some other plane. The formulae 

(16) x =f(u, v), y = <l>(u, v) 

establish a certain correspondence between the points of the two 
planes. We shall suppose 1) that the f unctions /(w, v) and </>(, v), 
together with their first partial derivatives, are continuous for all 
points (u, v) of the uv plane which lie within or on the boundary of 
a region A l bounded by a contour C l ; 2) that the equations (16) 
transform the region AI of the uv plane into a region A of the 
xy plane bounded by a contour C, and that a one-to-one correspond 
ence exists between the two regions and between the two contours 
in such a way that one and only one point of A 1 corresponds to any 
point of A ; 3) that the functional determinant A = D(f, <)/Z>(w, v) 
does not change sign inside of C lt though it may vanish at certain 
points of A i. 

Two cases may arise. When the point (u, v) describes the con 
tour C l in the positive sense the point (x, y) describes the contour C 
either in the positive or else in the negative sense without ever 
reversing the sense of its motion. We shall say that the corre 
spondence is direct or inverse, respectively, in the two cases. 

The area fl of the region A is given by the line integral 



Q = I 

J(.C 



taken along the contour C in the positive sense. In terms of the 
new variables u and v defined by (16) this becomes 

ft = I f(u, v) d<j>(u, v) , 
Ac,) 

where the new integral is to be taken along the contour C l in the 
positive sense, and where the sign -f- or the sign should be taken 



VI, 127] 



CHANGE OF VARIABLES 



265 



according as the correspondence is direct or inverse. Applying 
Green s theorem to the new integral with x = u, v = y, P = fd<f>/du, 
Q =/ d<f>/dv, we find 



c/u cv 



D(u, v) 

A\ 

dudv , 



whence 



or, applying the law of the mean to the double integral, 

D(f, *) 



(17) 



n = 



where (, rf) is a point inside the contour C l} and n l is the area of 
the region A v in the uv plane. It is clear that the sign -f or the 
sign should be taken according as A itself is positive or negative. 
Hence the correspondence is direct or inverse according as A is positive 
or negative. 

The formula (17) moreover establishes an analogy between func 
tional determinants and ordinary derivatives. For, suppose that the 
region A i approaches zero in all its dimensions, all its points approach 
ing a limiting point (u, v~). Then the region A will do the same, and 
the ratio of the two areas O and f^ approaches as its limit the abso 
lute value of the determinant A. Just as the ordinary derivative is 
the limit of the ratio of two linear infinitesimals, the functional 
determinant is thus seen to be the limit of the ratio of two infinites 
imal areas. From this point of view the formula (17) is the analogon 
of the law of the mean for derivatives. 

Remarks. The hypotheses which we have made concerning the correspondence 
between A and AI are not all independent. Thus, in order that the correspond 
ence should be one-to-one, it is necessary that A should not change sign in the 
regional of the uv plane. For, suppose that A vanishes along a curve 71 which 
divides the portion of AI where A is 
positive from the portion where A is 
negative. Let us consider a small arc 
mini of -yi and a small portion of AI 
which contains the arc mini. This 
portion is composed of two regions a\ 
and a\ which are separated by mini 
(Fig. 26). 

When the point (u, v) describes the Fio. 26 

region a\, where A is positive, the point 

(x, y) describes a region a bounded by a contour mnpm, and the two contours 
mi HI pi mi and mnpm are described simultaneously in the positive sense. When 
the point (, v) describes the region af, where A is negative, the point (x, y) 





266 



DOUBLE INTEGRALS 



[VI, 128 



describes a region a whose contour nmqr is described in the negative sense as 
n\m\q^n\ is described in the positive sense. The region a must therefore 
cover a part of the region a. Hence to any point (x, y) in the common part 
nrm correspond two points in the uv plane which lie on either side of the 
line mini. 

As an example consider the transformation X = x, Y = y 2 , for which A = 2 y. 
If the point (x, y) describes a closed region which encloses a segment a& of the 
x axis, it is evident that the point (X, Y) describes two regions both of which 
lie above the X axis and both of which are bounded by the same segment AB of 
that axis. A sheet of paper folded together along a straight line drawn upon it 
gives a clear idea of the nature of the region described by the point (X, Y}. 

The condition that A should preserve the same sign throughout AI is not suf 
ficient for one-to-one correspondence. In the example X = x 2 y 2 , Y = 2 xy, 
the Jacobian A = 4 (x 2 + y 2 ) is always positive. But if (r, 6) and (.R, w) are the 
polar coordinates of the points (x, y) and (X, F), respectively, the formulae of 
transformation may be written in the form R = r 2 , u = 2 0. As r varies from a 
to b (a < b) and varies from OtO7T + a(0<a< Tf/2), the point (.R, u) describes 
a circular ring bounded by two circles of radii a 2 and b 2 . But to every value of 
the angle u between and 2a correspond two values of 6, one of which lies 
between and a, the other between it and it + a. The region described by the 
point (X, Y) may be realized by forming a circular ring of paper which partially 
overlaps itself. 

128. Transformation of double integrals. First method. Retaining 
the hypotheses made above concerning the regions A and A l and the 
formulae (16), let us consider a function F(x, y) which is continuous 
in the region A. To any subdivision of the region A l into subregions 
a lf a 2 , , a n corresponds a subdivision of the region A into sub- 
regions a l} a 2 , , a n . Let to, and <r, be the areas of the two corre 
sponding subregions a, and a,., respectively. Then, by formula (17), 



(I),- = CTf 



D(u i} vj 



where , and v { are the coordinates of some point in the region a,. 
To this point (,-, v,-) corresponds a point x, =/(,-, v,), y,- = <(w,-, ^.-) 
of the region a,-. Hence, setting *(M, v) = -F[/(w, v), <(w, v)], we 
may write 

D(f, <# 



D(u i} Vi ) 



whence, passing to the limit, we obtain the formula 
(18) f f F(x, y) dx dy = I I F[f(u, v}, <f>(u, v) 

J J(A) J */Ui) 



D(u, v) 



dudv. 



VI, 128] 



CHANGE OF VARIABLES 



267 



Hence to perform a transformation in a double integral x and y should 
be replaced by their values as functions of the new variables u and v, 
and dx dy should be replaced by | A | du dv. We have seen already 
how the new field of integration is determined. 

In order to find the limits between which the integrations should 
be performed in the calculation of the new double integral, it is in 
general unnecessary to construct the contour C\ of the new field 
of integration A lf For, let us consider u and v as a system of 
curvilinear coordinates, and let one of the variables u and v in the 
formulae (16) be kept constant while the other varies. We obtain 
in this way two systems of curves u = const, and v = const. By 
the hypotheses made above, one and only one curve of each of these 
families passes through any 
given point of the region A. 
Let us suppose for definite- 
ness that a curve of the 
family v = const, meets the 
contour C in at most two 
points MI and MI which cor 
respond to values w t and u z 
of u (HI < w 2 ), and that each (< 

of the (v) curves which meets 7 ^//^^f^T^ 

the contour C lies between 
the two curves v a and 
v b (a<b~). In this case 
we should integrate first 
with regard to u, keeping v constant and letting u vary from ^ 
to w 2 , where u l and u z are in general functions of v, and then inte 
grate this result between the limits a and b. 

The double integral is therefore equal to the expression 




Fia. 27 



f do f V[/( 
U a \Ju. 



, V), 



A change of variables amounts essentially to a subdivision of the 
field of integration by means of the two systems of curves (u) and (v). 
Let w be the area of the curvilinear quadrilateral bounded by the 
curves (it), (u + du), (v), (v -f dv ), where du and dv are positive. 
To this quadrilateral corresponds in the uv plane a rectangle whose 
sides are du and dv. Then, by formula (17), w = | A(, 77)) du dv, where 
lies between u and u + du, and 77 between v and v + dv. The expres 
sion | b.(u, v) | du dv is called the element of area in the system of 



268 DOUBLE INTEGRALS [VI, 129 

coordinates (u, v~). The exact value of u> is o> = \\ A(M, v) \ + c \ du dv, 
where c approaches zero with du and dv. This infinitesimal may be 
neglected in finding the limit of the sum ^,F(x, y) w, for since A(M, v) 
is continuous, we may suppose the two (u) curves and the two 
(y) curves taken so close together that each of the e s is less in ab 
solute value than any preassigned positive number. Hence the abso 
lute value of the sum 2F(x, y^tdudv itself may be made less than 
any preassigned positive number. 

129. Examples. 1) Polar coordinates. Let us pass from rectangu 
lar to polar coordinates by means of the transformation x = p cos w, 
y = p sin GO. We obtain all the points of the xy plane as p varies 
from zero to + oo and u> from zero to 2?r. Here A = p ; hence the 
element of area is p da> dp, which is also evident geometrically. Let 
us try first to evaluate a double integral extended over a portion of 
the plane bounded by an arc AB which intersects a radius vector in 
at most one point, and by the two straight lines OA and OB which 
make angles ^ and to 2 with the x axis (Fig. 17, p. 189). Let 
R = <( w ) be the equation of the arc AB. In the field of integration 
o> varies from ^ to o> 2 and p from zero to R. Hence the double inte 
gral of a function f(x, y) has the value 



r 2 C R 

I M /O c 

i/ojj i/O 



cos GO, p sin CD) p dp . 



If the arc AB is a closed curve enclosing the origin, we should 
take the limits GO X = and co 2 = 2?r. Any field of integration can 
be divided into portions of the preceding types. Suppose, for 
instance, that the origin lies outside of the contour C of a given 
convex closed curve. Let OA and OB be the two tangents from 
the origin to this curve, and let RI =/ 1 (<o) and 7? 2 =/2( w ) be the 
equations of the two arcs ANB and A MB, respectively. For a 
given value of o> between o^ and o> 2 , p varies from RI to 7? 2 , and 
the value of the double integral is 



/* 

6?(0 I /(p 

, A 



cos GO, p sin w) p dp. 
2) Elliptic coordinates. Let us consider a family of confocal conies 



CHANGE OF VARIABLES 



269 



where X denotes an arbitrary parameter. Through every point of the plane pass 
two conies of this family, an ellipse and an hyperbola, for the equation (19) 




FIG. 28 



has one root X greater than c 2 , and another positive root p. less than c 2 , for any 
values of x and y. From (19) and from the analogous equation where X is 
replaced by p we find 



(20) 



y 



V(X - 



To avoid ambiguity, we shall consider only the first quadrant in the xy plane. 
This region corresponds point for point in a one-to-one manner to the region of 
the X/u plane which is bounded by the straight lines 

X = c 2 , M = 0, n = c z . 

It is evident from the formulae (20) that when the point (X, /u) describes the 
boundary of this region in the direction indicated by the arrows, the point (a;, y) 
describes the two axes Ox and Oy in the sense indicated by the arrows. The 
transformation is therefore inverse, which is verified by calculating A : 



= D(x, y) = 
D(X, M ) 



130. Transformation of double integrals. Second method. We shall 
now derive the general formula (18) by another method which 
depends solely upon the rule for calculating a double integral. We 
shall retain, however, the hypotheses made above concerning the 
correspondence between the points of the two regions A and A. 
If the formula is correct for two particular transformations 



x = f(n, v) , 



, v ) , 



it is evident that it is also correct for the transformation obtained 
by carrying out the two transformations in succession. This follows 
at once from the fundamental property of functional determinants 
(30) 



_ , 

D(u , v ) D(u, v} D(u , v ) 



270 



DOUBLE INTEGRALS 



[VI, 130 



Similarly, if the formula holds for several regions A, B, C, -, L, 
to which correspond the regions A l9 B 1) C lf , L 19 it also holds for 
the region A -{- B -\- C + + L. Finally, the formula holds if the 
transformation is a change of axes : 

x = x + x cos a y sin a, y = y + x sin a + y cos a. 
Here A = 1, and the equation 



ff 

J J(A) 



F(x, y} dx dy 



F(x + x cos y sin a, y -f x sin a + y cos a) dx dy 1 



is satisfied, since the two integrals represent the same volume. 

We shall proceed to prove the formula for the particular trans 
formation 

(21^ x = d>(x is 1 } 11 = ?/ 

which carries the region A into a region A which is included between 
the same parallels to the x axis, y = y and y = y. We shall sup 
pose that just one point of A corresponds to any given point of A and 

conversely. If a paral 
lel to the x axis meets 
the boundary C of the 
region A in at most two 
points, the same Avill be 
true for the boundary 
C" of the region A . To 
any pair of points m 

and m 1 on C whose or- 

x 

dinates are each y cor 
respond two points ?tt 
and m{ of the contour C . But the correspondence may be direct or 
inverse. To distinguish the two cases, let us remark that if c<f>/dx is 
positive, x increases with x , and the points m and m l and m and 
m{ lie as shown in Fig. 29 ; hence the correspondence is direct. On 
the other hand, if d<f>/dx is negative, the correspondence is inverse. 
Let us consider the first case, and let x , Xi, x n , x[ be the abscissae 
of the points ra , m^ m. , m{, respectively. Then, applying the for 
mula for change of variable in a simple integral, we find 




/in . 



FIG. 29 



f 

Jr 



F(x, 



(* , y ), y ] 



VI, 130] CHANGE OF VARIABLES 271 

where y and y are treated as constants. A single integration gives 
the formula 



r tfi r* 1 r y < c x( 

I dy\ F(x,y)dx=\ dy I F^( 

J J^O J* J u 



But the Jacobian A reduces in this case to d<f>/dx , and hence the 
preceding formula may be written in the form 

ff F(x, y}dxdy = ff F[A(x , y }, y ] *\dx dy . 

J J(A) J J(A ) 

This formula can be established in the same manner if d<f>/dx is 
negative, and evidently holds for a region of any form whatever. 

In an exactly similar manner it can be shown that the trans 
formation 

(22) x = z , y = t(x , y ) 
leads to the formula 

ff F(r, y)dxd U = ff F[.r , t(x , y )] I A | dx dy , 

J J(A) J J{4") 

where the new field of integration .1 corresponds point for point tc 
the region A. 

Let us now consider the general formulae of transformation 

(23) x =f(x l} y x ), y =f, (x,, y,) , 

where for the sake of simplicity (a-, y) and (x l} y^) denote the coor 
dinates of two corresponding points m and MI with respect to the 
same system of axes. Let A and A l be the two corresponding regions 
bounded by contours C and C 1} respectively. Then a third point m , 
whose coordinates are given in terms of those of m and M l by the 
relations x = x l} y = y, will describe an auxiliary region A , which 
for the moment we shall assume corresponds point for point to each 
of the two regions A and A^ The six quantities x, y, x lf y l} x , y 
satisfy the four equations 



whence we obtain the relations 

(24) x = sr l , y =/i(*nyi), 

which define a transformation of the type (22). From the equation 
y =/!(# , yj) we find a relation of the form y l = TT(X , y ) ; hence 
we may write 

(25) x =f(x , yO = 4>(x , y }, y = y . 



272 DOUBLE INTEGRALS [VI, 131 

The given transformation (23) amounts to a combination of the two 
transformations (24) and (25), for each of which the general formula 
holds. Therefore the same formula holds for the transformation (23). 

Remark. We assumed above that the region described by the 
point m corresponds point for point to each of the regions A and 
A v . At least, this can always be brought about. For, let us con 
sider the curves of the region A l which correspond to the straight 
lines parallel to the x axis in A. If these curves meet a parallel to 
the y axis in just one point, it is evident that just one point m of 
A will correspond to any given point m of A. Hence we need 
merely divide the region A t into parts so small that this condition 
is satisfied in each of them. If these curves were parallels to the 
y axis, AVO should begin by making a change of axes. 

131. Area of a curved surface. Let S be a region of a curved sur 
face free from singular points and bounded by a contour F. Let S 
be subdivided in any way whatever, let s { be one of the subregions 
bounded by a contour y i} and let m^ be a point of s t . Draw the tan 
gent plane to the surface S at the point m i} and suppose s,- taken so 
small that it is met in at most one point by any perpendicular to 
this plane. The contour y,- projects into a curve y- upon this plane ; 
we shall denote the area of the region of the tangent plane bounded 
by yl by o- f . As the number of subdivisions is increased indefinitely 
in such a way that each of them is infinitesimal in all its dimensions, 
the sum 2o\- approaches a limit, and this limit is called the area of 
the region S of the given surface. 

Let the rectangular coordinates x, y, z of a point of S be given in 
terms of two variable parameters u and v by means of the equations 

(26) x =/(M, v), y = <f>(u, v), z = if/(u, v) , 

in such a way that the region S of the surface corresponds point for 
point to a region R of the uv plane bounded by a closed contour C. 
We shall assume that the functions /, <j>, and if/, together with their 
first partial derivatives, are continuous in this region. Let R be 
subdivided, let r i be one of the subdivisions bounded by a contour c,-, 
and let w t - be the area of r { . To r { corresponds on S a subdivision s t 
bounded by a contour y f . Let a- t be the corresponding area upon the 
tangent plane defined as above, and let us try to find an expression 
for the ratio o^/o^. 

Let a { , (3^ y f be the direction cosines of the normal to the surface S 
at a point m f (ic t -, y t) z ( ~) of s t which corresponds to a point (u i} Vf) 



VI, 131] 



CHANGE OF VARIABLES 



273 



of r f . Let us take the point m t as a new origin, and as the new axes 
the normal at m { and two perpendicular lines m { X and m t Y in the 
tangent plane whose direction cosines with respect to the old axes are 
a , ft , y and a", ft", y", respectively. Let X, Y, Z be the coordinates 
of a point on the surface S with respect to the new axes. Then, 
by the well-known formulae for transformation of coordinates, we 
shall have 

X = a (x - *,.) + ?(y- y t ) +?*(*- *,) , 

F = a"(x - x { ) + ft"(y y t ) + y"(z f ) , 
Z = or, (x Xi) + fti (y ?/,) -f- y { (z ,-) . 

The area tr,- is the area of that portion of the A F plane which is 
bounded by the closed curve which the point (X, F) describes, as 
the point (u, v) describes the contour c . Hence, by 127, 

p(x, y) 

^- Hswr^r 

where u\ and v\ are the coordinates of some point inside of c i . An 
easy calculation now leads us to the form 




or, by the well-known relations between the nine direction cosines, 



=s 3. < Oi 



D(z,x) D(x,y) 

D(u[, v t ) T * D(u[, v 



><X> O 

Applying the general formula (17), we therefore obtain the equation 



*D(4O 

where u\ and v\ are the coordinates of a point of the region r { in the 
uv plane. If this region is very small, the point (M,-, v) is very near 
the point (?* f , v,.), and we may write 

T)fii >y\ T)fii v\ T)( f f\ Dff *r\ 

iJ\Jh Z) f^\J[t Z) . J \ z > ) _ u \ z i c ) i f 

\ ~r C> ) 

D(Ui, V t ) 



u, 



D(y, 



where the absolute value of 6 does not exceed unity. Since the 
derivatives of the functions /, <, and ^ are continuous in the 



274 DOUBLE INTEGRALS [VI, i;;i 

region R, we may assume that the regions r t have been taken so 
small that each of the quantities e,-, e-, e- is less than an arbitrarily 
preassigned number rj. Then the supplementary term will certainly 
be less in absolute value than 3^0, where O is the area of the 
region R. Hence that term approaches zero as the regions s t 
(and Tf) all approach zero in the manner described above, and the 
sum So-, approaches the double integral 



() 






(x, y) 



D(u, v) D(u, v) 



du dv , 



where a, ft, y are the direction cosines of the normal to the surface S 
at the point (u, v~). 

Let us calculate these direction cosines. The equation of the 
tangent plane ( 39) is 



whence 

a B y 1 



D(x, _ , ( 



D(u, v) D(u, v) D(u, v) 
Choosing the positive sign in the last ratio, we obtain the formula 

D (y> 



, Q - 

D(u, v) P D(u, v) 7 D(u, v) 



.D(1 

The well-known identity 

(aft - ba ) 2 + (be 1 - c& ) 2 + (ca f - ac ) 2 

which was employed by Lagrange, enables us to write the quantity 
under the radical in the form EG F 2 , where 

(27) 

the symbol S indicating that a; is to be replaced by y and z succes 
sively and the three resulting terms added. It follows that the area 
of the surface S is given by the double integral 



(28) A == / / 

J J(B 



- F*dudv. 

(B) 



VI, 132] CHANGE OF VARIABLES 275 

The functions E, F, and G play an important part in the theory 
of surfaces. Squaring the expressions for dx, dy, and dz and adding 
the results, we find 

(29) ds 2 = dx 2 + dy* + dz* = E du 2 + 2Fdu dv + G dv 2 . 

It is clear that these quantities E, F, and G do not depend upon 
the choice of axes, but solely upon the surface S itself and the inde 
pendent variables u and v. If the variables u and v and the sur 
face 5 are all real, it is evident that EG F 2 must be positive. 



132. Surface element. The expression V EG F 2 du dv is called the 
element of area of the surface S in the system of coordinates (u, v). 
The precise value of the area of a small portion of the siirface bounded 
by the curves (u), (u + du), (v~), (v -\- dv) is (j\EG F 2 + t)dudv, 
where e approaches zero with du and dv. It is evident, as above, 
that the term e du dv is negligible. 

Certain considerations of differential geometry confirm this result. 
For, if the portion of the surface in question be thought of as a small 
curvilinear parallelogram on the tangent plane to S at the point (u, v), 
its area will be equal, approximately, to the product of the lengths 
of its sides times the sine of the angle between the two curves (u) 
and (v~). If we further replace the increment of arc by the differ 
ential ds, the lengths of the sides, by formula (29), are ^/Edu and 
^/Gdv, if du and dv are taken positive. The direction parameters of 
the tangents to the two curves (u) and (v) are dx/du, dy/du, dz/du 
and dx/dv, dy/dv, dz/dv, respectively. Hence the angle a between 
them is given by the formula 

^ dx dx 
^ cu Gv F 



COS a = 



VI" 
iV ( 
\du 



whence sin a = V EG .F 2 /V EG. Forming the product mentioned, 
we find the same expression as that given above for the element of 
area. The formula for cos a shows that F = when and only when 
the two families of curves (u) and (y~) are orthogonal to each other. 
When the surface S reduces to a plane, the formulae just found 
reduce to the formulae found in 128. For, if we set if/(u, v) = 0, 
we find 



276 



DOUBLE INTEGRALS 



[VI, 132 



whence, by the rule for squaring a determinant, 

dx ex 
du dv 



du dv 



E F 
F G 



= EG - F 2 . 



Hence ^EG F 2 reduces to |A|. 

Examples. 1) To find the area of a region of a surface whose equa 
tion is z = f(x } y} which projects on the xy plane into a region R in 
which the function f(x, y), together with its derivatives p = df/dx and 
q = df/dy, is continuous. Taking x and y as the independent vari 
ables, we find E = \ + p 2 , F pq, G = 1 + q 2 , and the area in ques 
tion is given by the double integral 



(30) 



= ff 

J J(R) 



= ff 

J J(R 



(R) COS y 

where y is the acute angle between the z axis and the normal to the 
surface. 

2) To calculate the area of the region of a surface of revolution 
between two plane sections perpendicular to the axis of revolution. 
Let the axis of revolution be taken as the z axis, and let z = f(x) 
be the equation of the generating curve in the xz plane. Then the 
coordinates of a point on the surface are given by the equations 



z=f(p), 

where the independent variables p and o> are the polar coordinates of 
the projection of the point on the xy plane. In this case we have 



p-), F=0, G = P *. 

To find the area of the portion of the surface bounded by two plane 
sections perpendicular to the axis of revolution whose radii are p t and 
p 2 , respectively, p should be allowed to vary from Pl to p 2 (pi< p 2 ) and 
) from zero to 2?r. Hence the required area is given by the integral 



and can therefore be evaluated by a single quadrature. 
the arc of the generating curve, we have 

ds* = rf 2 + <fe = < 



If s denote 



VI, 133] IMPROPER INTEGRALS 277 

and the preceding formula may be written in the form 

f p * 

A = I 27Tp ds . 

Jpl 

The geometrical interpretation of this result is easy : 2jrp ds is 
the lateral area of a frustum of a cone whose slant height is ds and 
whose mean radius is p. Replacing the area between two sections 
whose distance from each other is infinitesimal by the lateral area 
of such a frustum of a cone, we should obtain precisely the above 
formula for A. 

For example, on the paraboloid of revolution generated by revolv 
ing the parabola x 9 = 2pz about the z axis the area of the section 
between the vertex and the circular plane section whose radius is r is 



III. GENERALIZATIONS OF DOUBLE INTEGRALS 
IMPROPER INTEGRALS SURFACE INTEGRALS 

133. Improper integrals. Let f(x, y) be a function which is con 
tinuous in the whole region of the plane which lies outside a closed 
contour F. The double integral of f(x, y) extended over the region 
between F and another closed curve C outside of F has a finite value. 
If this integral approaches one and the same limit no matter how 
C varies, provided merely that the distance from the origin to the 
nearest point of C becomes infinite, this limit is defined to be the 
value of the double integral extended over the whole region 
outside F. 

Let us assume for the moment that the function f(x, y] has a 
constant sign, say positive, outside F. In this case the limit of the 
double integral is independent of the form of the curves C. For, 
let Ci, C z , , C n , be a sequence of closed curves each of which 
encloses the preceding in such a way that the distance to the nearest 
point of C n becomes infinite with n. If the double integral / extended 
over the region between F and C n approaches a limit /, the same will 
be true for any other sequence of curves C{, C^, , C m , which 
satisfy the same conditions. For, if I m be the value of the double 
integral extended over the region between F and C m , n may be 
chosen so large that the curve C n entirely encloses C m , and wa 
shall have / < / < /. Moreover / increases with m. Hence I m 



278 DOUBLE INTEGRALS [VI, 133 

has a limit / < I. It follows in the same manner that I < I . Hence 
/ = I, i.e. the two limits are equal. 

As an example let us consider a function f(x, y), which outside a 
circle of radius r about the origin as center is of the form 



where the value of the numerator \f/(x, y~) remains between two posi 
tive numbers m and M. Choosing for the curves C the circles 
concentric to the above, the value of the double integral extended 
over the circular ring between the two circles of radii r and R is 
given by the definite integral 



\l/(p cos to, p sin o>)p dp 



/, C 
J, 



It therefore lies between the values of the two expressions 



By 90, the simple integral involved approaches a limit as R 
increases indefinitely, provided that 2a 1 > 1 or a > 1. But it 
becomes infinite with R if a < 1. 

If no closed curve can be found outside which the function /(a;, y) 
has a constant sign, it can be shown, as i 89, that the integral 
ffffa y)dxdy approaches a limit if the integral // f(x, y) \dxdy 
itself approaches a limit. But if the latter integral becomes infinite, 
the former integral is indeterminate. The following example, due 
to Cayley, is interesting. Let f(x, y) = sin (x 2 + if), and let us inte 
grate this function first over a square of side a formed by the axes 
and the two lines x = a, y = a. The value of this integral is 

r a r a 

I dx I sin (a; 2 + y z }dy 
Jo Jo 

= I sinx^dx x I cosy 2 dy+ I cosx*dx x I siny*dy. 
Jo Jo Jo Jo 

As a increases indefinitely, each of the integrals on the right has 
a limit, by 91. This limit can be shown to be V?r/2 in each case ; 
hence the limit of the whole right-hand side is TT. On the other 
hand, the double integral of the same function extended over the 
quarter circle bounded by the axes and the circle x 1 + y 2 = R 3 is 
equal to the expression 



/I, 134] IMPROPER INTEGRALS 279 



7T 

r* r 

\ du \ 

*/0 yo 



which, as R becomes infinite, oscillates between zero and 7r/2 and 
does not approach any limit whatever. 

We should define in a similar manner the double integral of a 
function f(x, y) which becomes infinite at a point or all along a line. 
First, we should remove the point (or the line) from the field of 
integration by surrounding it by a small contour (or by a contour 
very close to the line) which we should let dimmish indefinitely. 
For example, if the function f(x, y) can be written in the form 

f(x ) = _ ^(*> y) 





in the neighborhood of the point (a, b), where \}/(x, y) lies between 
two positive numbers m and M, the double integral of f(x, y) 
extended over a region about the point (a, b) which contains no 
other point of discontinuity has a finite value if and only if a is 
less than unity. 

134. The function B(p, q). We have assumed above that the contour C n 
recedes indefinitely in every direction. But it is evident that we may also sup 
pose that only a certain portion recedes to infinity. This is the case in the above 
example of Cayley s and also in the following example. Let us take the function 

/(x, y) = 4x 2 P- 1 y 2 - 1 e-* 2 -! 2 , 

where p and q are each positive. This function is continuous and positive in the 
first quadrant. Integrating first over the square of side a bounded by the axes 
and the lines x a and y = a, we find, for the value of the double integral, 

C 2x 2 J>-ie-* 2 dx x C a 2y*i-ie-*dy. 

Jo Jo 

Each of these integrals approaches a limit as a becomes infinite. For, by the 
definition of the function T(p) in 92, 

T(p)= f +C V-ie-<(K, 

Jo 

whence, setting t = x 2 , we find 

(31) r(p)= C + 2x*P-*e-**dx. 

Jo 

Hence the double integral approaches the limit T(p) T(q) as a becomes infinite. 
Let us now integrate over the quarter circle bounded by the axes and the 
circle z 2 + y 2 = R 2 . The value of the double integral in polar coordinates is 



f*R /* 

I 2p 2< -P + i)- l e-P dp x I 
Jo J 



280 DOUBLE INTEGRALS [VI, u 

As R becomes infinite this product approaches the limit 

T(p + q)B(p, q), 
where we have set 

rr 

(32) B(p,q) = C 2 2cos?P- l <l>sin 2 <i-i<t>d<}>. 

Jo 

Expressing the fact that these two limits must be the same, we find the equation 

(33) T(p)T(q) = r(p + q)B(p, q). 

The integral B(p, q} is called Euler s integral of the first kind. Setting t = sin 2 <, 
it may be written in the form 

(34) B(p, q)= f t9-i(l - t)v- 1 dt. 

JQ 

The formula (33) reduces the calculation of the function B(p, q) to the calcu 
lation of the function T. For example, setting p q = 1/2, we find 



whence F(l/2) = vV. Hence the formula (31) gives 



f. 



2 
In general, setting q = 1 p and taking p between and 1 , we find 



t l-p) = f (~ 

Jo \ l 



T(p)T(l-p) = 
We shall see later that the value of this integral is jr/sin pit. 

135. Surface integrals. The definition of surface integrals is analogous to that 
of line integrals. Let S be a region of a surface bounded by one or more curves F. 
We shall assume that the surface has two distinct sides in such a way that if one 
side be painted red and the other blue, for instance, it will be impossible to pass 
from the red side to the blue side along a continuous path which lies on the sur 
face and which does not cross one of the bounding curves.* Let us think of S as 
a material surface having a certain thickness, and let m and m be two points 
near each other on opposite sides of the surface. At m let us draw that half of 
the normal mn to the surface which does not pierce the surface. The direction 
thus defined upon the normal will be said, for brevity, to correspond to that side 
of the surface on which m lies. The direction of the normal which corresponds 
to the other side of the surface at the point m will be opposite to the direction 
just defined. 

Let z = <f>(x, y) be the equation of the given surface, and let -S be a region of 
this surface bounded by a contour F. We shall assume that the surface is met 
in at most one point by any parallel to the z axis, and that the function 0(z, y) 

* It is very easy to form a surface which does not satisfy this condition. We need 
only deform a rectangular sheet of paper ABCD by pasting the side B C to the side AD 
in such a way that the point C coincides with A and the point B with D. 



VI, 135] SURFACE INTEGRALS 281 

is continuous inside the region A of the xy plane which is bounded by the curve C 
into which T projects. It is evident that this surface has two sides for which 
the corresponding directions of the normal make, respectively, acute and obtuse 
angles with the positive direction of the z axis. We shall call that side whose 
corresponding normal makes an acute angle with the positive z axis the upper 
side. Now let P(x, y, z) be a function ofi the three variables x, y, and z which 
is continuous in a certain region of space which contains the region S of the sur 
face. If z be replaced in this function by <(x, y), there results a certain function 
P [x, y, <p(x, y)] of x and y alone ; and it is natural by analogy with line integrals 
to call the double integral of this function extended over the region A, 



(35) f f P [x, y, 0(x, y)] dx dy , 

v <s (A) 



the surface integral of the function P(x, y, z) taken over the region S of the given 
surface. Suppose the coordinates x, y, and z of a point of S given in terms of two 
auxiliary variables u and v in such a way that the portion S of the surface corre 
sponds point for point in a one-to-one manner to a region R of the uv plane. Let 
da- be the surface element of the surface S, and 7 the acute angle between the posi 
tive z axis and the normal to the upper side of S. Then the preceding double 
integral, by 131-132, is equal to the double integral 



(36) rr P(x, y, z)cos7d<r, 



where x, y, and z are to be expressed in terms of u and v. This new expression 
is, however, more general than the former, for cos 7 may take on either of two 
values according to which side of the surface is chosen. When the acute angle 7 
is chosen, as above, the double integral (35) or (36) is called the surface integral 

(37) 

extended over the upper side of the surface S. But if 7 be taken as the obtuse 
angle, every element of the double integral will be changed in sign, and the new 
double integral would be called the surface integral / / Pdxdy extended over the 
lower side of <S. In general, the surface integral// Pdx dy is equal to the double 
integral (35) according as it is extended over the upper or the lower side of S. 

This definition enables us to complete the analogy between simple a^id double 
integrals. Thus a simple integral changes sign when the limits are interchanged, 
while nothing similar has been developed for double integrals. With the gen 
eralized definition of double integrals, we may say that the integral///(x, y) dx dy 
previously considered is the surface integral extended over the upper side of the 
xy plane, while the same integral with its sign changed represents the surface 
integral taken over the under side. The two senses of motion for a simple inte 
gral thus correspond to the two sides of the xy plane for a double integral. 

The expression (36) for a surface integral evidently does not require that the 
surface should be met in at most one point by any parallel to the z axis. In the 
same manner we might define the surface integrals 

ff Q(z, y, z) dy dz, ff R ( x v* z ) dz dx 



282 



DOUBLE INTEGRALS 



[VI, 136 



and the more general integral 

f fp(x, y, z)dxdy + Q(x, y, z)dydz + R(x, y, z)dzdx. 
This latter integral may also be written in the form 

C f [Pcos? + Qcosa + JJcos/3]d<r, 

where a, , 7 are the direction angles of the direction of the normal which cor 
responds to the side of the surface selected. 

Surface integrals are especially important in Mathematical Physics. 

136. Stokes* theorem. Let L be a skew curve along which the functions 
P(x, y, z), Q(x, y, z), R(x, y, z) are continuous. Then the definition of the line 
integral 

Pdx + Qdy +Rdz 



C 

J( 



(L) 

taken along the line L is similar to that given in 93 for a line integral taken 
along a plane curve, and we shall not go into the matter in detail. If the curve L 
is closed, the integral evidently may be broken up into the sum of three line inte 
grals taken over closed plane curves. Applying Green s theorem to each of these, 
it is evident that we may replace the line integral by the sum of three double 
integrals. The introduction of surface integrals enables us to state this result in 
very compact form. 

Let us consider a two-sided piece S of a surface which we shall suppose for 
definiteness to be bounded by a single curve P. To each side of the surface 
corresponds a definite sense of direct motion along the contour r. We shall 
assume the following convention : At any point M of the contour let us draw 
that half of the normal Mn which corresponds to the side of the surface under 
consideration, and let us imagine an observer with his head at n and his feet at M ; 

we shall say that that is the positive sense 
of motion which the observer must take in 
order to have the region S at his left hand. 
Thus to the two sides of the surface corre 
spond two opposite senses of motion along 
the contour F. 

Let us first consider a region S of a sur 
face which is met in at most one point by 
any parallel to the z axis, and let us suppose 
the trihedron Oxyz placed as in Fig. 30, 
where the plane of the paper is the yz plane 
and the x axis extends toward the observer. 
To the boundary F of S will correspond a 
closed contour C in the xy plane ; and these 
two curves are described simultaneously in 
the sense indicated by the arrows. Let 

z = /(x, y) be the equation of the given surface, and let P(x, y, z) be a function 
which is continuous in a region of space which contains S. Then the line inte 
gral f r P(x, y, z) dx is identical with the line integral 




VI, 136] SURFACE INTEGRALS 283 

/ P[x, y, <p(x, y)]dx 

/(C) 

taken along the plane curve C. Let us apply Green s theorem ( 126) to this 
latter integral. Setting 



P(x, y) = P[z, y, <(>(x, y)] 
for definiteness, we find 



, y) _ dP dP_ d<fi _8P_ S 



y cy cz cy cy cz cosy 

where or, /3, 7 are the direction angles of the normal to the upper side of S. 
Hence, by Green s theorem, 



-D, - ^j 
P(x, y)dx 



C C 
= I 

J J (A 



dP 



cz d V cos y 



where the double integral is to be taken over the region A of the xy plane 
bounded by the contour C. But the right-hand side is simply the surface 
integral 



cos /3 cos 7 ) do- 

dz dy I 

extended over the upper side of S ; and hence we may write 

f P(x, ?/, z)dx = / I --dzdx dxdy. 

J(F) J J (S) cz dy 

This formula evidently holds also when the surface integral is taken over the 
other side of <S, if the line integral is taken in the other direction along F. And 
it also holds, as does Green s theorem, no matter what the form of the surface 
may be. By cyclic permutation of x, y, and z we obtain the following analogous 
formulae : 



f Q(x, y,z)dy= 

J ^ 

C R(x,y,z)dz= C i dydz- dzdx. 

J<n J J (S) sy dx 

Adding the three, we obtain Stokes theorem in its general form : 

/ 
J P(x, y, z)dx + Q(z, y, z)dy + R(x, y, z)dz 

dP\, ^(dE dQ\. /d ^ 

-- )dxdy + ( --- 2 }dydz + ( --- \dzdx. 

ty/ 






The sense in which T is described and the side of the surface over which the 
double integral is taken correspond according to the convention made above. 



284 DOUBLE INTEGRALS [VI, 137 



IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS 

137. Volumes. Let us consider, as above, a region of space bounded 
by the xy plane, a surface S above that plane, and a cylinder whose 
generators are parallel to the z axis. We shall suppose that the 
section of the cylinder by the plane z is a contour similar to 
that drawn in Fig. 25, composed of two parallels to the y axis and two 
curvilinear arcs APB and A QB . If % f(x, y) is the equation of the 
surface S, the volume in question is given, by 124, by the integral 



r b rvt 
V = \ dx I f(x, y)dy. 

Jo. Jy, 



Now the integral f " 2 /(o:, y}dy represents the area A of a section of 
this volume by a plane parallel to the yz plane. Hence the preceding 
formula may be written in the form 



(39) V=f 

U a 



b 

A dx. 



The volume of a solid bounded in any way whatever is equal 
to the algebraic sum of several volumes bounded as above. For 
instance, to find the volume of a solid bounded by a convex closed 
surface we should circumscribe the solid by a cylinder whose gen 
erators are parallel to the z axis and then find the difference between 
two volumes like the preceding. Hence the formula (39) holds for 
any volume which lies between two parallel planes x = a and x = b 
(a < I) and which is bounded by any surface whatever, where A 
denotes the area of a section made by a plane parallel to the two 
given planes. Let us suppose the interval (a, ) subdivided by the 
points a, x l} x z , , x n _ l} b, and let A , A u , A ; , be the areas 
of the sections made by the planes x = a, x = x lt -, respectively. 
Then the definite integral $*k dx is the limit of the sum 



The geometrical meaning of this result is apparent. For A,-_! (#,- a;,-_i), 
for instance, represents the volume of a right cylinder whose base is 
the section of the given solid by the plane x = a-,-_i and whose height 
is the distance between two consecutive sections. Hence the volume 
of the given solid is the limit of the sum of such infinitesimal cylin 
ders. This fact is in conformity with the ordinary crude notion of 
volume. 



VI, 138] APPLICATIONS 285 

If the value of the area A be known as a fnnction of x, the vol 
ume to be evaluated may be found by a single quadrature. As an 
example let us try to find the volume of a portion of a solid of revo 
lution between two planes perpendicular to the axis of revolution. 
Let this axis be the x axis and let z = f(x) be the equation of the 
generating curve in the xz plane. The section made by a plane par 
allel to the yz plane is a circle of radius f(x). Hence the required 
volume is given by the integral TT^ [/(x)] 2 er. 

Again, let us try to find the volume of the portion of the ellipsoid 

n n n 

^4.^4. f! = i 

a* b* c 2 

bounded by the two planes x = x , x = X. The section made by a 
plane parallel to the plane x = is an ellipse whose semiaxes are 
b Vl x 2 /a 2 and c Vl x 2 /a 2 . Hence the volume sought is 

a-A / Y 8 - 

* 



V= I Trbc l- 
cA \ 

To find the total volume we should set x = a and X = a, which 
gives the value %irabc. 

138. Ruled surface. Prismoidal formula. When the area A is an integral 
function of the second degree in z, the volume may be expressed very simply 
in terms of the areas B and B of the bounding sections, the area 6 of the mean 
section, and the distance h between the two bounding sections. If the mean 
section be the plane of yz, we have 

V = C + (te 2 + 2mx + n) dx = 21 + 2na. 

J a 3 

But we also have 

h = 2a, b = n, B = Itf + 2ma + n , B = la* - 2ma + n , 
whence n = &, a = h/2, 2Za 2 = B + B 26. These equations lead to the formula 

(40) F=^[B + B + 4&], 

o 

which is called the prismoidal formula. 

This formula holds in particular for any solid bounded by a ruled surface and 
two parallel planes, including as a special case the so-called prismoid.* For, 
let y = ax + p and z = bx + q be the equations of a variable straight line, where 
a, 6, p, and q are continuous functions of a variable parameter t which resume 
their initial values when t increases from t Q to T. This straight line describes 

* A prismoid is a solid bounded by any number of planes, two of which are paral 
lel and contain all the vertices. TRANS. 



286 



DOUBLE INTEGRALS 



[VI, 139 



a ruled surface, and the area of the section made by a plane parallel to the plane 
x = is given, by 94, by the integral 



/T 

= I (ax + p)(b x + q )dt, 

Jto 



where a , & , c , d denote the derivatives of a, 6, c, d with respect to t. These 
derivatives may even be discontinuous for a finite number of values between t 
and T, which will be the case when the lateral boundary consists of portions of 
several ruled surfaces. The expression for A may be written in the form 

r r r r r r 

A = x 2 / ab dt + x I (aq + pb )dt + \ pq dt, 
Jt n Jt Jt 

where the integrals on the right are evidently independent of x. Hence the 
formula (40) holds for the volume of the given solid. It is worthy of notice that 
the same formula also gives the volumes of most of the solids of elementary geometry . 

139. Viviani s problem. Let C be a circle described with a radius OA (= R) 
of a given sphere as diameter, and let us try to find the volume of the portion 
of the sphere inside a circular cylinder whose right section is the circle C. 
Taking the origin at the center of the sphere, one fourth the required volume 
is given by the double integral 

- 



- z - 2/ 2 dxdy 



extended over a semicircle described on OA as diameter. Passing to polar coor 
dinates p and w, the angle u varies from to it/2, and p from to R cos w. Hence 
we find 





FIG. 31 



v l rim m \* RS i* 2 \ 

= - I (R 3 R 3 Bin* w) du = ( --- ) 
4 3 Jo 3 \2 3/ 

If this volume and the volume inside the cylinder 
which is symmetrical to this one with respect to 
the z axis be subtracted from the volume of the 
whole sphere, the remainder is 



- 

3 3 \2 

Again, the area ft of the portion of the sur 
face of the sphere inside the given cylinder is 



dxdy. 



Replacing p and q by their values x/z and y/z, respectively, and passing to 
polar coordinates, we find 



VI, 140] APPLICATIONS 287 

f ftRcoiui r> , * 

* I Rf>d(> = 4 f 2 - 

Jo V^ 2 -, 2 Jo 



tt = 4fl 2 f 2 (l-sinw)dw = 4R 2 (- - 

Jo a 



Subtracting the area enclosed by the two cylinders from the whole area of the 
sphere, the remainder is 

it 

^ 

140. Evaluation of particular definite integrals. The theorems estab 
lished above, in particular the theorem regarding differentiation 
under the integral sign, sometimes enable us to evaluate certain defi 
nite integrals without knowing the corresponding indefinite integrals 
We proceed to give a few examples. 

Setting 

A = F(a\ = r* log ( 1 + aX ^ dx 

the formula for differentiation under the integral sign gives 

dA _ log (1+a 2 ) f xdx 

da ~ 1 + a 2 



Breaking up this integrand into partial fractions, we find 
x 1 Ix + a a 



1+ax 
whence 






x dx log (1 + <* 2 ) . 

- fai 



It follows that 



a log(l4- 2 ) 

- arc tan - 



o/-i i 
rfa 1+a 2 2(1+ a 2 ) 

whence, observing that ,4 vanishes when a = 0, we may write 



r log (i+<> ., r g a 

A = \ 0/1 I 2\ ^ + I T~i - "2 

Jo 2(1+ a 2 ) J 1+a 2 



Integrating the first of these integrals by parts, we finally find 
A = - arc tan a log (1 + a 2 ) . 

m 



288 DOUBLE INTEGRALS [VI, 140 

Again, consider the function x v . This function is continuous 
when x lies between and 1 and y between any two positive 
numbers a and b. Hence, by the general formula of 123, 



/i /-& /?> /* \ 

dx I x v dy = I dy I x y dx. 
t/a J a UQ 

r l 

Jo 



But 

>i 

ipy flf 

I ,, I 1 I ,, i 1 

o Ly + I J y + L 

hence the value of the right-hand side of the previous equation is 

- = log 
On the other hand, we have 

r 







log a; 
whence 

= log 



log x 

In general, suppose that P(x, y) and Q(x, ?/) are two functions 
which satisfy the relation dP/dy = dQ/dx, and that x , x^ y , y^ are 
given constants. Then, by the general formula for integration 
under the integral sign, we shall have 



p ap p p0Q 

c?o; I -dy=\ dy \ -^-dx, 

J* ty J, Jo J J* ^ 

or 

r*i r^ 

(41) 1 [P(x, y,}-P(x, y )-]dx= I [Q(^ 15 y)- Q 

^^o ^"o 



Cauchy deduced the values of a large number of definite inte 
grals from this formula. It is also closely and simply related to 
Green s theorem, of which it is essentially only a special case. 
For it may be derived by applying Green s theorem to the line 
integral fPdx + Qdy taken along the boundary of the rectangle 
formed by the lines x = X Q , x x^ y = y , y = y l . 

In the following example the definite integral is evaluated by a 
special device. The integral 



= I log (1 2a cos x + n- 2 ) dx 
Jo 



VI, 110] APPLICATIONS 289 

has a finite value if \a\ is different from unity. This function 
F(<x) has the following properties. 

1) F(- a) = F(a). For 

F(- a) = I log (1 + 2a cos x + a- 2 ) dx, 
Jo 

or, making the substitution x = TT y, 

F(- a) I log (1 2a cos y + a 2 ) dy = F(a) . 
Jo 

2) F(a 2 ) = 2F(a). For we may set 

2F() = F() + F(-a), 
whence 

2F(a) = I [log (1 2a cos x + a 2 ) + log (1 + 2a COS x + a 2 )] dx 
Jo 

= I log (I - 2a* cos 2x + a*)dx. 
Jo 

If we now make the substitution 2x = y, this becomes 

1 C" 
2F(a) = - I \og(l-2a i cosy + a 4 )dy 

* Jo 

i r 2 * 

+ - I log (1 - 2a 2 cos y + a 4 ) dy. 

Making a second substitution y = lit z in the last integral, we 
find 

X2rr f*n 

\og(l-2a 2 cosy + a*)dy= I log (1 - 2 2 cos z + a*)dz, 
Jo 



which leads to the formula 



From this result we have, successively, 

F(a) = \ F(a 2 ) = \ F(a") = = 

If | a | is less than unity, a 2 " approaches zero as n becomes infinite. 
The same is true of F(a 2 "), for the logarithm approaches zero. 
Hence, if | a| < 1, we have F(a) = 0. 



290 DOUBLE INTEGRALS [VI, 141 

If | a | is greater than unity, let us set a = I//?. Then we find 



F(a) = f 
Jo 



2 cos x 1 



= 1 log(l-2/?cos;r 
Jo 



where \ft\ is less than unity. Hence we have in this case 
F(a) = TT log ft 2 = TT log a 2 . 

Finally, it can be shown by the aid of Ex. 6, p. 205, that F( 1) = ; 
hence F(a) is continuous for all values of a. 

141. Approximate value of logF(n + l). A great variety of devices may be 
employed to find either the exact or at least an approximate value of a definite 
integral. We proceed to give an example. We have, by definition, 

n +00 

T(n + l) = I x n e- x dx. 
Jo 

The function x n e~ x assumes its maximum value n n e~ n f or x = n. As x increases 
from zero to n, x n e~ x increases from zero to n n e~ n (n>0), and when x increases 
from n to + oo, x n e~ x decreases from n n er n to zero. Likewise, the function 
n n er n e- tt increases from zero to n n er n as t increases from oo to zero, and 
decreases from n n e~ n to zero as t increases from zero to + oo. Hence, by the 
substitution 



n -* = 



(42) 

the values of x and t correspond in such a way that as t increases from oo 
to + oo, x increases from zero to + oo. 

It remains to calculate dx/dt. Taking the logarithmic derivative of each side 
of (42), we find 

dx, 2tx 

dt x n 
We have also, by (42), the equation 

tfl = x n n log f - j 

W 

For simplicity let us set x = n + z, and then develop log (1 + z/n) by Taylor s 
theorem with a remainder after two terms. Substituting this expansion in the 
value for < 2 , we find 

nz 2 



[2 
n 



where 6 lies between zero and unity. From this we find, successively, 



x - 



VI, 142] APPLICATIONS 291 

whence, applying the formula for change of variable, 

Y(n + 1) = 2n n e-" */- C "e-^dt + 2n"er n C *V< 2 (1 

\ / I/ 00 V 00 

The first integral is 

,_ 

e- <2 dt = Vnr. 



As for the second integral, though we cannot evaluate it exactly, since we do 
not know 0, we can at least locate its value between certain fixed limits. For 
all its elements are negative between <x> and zero, and they are all positive 
between zero and + oo. Moreover each of the integrals f_ n , / +cc is ^ ess m 
absolute value than / + ter^dt = 1/2. It follows that 



(43) F(n + 1) = \/2nn n e~ n ( - 



V*n/ 
where u lies between 1 and + 1- 

If n is very large, w/V2n is very small. Hence, if we take 



= n"e- 

as an approximate value of T(n + 1), our error is relatively small, though the 
actual error may be considerable. Taking the logarithm of each side of (43), we 
find the formula 

(44) log r(n + 1) = (n + 1) logn - n + 1 log(27r) + e, 

where e is very small when n is very large. Neglecting e, we have an expression 
which is called the asymptotic value of logT(n + 1). This formula is inter 
esting as giving us an idea of the order of magnitude of a factorial. 

142. D Alembert s theorem. The formula for integration under the integral 
sign applies to any function /(x, y) which is continuous in the rectangle of inte 
gration. Hence, if two different results are obtained by two different methods 
of integrating the function /(x, y), we may conclude that the function /(x, y) is 
discontinuous for at least one point in the field of integration. Gauss deduced 
from this fact an elegant demonstration of d Alembert s theorem. 

Let F(z) be an integral polynomial of degree m in z. We shall assume for 
definiteness that all its coefficients are real. Replacing z by p(cosw + isinw), 
and separating the real and the imaginary parts, we have 

F(z) = P + iQ, 
where 

P = A p m cosm<,} + Aip m - l cos(m !)&> 

Q = A p m smmw + A\p m ~ l sin(m 1) w 



If we set V = arc tan (P/Q), we shall have 

aP_ p aQ q^-P^ 

8V dp dp 8V 8u du 

a/T = P*+ Q2 ~^ ~ P2 + Q2 

and it is evident, without actually carrying out the calculation, that the second 

derivative is of the form 

&V M 



292 DOUBLE INTEGRALS [VI, Exs. 

where M is a continuous function of p and u. This second derivative can only 
be discontinuous for values of p and <a for which P and Q vanish simultaneously, 
that is to say, for the roots of the equation F(z) = 0. Hence, if we can show that 
the two integrals 



(4o) 



r\ c R w* C R * c"*r* 

I du I -dp, I dp I du 

Jo Jo d P du Jo Jo d P Cu 



are unequal for a given value of R, we may conclude that the equation F(z) = 
has at least one root whose absolute value is less than R. But the second inte 
gral is always zero, for 



f" 

Jo 



^ 

au = I 



and dV/dp is a periodic function of w, of period 2x. Calculating the first inte 
gral in a similar manner, we find 



X 



and it is easy to show that dV/du is of the form 

dV _ mA?) p" m + - 
du Ag p 2m -f 

where the degree of the terms not written down is less than 2m in p, and where 
the numerator contains no term which does not involve p. As p increases indefi 
nitely, the right-hand side approaches m. Hence R may be chosen so large 
that the value of cV/dw, for p = R, is equal to m + e, where e is less than m 
in absolute value. The integral f Q 2n ( m + e) du is evidently negative, and 
hence the first of the integrals (45) cannot be zero. 

EXERCISES 

1. At any point of the catenary defined in rectangular coordinates by the 
equation 

0/5 - 

y = - I e a + e 
2 \ 

let us draw the tangent and extend it until it meets the x axis at a point T. 
Revolving the whole figure about the x axis, find the difference between the areas 
described by the arc AM of the catenary, where A is the vertex of the catenary, 
and that described by the tangent M T (I) as a function of the abscissa of the 
point M, (2) as a function of the abscissa of the point T. 

[Licence, Paris, 1889.] 

2. Using the usual system of trirectangular coordinates, let a ruled surface 
be formed as follows : The plane zOA revolves about the x axis, while the gen 
erating line -D, which lies in this plane, makes with the z axis a constant angle 
whose tangent is X and cuts off on OA an intercept OC equal to Xa0, where a 
is a given length and 6 is the angle between the two planes zOx and zOA. 



VI.Exs.] EXERCISES 293 

1) Find the volume of the solid bounded by the ruled surface and the planes 
xOy, zOx, and zOA, where the angle 6 between the last two is less than 2n. 

2) Find the area of the portion of the surface bounded by the planes xOy, 

zOx, zOA. 

[Licence, Paris, July, 1882.] 

3. Find the volume of the solid bounded by the xy plane, the cylinder 
&2 X 2 _|_ a 2y2 a 2&2 ? anc [ the elliptic paraboloid whose equation in rectangular 

coordinates is 

2z _ & y* 

c p 2 q* 

[Licence, Paris, 1882.] 

4. Find the area of the curvilinear quadrilateral bounded by the four con- 
focal conies of the family 



which are determined by giving X the values c 2 /3, 2c 2 /3, 4c a /3, 5c 2 /3, respectively. 

[Licence, Besan^on, 1885.] 

5. Consider the curve 

y = \/2 (sin x cos x) , 

where x and y are the rectangular coordinates of a point, and where x varies 
from 7T/4 to 5?r/4. Find : 

1) the area between this curve and the x axis ; 

2) the volume of the solid generated by revolving the curve about the x axis ; 

3) the lateral area of the same solid. 

[Licence, Montpellier, 1898.] 

6. In an ordinary rectangular coordinate plane let A and B be any two 
points on the y axis, and let AMB be any curve joining A and B which, together 
with the line AB, forms the boundary of a region AMBA whose area is a pre- 
assigned quantity S. Find the value of the following definite integral taken 
over the curve AMB : 

my] dx + [<t> (y}e x - m]dy, 

where m is a constant, and where the function <f>(y), together with its derivative 

<t> (y), is continuous. 

[Licence, Nancy, 1895.] 

7. By calculating the double integral 

> + 00 / +00 

/O 

in two different ways, show that, provided that a is not zero, 

+ 00 . 

sin ax , , ft 



ft + 00 ft +00 

I e- x ! sinaxdydx 
Jo Jo 



8. Find the area of the lateral surface of the portion of an ellipsoid of revo 
lution or of an hyperboloid of revolution which is bounded by two planes perpen 
dicular to the axis of revolution. 



294 



DOUBLE INTEGRALS 



[VI, Exs. 



9*. To find the area of an ellipsoid with three unequal axes. Half of the total 
area A is given by the double integral 




1- 



X 2 - 



- C" 



dxdy 



extended over the interior of the ellipse 6 2 x 2 + a 2 ?/ 2 = a 2 ft 2 . Among the methods 
employed to reduce this double integral to elliptic integrals, one of the simplest, 
due to Catalan, consists in the transformation used in 125. Denoting the 
integrand of the double integral by v, and letting v vary from 1 to + cc, it is 
easy to show that the double integral is equal to the limit, as I becomes infinite, 

of the difference 

il 

7tabl(P-l) 



nab 




This expression is an undetermined form ; but we may write 




- -- > 





and hence the limit considered above is readily seen to be 

+00 

/.2 / 

Ttab 




ab 




dv 



1,2 J ( V 2_l + ^W U 2_1 + C2 \ 

i \ V a 2 \ &*/ - 



10*. If from the center of an ellipsoid whose semiaxes are a, 6, c a perpen 
dicular be let fall upon the tangent plane to the ellipsoid, the area of the surface 
which is the locus of the foot of the perpendicular is equal to the area of an 
ellipsoid whose semiaxes are be/a, ac/b, ab/c. 

[WILLIAM ROBERTS, Journal de Liouville, Vol. XI, 1st series, p. 81.] 



VI, EXB.] EXERCISES 295 

11. Evaluate the double integral of the expression 

(* - V)"f(v) 

extended over the interior of the triangle bounded by the straight lines y = i , 
y = x, and x = X in two different ways, and thereby establish the formula 



dx (x - y) n f(y) dy = I y ^ f(y) dy . 

From this result deduce the relation 

x- C X f(x)dx= l f\x - y)f(y)dy. 
J *9 (n-l)!^x 



In a similar manner derive the formula 



X f(x) dx = \ - f *(x* - 2/ 2 ) 
2 . 4 . 6 2n Jr 



and verify these formulae by means of the law for differentiation under the 
integral sign. 



CHAPTER VII 

MULTIPLE INTEGRALS 
INTEGRATION OF TOTAL DIFFERENTIALS 

I. MULTIPLE INTEGRALS CHANGE OF VARIABLES 

143. Triple integrals. Let F(x, y, z) be a function of the three 
variables x, y, z which is continuous for all points M, whose rec 
tangular coordinates are (cc, y, z), in a finite region of space () 
bounded by one or more closed surfaces. Let this region be sub 
divided into a number of subregions (e^, (e 2 ~), , (e n ), whose vol 
umes are v l9 v z , , v n , and let (., 17,-, ; ) be the coordinates of any 
point m,- of the subregion (e,-). Then the sum 

(i) 

approaches a limit as the number of the subregions (e.) is increased 
indefinitely in such a way that the maximum diameter of each of 
them approaches zero. This limit is called the triple integral of 
the function F(x, y, z) extended throughout the region (), and 
is represented by the symbol 

(2) J J J F(x, y, z} dx dy dz . 

The proof that this limit exists is practically a repetition of the 
proof given above in the case of double integrals. 

Triple integrals arise in various problems of Mechanics, for 
instance in finding the mass or the center of gravity of a solid 
body. Suppose the region () filled with a heterogeneous sub 
stance, and let p.(x, y, z) be the density at any point, that is to say, 
the limit of the^ratio of the mass inside an infinitesimal sphere about 
the point (x, y, ) as center to the volume of the sphere. If ^ and /x 2 
are the maximum and the minimum value of /t in the subregion (e^), 
it is evident that the mass inside that subregion lies between ^v, 
and fujv,-; hence it is equal to v,./i(,., 17.-, ,.), where (,-, 7;,-, ,-) is a 
suitably chosen point of the subregion (e ( ). It follows that the total 

296 



VII, 143] INTRODUCTION CHANGE OF VARIABLES 297 

mass is equal to the triple integral fffp dx dy dz extended through 
out the region (). 

The evaluation of a triple integral may be reduced to the suc 
cessive evaluation of three simple integrals. Let us suppose first 
that the region () is a rectangular parallelepiped bounded by the 
six planes x = x , x = X, y y , y F, z = z , z = Z. Let (E) 
be divided into smaller parallelepipeds by planes parallel to the 
three coordinate planes. The volume of one of the latter is 
(x f o^j) (y k y k _ l ) (z t ,_!), and we have to find the limit of 
the sum 

(3) S = 

where the point (, w , ij ikl , lW ) is any point inside the corresponding 
parallelepiped. Let us evaluate first that part of S which arises 
from the column of elements bounded by the four planes 



taking all the points ( ikl , y M , iw ) upon the straight line x = ar.-.j, 
y = y/c-i- This column of parallelepipeds gives rise to the sum 

(x { - x i _ l )(y t - yt_i)[^(*,-_i, y^i, Ci)(i - ) H ], 

and, as in 123, the s may be chosen in such a way that the 
quantity inside the bracket will be equal to the simple integral 

*(**-! yt-i) = I F( x i-i> yt-u *) d *- 
Jz 

It only remains to find the limit of the sum 



But this limit is precisely the double integral 

$(x, ?/) dx dy 



extended over the rectangle formed by the lines x = x , x = X, 
y = 2/o > y = Y. Hence the triple integral is equal to 

I dx I *(x, y)dy, 
J*, Jv 

or, replacing &(x, y) by its value, 

pX f*Y f*Z 

(4) I dx dy F(x,y,z)dz. 

J *,* Jvn *J *a. 



298 



MULTIPLE INTEGRALS 



[VII, 144 



The meaning of this symbol is perfectly obvious. During the first 
integration x and y are to be regarded as constants. The result will be 
a function of x and y, which is then to be integrated between the limits 
?/ and F, x being regarded as a constant and y as a variable. The 
result of this second integration is a function of x alone, and the last 
step is the integration of this function between the limits x and X. 

There are evidently as many ways of performing this evaluation 
as there are permutations on three letters, that is, six. For instance, 
the triple integral is equivalent to 



rZ r>X f*Y r> 

I dz I dx I F(x, y, z)dy = I 

J* J* J J*a 



where *() denotes the double integral of F(x, y, z) extended over 
the rectangle formed by the lines x = x , x = X, y = y , y = Y. We 
might rediscover this formula by commencing with the part of the 
sum S which arises from the layer of parallelepipeds bounded by the 
two planes z = z l _ l , z = z ( . Choosing the points (, 77, ) suitably, 
the part of S which arises from this layer is 



and the rest of the reasoning is similar to that above. 



144. 

manner 



Let us now consider a region of space bounded in any 
whatever, and let us divide it into subregions such that any 

line parallel to a suitably chosen 
fixed line meets the surface which 
bounds any subregion in at most 
two points. We may evidently 
restrict ourselves without loss of 
generality to the case in which a 
line parallel to the z axis meets 
the surface in at most two points. 
The points upon the bounding 
surface project upon the xy plane 
into the points of a region A 
bounded by a closed contour C. 
To every point (x, y) inside C cor 
respond two points on the bound- 
ing surface whose coordinates are 




FIG. 32 



i and 



= fa( x > y}- We shall suppose that the functions 
are continuous inside C, and that <j> l <<j> t . Let us now 



VII, 144] INTRODUCTION CHANGE OF VARIABLES 299 

divide the region under consideration by planes parallel to the coor 
dinate planes. Some of the subdivisions will be portions of paral 
lelepipeds. The part of the sum (1) which arises from the column 
of elements bounded by the four planes x = Xf_ l} x = x { , y = y k _i, 
y = y k is equal, by 124, to the expression 

(x f - ,._!> (y k - y t _,) I J F(x { _ l , y t _ lt z) dz + f , k J , 

where the absolute value of e ik may be made less than any preassigned 
number c by choosing the parallel planes sufficiently near together. 
The sum 



approaches zero as a limit, and the triple integral in question is 
therefore equal to the double integral 

<(x, y) dx dy 

extended over the region (J) bounded by the contour C, where the 
function 3>(z, y) is denned by the equation 



r** 

= I F (*> y> *)** 

Jz. 



If a line parallel to the y axis meets the contour C in at most two 
points whose coordinates are y = ^ (x) and y = fa (x), respectively, 
while x varies from x l to a; 2 , the triple integral may also be written 
in the form 



r x t rt r z t 
(5) / dx \ dy I F(x, y, z)dz. 

i/Xj J y^ Jz l 



The limits z 1 and z 2 depend upon both x and y, the limits y t and y z 
are functions of x alone, and finally the limits x l and a? 2 are constants. 
We may invert the order of the integrations as for double inte 
grals, but the limits are in general totally different for different 
orders of integration. 

Note. If ^(x) be the function of x given by the double integral 
/ r z * 

*() =| dy I F(x, y, z) dz 

Jv, J*, 



300 MULTIPLE INTEGRALS [VII, 145 

extended over the section of the given region by a plane parallel to 
the yz plane whose abscissa is x, the formula (5) may be written 



This is the result we should have obtained by starting with the 
layer of subregions bounded by the two planes x = Xf_ l} x = x t . 
Choosing the points (, r), ) suitably, this layer contributes to the 
total sum the quantity 



Example. Let us evaluate the triple integral fffz dx dy dz extended through 
out that eighth of the sphere x 2 + y* + z 2 = fi* which lies in the first octant. If 
we integrate first with regard to z, then with regard GO y, and finally with regard 
to x, the limits are as follows : x and y being given, z may vary from zero to 
V-R 2 x 2 y 2 ; x being given, y may vary from zero to V.R 2 x 2 ; and x itself 
may vary from zero to B. Hence the integral in question has the value 



/*// (*R ~V2-:r2 />V ff2-x2-y4 

I I I zdxdydz = I dx I dy I zdz, 

J J J Jo Jo Jo 

whence we find successively 



f 

Jo 



zdz = ( 2 -x 2 -y 2 ), 



if 

2 Jo 



and it merely remains to calculate the definite integral \f K (R* x 2 )^dx, which, 
by the substitution x = B cos <f>, takes the form 



Hence the value of the given triple integral is, by 116, x 



145. Change of variables. Let 
(6) 



x = /(?/,, v, w), 
y = <(?*, v, w}, 
z = $(u, v, w) , 



be formulae of transformation which establish a one-to-one corre 
spondence between the points of the region (E~) and those of another 
region (^i). We shall think of u, v, and w as the rectangular coor 
dinates of a point with respect to another system of rectangular 



VII, 145] INTRODUCTION CHANGE OF VARIABLES 301 



coordinates, in general different from the first. If F(x, y, ) is a 
continuous function throughout the region (), we shall always have 



(7) 



mF(x, y, 2) dx dy dz 
;> 

mF[f(u, v, w), 
i> 



D(f, 



D(u, v, 



dudv dw, 



where the two integrals are extended throughout the regions (E) 
and (E-i), respectively. This is the formula for change of variables 
in triple integrals. 

In order to show that the formula (7) always holds, we shall 
commence by remarking that if it holds for two or more particular 
transformations, it will hold also for the transformation obtained by 
carrying out these transformations in succession, by the well-known 
properties of the functional determinant ( 29). If it is applicable 
to several regions of space, it is also applicable to the region obtained 
by combining them. We shall now proceed to show, as we did for 
double integrals, that the formula holds for a transformation which 
leaves all but one of the independent variables unchanged, for 
example, for a transformation of the form 

(8) x = x 1 y = ?/ a = \!/(x y 1 z } 

We shall suppose that the two points M(x, y, z) and M (x , y ,z } are 
referred to the same system of rectangular axes, and that a parallel 
to the z axis meets the surface which 
bounds the region (E) in at most two 
points. The formulae (8) establish a corre 
spondence between this surface and another 
surface which bounds the region (E ). The 
cylinder circumscribed about the two sur 
faces with its generators parallel to the 
z axis cuts the plane z = along a closed 
curve C. Every point m. of the region A 
inside the contour C is the projection of 
two points m^ and m 2 of the first surface, 
whose coordinates are z 1 and z 2 , respectively, and also of two 
points m[ and m 2 of the second surface, whose coordinates are z[ 
and z 2 , respectively. Let us choose the notation in such a way 
that z l <z 3 , and z[<z 2 . The formulae (8) transform the point m^ 
into the point m{, or else into the point m 2 . To distinguish the 
two cases, we need merely consider the sign of d\f//dz . If dty/dz is 




C 



FIG. 33 



302 



MULTIPLE INTEGRALS 



[VII, 145 



positive, z increases with % , and the points m l and m 2 go into the 
points m{ and ra^, respectively. On the other hand, if c\(//dz is 
negative, z decreases as 2 increases, and m t and m 2 go into m s and 
m{, respectively. In the previous case we shall have 

I F(x, y, z)dz = I F^x, y, t(x, y, 2 )] ^ dz , 

*/Zi Jz i 

whereas in the second case 

/"**> /* 2 9 / 

I F(x, y, z) dz = - I F\x, y, $(x, y, **)] jfc dz . 

Jz^ Jz{ 



In either case we may write 



(9) f *F(x, y, z)dz = f V|>, y, t(x, y, z )] 

C/Zj *) ZJ 



dz . 



If we now consider the double integrals of the two sides of this 
equation over the region A, the double integral of the left-hand side, 



II dxdy I F(x, y, z)dz, 

J J(A) Je t 



is precisely the triple integral fffF(x, y, z) dx dy dz extended through 
out the region (E). Likewise, the double integral of the right-hand 
side of (9) is equal to the triple integral of 



F[x , y , 



extended throughout the region (E ~), which readily follows when 
x and y are replaced by x and y , respectively. Hence we have in 
this particular case 



J J J (E) 



F(x, y, z) dx dy dz 



dx di dz . 



But in this case the determinant D(x, y, z)/D(x , y , z 1 } reduces to 
d(j//dz . Hence the formula (7) holds for the transformation (8). 

Again, the general formula (7) holds for a transformation of the 
type 

(10) *=/(* , y ,*% y = <t>(x ,y<,zi), 



VII, 145] INTRODUCTION CHANGE OF VARIABLES 303 

where the variable z remains unchanged. We shall suppose that 
the formulae (10) establish a one-to-one correspondence between 
the points of two regions (E) and ( ), and in particular that the 
sections R and R made in (E) and ( ), respectively, by any 
plane parallel to the xy plane correspond in a one-to-one manner. 
Then by the formulae for transformation of double integrals we 
shall have 




(11) 

I = / / RT fi-r. i/ ?. \ A*(r . -j/ j- Y f. l 

D(x , y ) 



, ^ dx dy . 



The two members of this equation are functions of the variable 
z = z alone. Integrating both sides again between the limits z v 
and 2 , between which z can vary in the region (), we find the 
formula 



Jill F ( x > y, z ) dx d v dz 
J J J(E) 
= rrr F^^^,^^,^,^ 

J J J(E ) 



D(x ,y<) 



dx dy dz . 



But in this case D(x, y, z)/D(x , y , ) = D(x, y)/D(x , y ~). Hence 
the formula (7) holds for the transformation (10) also. 

We shall now show that any change of variables whatever 

(13) x = f(x l} y l} zj, y = <j>( Xl , y l} zj, z = ^(x lt y 1} j) 

may be obtained by a combination of the preceding transformations. 
For, let us set x = x l} y = y lt z = z. Then the last equation of 

(13) may be written z = if/(x , y , z^, whence z v = 7r(x , y , z ~). 
Hence the equations (13) may be replaced by the six equations 

(14) x =/[* , y , TT(X<, y } )], y = ftx , V <, ir(x t y , )], z = z , 

(15) x = xi, y = yi, * = <j>(xi, yi, ,). 

The general formula (7) holds, as we have seen, for each of the 
transformations (14) and (15). Hence it holds for the transforma 
tion (13) also. 

We might have replaced the general transformation (13), as the 
reader can easily show, by a sequence of three transformations of 
the type (8). 



304 



MULTIPLE INTEGRALS 



[VII, 146 



146. Element of volume. Setting F(x, y, K) = I in the formula (7), 
we find 



//***// 



du dv dw . 



The left-hand side of this equation is the volume of the region ("). 
Applying the law of the mean to the integral on the right, we find 
the relation 



(16) 



D(u, v, 



where Vi is the volume of (^i), and , rj, are the coordinates of some 
point in (#1). This formula is exactly analogous to formula (17), 
Chapter VI. It shows that the functional determinant is the limit 
of the ratio of two corresponding infinitesimal volumes. 

If one of the variables u, v, w in (6) be assigned a constant value, 
while the others are allowed to vary, we obtain three families of 
surfaces, u = const., v = const., w = const., by means of which the 
region (T) may be divided into subregions analogous to the paral- 
lelopipeds used above, each of which is bounded by six curved faces. 
The volume of one of these subregions bounded by the surfaces 
(u), (u + du), (v}, (v + dv), (w), (w + dw} is, by (16), 



AF = 



. -j- c f c?< c?v dw , 
D(w, v, w) J 



where du, dv, and div are positive increments, and where c is infini 
tesimal with du, dv, and dw. The term e du dv dw may be neglected, 
as has been explained several times ( 128). The product 



(17) 



dV = 



D(u, v, w} 



is the principal part of the infinitesimal AF, and is called the element 
of volume in the system of curvilinear coordinates (u, v, w). 

Let c?s 2 be the square of the linear element in the same system of 
coordinates. Then, from (6), 



o/ ^i /* ^ -P 

= ~- - - 



dx = ~- du 
cu 



% ~JL 

dw. dy = -r- du 



dz = -^- du 
cu 



=H l du*+H. t dv*+H 3 dw 2 +2F 1 dvdw+2F. t dudiv+2F 3 dudv, 



vv Ow 

whence, squaring and adding, we find 



VII, 14fi] INTRODUCTION CHANGE OF VARIABLES 305 
the notation employed being 

(19) 

(7 IT C OC G OT (JOT O *y 

dw du dw ^ du dv 

where the symbol AJ means, as usual, that x is to be replaced by y 
and z successively and the resulting terms then added. 

The formula for dV is easily deduced from this formula for ds*. 
For, squaring the functional determinant by the usual rule, we find 



dx dy 


dz 


2 


du du 


du 








ffl 


dx d y 


d_z 


T"t 

1 n 


dv do 


CV 


* 3 






F, 


dx dy 


dz 


- 1 2 


dw dw 


div 





F l 



F, 



= M, 



whence the element of volume is equal to \M du dv dw. 

Let us consider in particular the very important case in which 
the coordinate surfaces (u), (?;), (w) form a triply orthogonal system, 
that is to say, in which the three surfaces which pass through any 
point in space intersect in pairs at right angles. The tangents to 
the three curves in which the surfaces intersect in pairs form a tri- 
rectangular trihedron. It follows that we must have FI = 0, F 2 = 0, 
^ = 0; and these conditions are also sufficient. The formulae for 
dV and ds* then take the simple forms 



a. 

(20) ds 2 = H l du* + H 2 dv 2 + H s dw 2 , 



dV = 



These formulae may also be derived from certain considerations of 
infinitesimal geometry. Let us suppose du, dv, and dw very small, 
and let us substitute in place of the small subregion defined above a 
small parallelopiped with plane faces. Neglecting infinitesimals of 
higher order, the three adjacent edges of the parallelopiped may be 
taken to be \fn\ dii, ^/If 2 dv, and \/^ dw, respectively. The for 
mulae (20) express the fact that the linear element and the element of 
volume are equal to the diagonal and the volume of this parallelo 
piped, respectively. The area ^/H 1 H 2 du dv of one of the faces repre 
sents in a similar manner the element of area of the surface (w). 

As an example consider the transformation to polar coordinates 



(21) x = p sin0cos<, y = p sin 



z = 



p cos 6, 



306 



MULTIPLE INTEGRALS 



[VII, 146 



where p denotes the distance of the point M(x, y, z) from the origin, 
6 the angle between OM and the positive z axis, and <f> the angle 
which the projection of OM on the xy plane makes with the positive 
x axis. In order to reach all points in space, it is sufficient to let p 
vary from zero to + <x>, from zero to TT, and < from zero to 2?r. 
From (21) we find 

(22) ds 2 = dp 2 + P 2 d6 2 + P 2 sin 2 0d<j> 2 , 
whence 

(23) dV = p 2 sin 6 dp dO d<f> . 

These formulae may be derived without any calculation, however. 
The three families of surfaces (/a), (0), (<) are concentric spheres 

about the origin, cones of revolution 
about the z axis with their vertices 
at the origin, and planes through 
the z axis, respectively. These 
surfaces evidently form a triply 
orthogonal system, and the dimen 
sions of the elementary subregion 
y are seen from the figure to be dp, 
p dO, p sin d<j> ; the formulae (22) 
and (23) now follow immediately. 
To calculate in terms of the va 
riables p, 0, and < a triple integral 

extended throughout a region bounded by a closed surface S, which 
contains the origin and which is met in at most one point by a radius 
vector through the origin, p should be allowed to vary from zero to R, 
where R = /(0, <f>) is the equation of the surface ; 6 from zero to TT ; 
and < from zero to 2?r. For example, the volume of such a surface is 




FIG. 34 



V= I d<j> I dO I P 2 sinOdp. 
Jo Jo Jo 



The first integration can always be performed, and we may write 

R 3 sin 



- c w d+ r 

i/O t/0 



dQ. 



Occasional use is made of cylindrical coordinates r, <a, and z defined 
by the equations x = r cos to, y = r sin w, z = z. It is evident that 



and 



dV = r dw dr dz . 



VII, 147] INTRODUCTION CHANGE OF VARIABLES 307 



147. Elliptic coordinates. The surfaces represented by the equation 



(24) 



X a X b X c 



-1=0, 



where X is a variable parameter and a > b > c> 0, form a family of confocal 
conies. Through every point in space there pass three surfaces of this family, 
an ellipsoid, a parted hyperboloid, and an unparted hyperboloid. For the equa 
tion (24) always has one root \i which lies between b and c, another root X 2 
between a and 6, and a third root X 3 greater than a. These three roots \i, X 2 , Xg 
are called the elliptic coordinates of the point whose rectangular coordinates are 
(x, y, z). Any two surfaces of the family intersect at right angles : if X be given 
the values Xi and X 2 , for instance, in (24), and the resulting equations be sub 
tracted, a division by Xi X 2 gives 



(25) 



- a) 



- 6)(X 2 - b) 



- c)(X 2 - c) 



= 0, 



which shows that the two surfaces (Xi) and (X 2 ) are orthogonal. 

In order to obtain x, y, and z as functions of Xi , X 2 , X 3 , we may note that the 
relation 

(X - o)(X - 6)(X - c) - x 2 (X - 6)(X - c) - y 2 (X - c)(X - a) - z 2 (X - a)(\ - b) 

= (X - Xi)(X - X 2 )(X - X 3 ) 

is identically satisfied. Setting X = a, X = 6, X = c, successively, in this equa 
tion, we obtain the values 

(X 3 - a)(a-X 1 )(q-X 2 ) 





(26) 



(a b)(a c) 



(a _ b)(b - c) 



(X 8 _ C )(X, 



(a c)(b c) 
whence, taking the logarithmic derivatives, 



, x 
dx = - 



2 V Xi - a 



X 2 a X 3 



^) 



*-5 






+ 



d\ 3 



b X 2 b X 3 b 






UZ I -p ~T I * 

2 \Xi c X 2 c X s c/ 

Forming the sum of the squares, the terms in dXidX 2 , dX 2 dX 8 , dX 3 dXi must dis 
appear by means of (25) and similar relations. Hence the coefficient of dXj is 



4 L(Xi - a) 2 (\! - 6) 2 (X! - c) 2 
or, replacing x, y, z by their values and simplifying, 

(27) 1 (X, - XQft, - XQ 

4(X 1 -a)(X 1 -6)(X 1 -c) 



308 MULTIPLE INTEGRALS [VII, 148 

The coefficients 3f 2 and 3f 8 of d\\ and d\g, respectively, may be obtained from 
this expression by cyclic permutation of the letters. The element of volume is 



therefore -VM i M 2 M z d\i d\ 2 d\ a . 

148. Dirichlet s integrals. Consider the triple integral 
xpyiz r (l x y z) dxdydz 



taken throughout the interior of the tetrahedron formed by the four planes 
x = 0, 2/ = 0, z = 0, x + y + z = 1. Let us set 

x + y + z = , y + z = TJ , z = |r;f , 
where , r/, f are three new variables. These formulae may be written in the form 



and the inverse transformation is 



When x, y, and z are all positive and x + y + z is less than unity, , 17, and f all 
lie between zero and unity. Conversely, if , 17, and f all lie between zero and 
unity, x, y, and z are all positive and x + y + z is less than unity. The tetra 
hedron therefore goes over into a cube. 

In order to calculate the functional determinant, let us introduce the auxiliary 
transformation X=f, F=)j, Z = ?;f , which gives x = X F, y = F Z , 
z = Z. Hence the functional determinant has the value 



D(x, y, z) = D(x, 77, z) D(JT, r, Z) = 
D(, T,, f) D(X, Y, Z) J>(|, r,, ^ 

and the given triple integral becomes 

f dg f *i rV + +r+2 (i- O i 7+r+i a- ^ra- f)df. 

t/O /0 /0 

The integrand is the product of a function of f, a function of 77, and a func 
tion of f. Hence the triple integral may be written in the form 



f p + 9 + r + 2(l _ lyfy X C 7 9 + - + l(l - T ? )?dr, X f 

Jo /o /o 



or, introducing T functions (see (33), p. 280), 

l) T(q + r + 2)T(p+l) 



T(p + q + r + s + 4) Y(p + q + r + 3) T(q + r + 2) 

Canceling the common factors, the value of the given triple integral is finally 
found to be 

(28) r(p+i)r( g +i)r(r+pr(+i) 



VII, 149] INTRODUCTION CHANGE OF VARIABLES 309 

149. Green s theorem.* A formula entirely analogous to (15), 126, may be 
derived for triple integrals. Let us first consider a closed surface S which is 
met in at most two points by a parallel to the z axis, and a function R(x, y, z) 
which, together with dR/dz, is continuous throughout the interior of this surface. 
All the points of the surface S project into points of a region A of the xy plane 
which is bounded by a closed contour C. To every point of A inside C corre 
spond two points of S whose coordinates are z\ = <f>\ (x, y) and z 2 = <j> 2 (x, y). 
The surface S is thus divided into two distinct portions /Si and 83 . We shall 
suppose that z\ is less than z 2 . 

Let us now consider the triple integral 

dxdydz 
dz 

taken throughout the region bounded by the closed surface S. A first integra 
tion may be performed with regard to z between the limits z\ and z 2 ( 144), 
which gives R(x, y, z 2 ) - R(x, y, z t ). The given triple integral is therefore 
equal to the double integral 



J J[Jfc(, y, z 2 ) - R(x, y, Zi)]dxdy 



over the region A. But the double integral f f R(x, y, z 2 )dxdy is equal to the 
surface integral ( 135) 

R(x, y, z)dxdy 



taken over the upper side of the surface /S 2 . Likewise, the double integral of 
R(x, y, Zi) with its sign changed is the surface integral 

ff R(x,y,z)dxdy 
J J(sj 

taken over the lower side of <Si . Adding these two integrals, we may write 

/ / / aT* 6 ^* = ff s R ^ y Z ) dxd 2/ 

where the surface integral is to be extended over the whole exterior of the sur 
face S. 

By the methods already used several times in similar cases this formula may 
be extended to the case of a region bounded by a surface of any form whatever. 
Again, permuting the letters z, y, and z, we obtain the analogous formulae 



III ^ dxd y dz 

III -Qdxdydz= C C Q(x, y, z)dzdx. 
JJJ** J J <^ 



* Occasionally called Ostrogradsky s theorem. The theorem of 126 is sometimes 
called Riemann s theorem. But the title Green s theorem is more clearly established 
and seems to be the more fitting. See Ency. der Math. Wiss., II, A, 7, b and c. 
TRANS. 



310 MULTIPLE INTEGRALS [VII, 150 

Adding these three formulae, we finally find the general Green s theorem for 
triple integrals : 




(29) 

= C I P(x, y, z) dy dz + Q(x, y, z) dz dx + R(x, y, z)dxdy, 

J J(S) 

where the surface integrals are to be taken, as before, over the exterior of the 
bounding surface. 

If, for example, we set P = x, Q = R = QorQ = y, P R = Q or R = z, 
P = Q = 0, it is evident that the volume of the solid bounded by S is equal to 
any one of the surface integrals 

(29 ) CC xdydz, CC ydzdx, CC zdxdy. 

J J(S) J J(S) J J(S) 

150. Multiple integrals. The purely analytical definitions which have been 
given for double and triple integrals may be extended to any number of vari 
ables. We shall restrict ourselves to a sketch of the general process. 

Let Xi , x 2 t %n be n independent variables. We shall say for brevity 
that a system of values x\ , x% , , x n of these variables represents a point in 
space of n dimensions. Any equation F(x\, x 2 , , x tl ) = 0, whose first member 
is a continuous function, will be said to represent a surface; and if F is of the 
first degree, the equation will be said to represent a plane. Let us consider the 
totality of all points whose coordinates satisfy certain inequalities of the form 

(30) ti(xi,X3,-.-,x H )<0, i = l, 2, .-., k. 

We shall say that the totality of these points forms a domain D in space of n 
dimensions. If for all the points of this domain the absolute value of each of 
the coordinates x,- is less than a fixed number, we shall say that the domain D is 
finite. If the inequalities which define D are of the form 

(31) xJ^X!<x}, x<;x2<x 2 , -, <<x n ^xj,, 

we shall call the domain a prismoid, and we shall say that the n positive quan 
tities x\ xf are the dimensions of this prismoid. Finally, we shall say that a 
point of the domain D lies on the frontier of the domain if at least one of the 
functions fr in (30) vanishes at that point. 

Now let Z) be a finite domain, and let f(x\ , x 2 , , x n ) be a function which 
is continuous in that domain. Suppose D divided into subdomains by planes 
parallel to the planes x t - = (t = 1, 2, , n), and consider any one of the pris- 
moids determined by these planes which lies entirely inside the domain D. 
Let Axi , Ax 2 , , Ax n be the dimensions of this prismoid, and let i , 2 , , n 
be the coordinates of some point of the prismoid. Then the sum 



(32) S = S/(fc, 2 ,--,) AX! Ax 2 Ax , 

formed for all the prismoids which lie entirely inside the domain Z), approaches 
a limit I as the number of the prismoids is increased indefinitely in such a way 



VII, 150] INTRODUCTION CHANGE OF VARIABLES 311 

that all of the dimensions of each of them approach zero. We shall call this 
limit I the n-tuple integral of f(x\ , x 2 , , z,,) taken in the domain D and shall 
denote it by the symbol 

1 =ff- "fffai *2, , z n )dzidz 2 dx n . 

The evaluation of an n-tuple integral may be reduced to the evaluation of 
n successive simple integrals. In order to show this in general, we need only 
show that if it is true for an (n l)-tuple integral, it will also be true for an 
n-tuple integral. For this purpose let us consider any point (xi , x 2 , , x n ) 
of D. Discarding the variable z n for the moment, the point (x\ , x 2 , , x n _ i) evi 
dently describes a domain D in space of (n 1) dimensions. We shall suppose 
that to any point (x 1? X2, , x n _i) inside of JK there correspond just two 
points on the frontier of Z), whose coordinates are (xi, X2, , x n _i; x^) and 
(xi, x 2 , , x n _i ; x^ 2) ), where the coordinates x^ and x^ 2) are continuous func 
tions of the n 1 variables x\ , x 2 , , x n -i inside the domain IX. If this con 
dition were not satisfied, we should divide the domain D into domains so small 
that the condition would be met by each of the partial domains. Let us now 
consider the column of prismoids of the domain D which correspond to the 
same point (xi , x 2 , , x n -i)- It is easy to show, as we did in the similar case 
treated in 124, that the part of S which arises from this column of prismoids is 

r r *w ~\ 

Ax 1 Ax 2 ---Ax n _ i J (i) /(x 1 ,x 2 , --^x^dXn + e , 

where | e | may be made smaller than any positive number whatever by choos 
ing the quantities Ax* sufficiently small. If we now set 



r 
(33) *(xi, x 2 , -, x n -i)=J (1) 



it is clear that the integral I will be equal to the limit of the sum 

S*(xi, x 2 , , x n _i)AxiAx 2 - Ax n _i, 
that is, to the (n l)-tuple integral 

(34) 1 = fff- --J^Xi, x 2 , , x n -i)dxi---dx n -i, 

in the domain ZK. The law having been supposed to hold for an (n l)-tuple 
integral, it is evident, by mathematical induction, that it holds in general. 

We might have proceeded differently. Consider the totality of points 
(xi , za , , Xn) for which the coordinate x n has a fixed value. Then the 
point (xi, x 2 , , x n _i) describes a domain 8 in space of (n 1) dimensions, 
and it is easy to show that the n-tuple integral I is also equal to the expression 

(35) I 

where 0(x n ) is the (. l)-tuple integral /// -ffdxi dxn-i extended through 
out the domain 5. Whatever be the method of carrying out the process, the limits 
for the various integrations depend upon the nature of the domain D, and 



312 MULTIPLE INTEGRALS [VII, 150 

vary in general for different orders of integration. An exception exists in case 
D is a prismoid denned by inequalities of the form 



The multiple integral is then of the form 



and the order in which the integrations are performed may be permuted in any 
way whatever without altering the limits which correspond to each of the 
variables. 

The formula for change of variables also may be extended to n-tuple integrals. 
Let 

(36) Xi = 0,-(zi, zg, , x n ), t = l, 2, -,n, 

be formulae of transformation which establish a one-to-one correspondence between 
the points (xi , z 2 .,--, x n ) of a domain jy and the points (xi , x 2 , , x n ) of a 
domain D. Then we shall have 



(37) 



The proof is similar to that given in analogous cases above. A sketch of the 
argument is all that we shall attempt here. 

1) If (37) holds for each of two transformations, it also holds for the trans 
formation obtained by carrying out the two in succession. 

2) Any change of variables may be obtained by combining two transforma 
tions of the following types : 

(38) Xi = xf, x 2 = Xa, , x n _i = x;,_i, x n n (xi, xg, , x;,), 

(39) zi 




3) The formula (37) holds for a transformation of the type (38), since the 
given n-tuple integral may be written in the form (34). It also holds for any 
transformation of the form (39), by the second form (35) in which the multiple 
integral may be written. These conclusions are based on the assumption that 
(37) holds for an (n l)-tuple integral. The usual reasoning by mathematical 
induction establishes the formula in general. 

As an example let us try to evaluate the definite integral 

I = CC- f xf x? 2 x" (1 - xi - x 2 ----- x n f dxi dx 2 --- dx n , 

where ai, cr 2 , , a n , are certain positive constants, and the integral is to be 
extended throughout the domain D defined by the inequalities 

G<X!, 0<z 2 , , 0<x n , xi + x 2 + +x n < 1. 
The transformation 



VII, 151] 



TOTAL DIFFERENTIALS 



313 



carries D into a new domain IX defined by the inequalities 

0<fi^l, 0<&<1, --, <<!, 
and it is easy to show as in 148 that the value of the functional determinant is 

, X 2 , -, X n ) _ t n-i fc -2 



The new integrand is therefore of the form 

^ 1 + - + . + -i^ 1 + - + . + -*...^ (1 _ f y (1 _ &) 1 ...(1 _&,)-., 
and the given integral may be expressed, as before, in terms of T functions : 
(40) I = T( 



II. INTEGRATION OF TOTAL DIFFERENTIALS 

151. General method. Let P(x, y) and Q(x, y~) be two functions of 
the two independent variables x and y. Then the expression 

Pdx + Qdy 

is not in general the total differential of a single function of the two 
variables x and y. For we have seen that the equation 

(41) du = Pdx + Qdy 
is equivalent to the two distinct equations 

(42) = *<-,), = (*,). . 

Differentiating the first of these equations with respect to y and the 
second with respect to x, it appears that u(x, y) must satisfy each 
of the equations 

C 2 u _ dP(x, y) 8 2 u _ dQ(x, y) 

cxdy dy Cydx dx 

A necessary condition that a function u(x, y) should exist which 
satisfies these requirements is that the equation 

(43) _.! 

Cy dx 

should be identically satisfied. 

This condition is also sufficient. For there exist an infinite 
number of functions u(x, y) for which the first of equations (42) 
is satisfied. All these functions are given by the formula 



= / P(x,y)dj. 

Jx n 



314 MULTIPLE INTEGRALS [VII, 151 

where x is an arbitrary constant and Fis an arbitrary function of y. 
In order that this function u(x, y) should satisfy the equation (41), 
it is necessary and sufficient that its partial derivative with respect 
to x should be equal to Q(x, y), that is, that the equation 



f 

Jx n 



dp 



_ < fe + _=^ y) 

should be satisfied. But by the assumed relation (43) we have 

f tx 7\T> f* x si f) 

I dx = I -z-dx = Q(x, y) - Q(x , y) , 

I Cy I CX 

whence the preceding relation reduces to 



The right-hand side of this equation is independent of x. Hence 
there are an infinite number of functions of y which satisfy the 
equation, and they are all given by the formula 



= f 

An 



Q(o, 



where y is an arbitrary value of y, and C is an arbitrary constant. 
It follows that there are an infinite number of functions u(x, y) 
which satisfy the equation (41). They are all given by the formula 



(44) u = f P(x, y} dx + I Q(x , 

Jx J /Q 



and differ from each other only by the additive constant C. 
Consider, for example, the pair of functions 

a; + my y mx 

~ x 2 + y* x 2 + y 2 

which satisfy the condition (43). Setting x = and y = 1, the 
formula for u gives 

C*x + my. C dy , 
u=l , 2 dx+ I -^- + C, 
Jo x* + y 2 J l y 

whence, performing the indicated integrations, we find 

1 F x~\ x 

u = - [log(a; 2 + y 2 )]* + m arc tan - + log y + C, 

* L yjo 

or, simplifying, 

1 x 

u = - log(cc 2 + y 2 ) -f m arc tan - + C. 

* y 



VII, 151] TOTAL DIFFERENTIALS 315 

The preceding method may be extended to any number of inde 
pendent variables. We shall give the reasoning for three variables. 
Let P, Q, and R be three functions of x, y, and z. Then the total 
differential equation 

(45) du = Pdx + Q dy + R dz 
is equivalent to the three distinct equations 

(46) = P, = Q, = R. 

Calculating the three derivatives d^u/dxdy, d 2 u/dydz, d^u/dzdx in 
two different ways, we find the three following equations as neces 
sary conditions for the existence of the function u : 

d_P_d_Q, 2Q = fL^ <LR = d JL. 

dy dx dz dy dx dz 

Conversely, let us suppose these equations satisfied. Then, by the 
first, there exist an infinite number of functions u(x, y, z) whose 
partial derivatives with respect to x and y are equal to P and Q, 
respectively, and they are all given by the formula 

u = I P(x, y, z)dx + I Q(x , y, z)dy + Z, 

where Z denotes an arbitrary function of z. In order that the deriva 
tive du/dz should be equal to R, it is necessary and sufficient that 
the equation 

should be satisfied. Making use of the relations (47), which were 
assumed to hold, this condition reduces to the equation 

R(x, y, z) - R(x , y, z} + R(x , y, z) - R(x , y , z) + = R(x, y, z} , 

= R(x ,y ,z). 

It follows that an infinite number of functions u(x, y, z) exist 
which satisfy the equation (45). They are all given by the formula 

f* x pv f* z 

(48) u = I P(x, y, z)dx + / Q(x , y, z)dy+ \ R(x ,y , z)dz + C, 

Jx K Jy c> z a 

where x , y n , z a are three arbitrary numerical values, and C is an 
arbitrary constant. 



316 MULTIPLE INTEGRALS [VII, 152 

152. The integral ^ x>y) Pdx + Qdy. The same subject may be 
treated from a different point of view, which gives deeper insight 
into the question and leads to new results. Let P(x, y) and Q(x, y} 
be two functions which, together with their first derivatives, are 
continuous in a region A bounded by a single closed contour C. 
It may happen that the region A embraces the whole plane, in 
which case the contour C would be supposed to have receded to 
infinity. The line integral 

I Pdx + Qdy 

taken along any path D which lies in A will depend in general upon 
the path of integration. Let us first try to find the conditions under 
which this integral depends only upon the coordinates of the extremi 
ties (x , y ) and (x i} y v ) of the path. Let M and N be any two points 
of region A, and let L and L be any two paths which connect these 
two points without intersecting each other between the extremities. 
Taken together they form a closed contour. In order that the values 
of the line integral taken along these two paths L and L should be 
equal, it is evidently necessary and sufficient that the integral taken 
around the closed contour formed by the two curves, proceeding 
always in the same sense, should be zero. Hence the question at 
issue is exactly equivalent to the following : What are the conditions 
under which the line integral 



f 



Pdx + Qdy 



taken around any closed contour whatever which lies in the region A 
should vanish ? 

The answer to this question is an immediate result of Green s 
theorem : 



Pta + a Ay =//(|f - g 



(49) 

where C is any closed contour which lies in A, and where the double 
integral is to be extended over the whole interior of C. It is clear 
that if the functions P and Q satisfy the equation 



the line integral on the left will always vanish. This condition is 
also necessary. For, if dP/dy dQ/dx were not identically zero 



VII, 152] TOTAL DIFFERENTIALS 317 

in the region A, since it is a continuous function, it would surely be 
possible to find a region a so small that its sign would be constant 
inside of a. But in that case the line integral taken around the 
boundary of a would not be zero, by (49). 

If the condition (43 ) is identically satisfied, the values of the 
integral taken along two paths L and V between the same two 
points M and N are equal provided the two paths do not intersect 
between M and N. It is easy to see that the same thing is true 
even when the two paths intersect any number of times between M 
and N. For in that case it would be necessary only to compare 
the values of the integral taken along the paths L and L with its 
value taken along a third path L", which intersects neither of the 
preceding except at M and N. 

Let us now suppose that one of the extremities of the path of 
integration is a fixed point (x 0) y }, while the other extremity is a 
variable point (x, y) of A. Then the integral 

X(*, V) 
Pdx + Qdy 
o- Vo> 

taken along an arbitrary path depends only upon the coordinates 
(x, y*) of the variable extremity. The partial derivatives of this 
function are precisely P(x, y) and Q(x, y). For example, we have 

s*(x + A:r, y) 

F(x + As, y) = F(x, y} + I P(x, y} dx, 

/(* v) 

for we may suppose that the path of integration goes from (x , y ) 
to (x, y), and then from (x, y~) to (x -f Ace, y) along a line parallel to 
the x axis, along which dy = 0. Applying the law of the mean, we 
may write 

F(x + Ace, ?/) F( 
v - -* - L 



= P(x + 0Az, y}, 0<0<1. 

u 

Taking the limit when Ace approaches zero, this gives F x = P. 
Similarly, F y = Q. The line integral F(x, y}, therefore, satisfies the 
total differential equation (41), and the general integral of this 
equation is given by adding to F(x, y) an arbitrary constant. 

This new formula is more general than the formula (44) in that 
the path of integration is still arbitrary. It is easy to deduce (44) 
from the new form. To avoid ambiguity, let (x , y ) and (a^, yi) be 
the coordinates of the two extremities, and let the path of integra 
tion be the two straight lines x = x , y = y^ Along the former, 



318 



MULTIPLE INTEGRALS 



[VII, 153 



x = x , dx = 0, and y varies from y to y t . Along the second, 



y = y l} dy = 0, and x varies from x to 
is equal to 



Hence the integral (50) 



r, 

I 

J;/ 



which differs from (44) only in notation. 

But it might be more advantageous to consider another path of 
integration. Let x = /(), y = <}>() be the equations of a curve 
joining (x , y ) and (x lt y^, and let t be supposed to vary con 
tinuously from t to t l as the point (x, y) describes the curve 
between its two extremities. Then we shall have 



f l pdx + Qdy = I 
/<%, > Jf 



where there remains but a single quadrature. If the path be 
a straight line, for example, we should set x = x + tfa #), 
y = y + t(y l y,,), and we should let t vary from to 1. 

Conversely, if a particular integral (x, y) of the equation (41) 
be known, the line integral is given by the formula 



L 



(x. y) 



= 3>(x, y) 3>(x , y c ), 



which is analogous to the equation (6) of Chapter IV. 

153. Periods. More general cases may be investigated. In the 
first place, Green s theorem applies to regions bounded by several 
contours. Let us consider for defmiteness a region A bounded by 
an exterior contour C and two contours C" and 
C" which lie inside the first (Fig. 35). Let P 
and Q be two functions which, together with 
their first derivatives, are continuous in this 
region. (The regions inside the contours C 
and C" should not be considered as parts of 
the region A, and no hypothesis whatever is 
made regarding P and Q inside these regions.) 
Let the contours C and C" be joined to the contour C by trans 
versals ab and cd. We thus obtain a closed contour abmcdndcpbaqa, 
or F, which may be described at one stroke. Applying Green s 
theorem to the region bounded by this contour, the line integrals 




VII, 153] 



TOTAL DIFFERENTIALS 



319 



which arise from the transversals ab and cd cancel out, since each 
of them is described twice in opposite directions. It follows that 



f 



Pdx + Qdy 



f/Y-- 

JJ \&; 



r/0 u 



where the line integral is to be taken along the whole boundary of 
the region A, i.e. along the three contours C, C , and C", in the senses 
indicated by the arrows, respectively, these being such that the 
region j\ always lies on the left. 

If the functions P and Q satisfy the relation dQ/dx = dP/dy in 
the region A, the double integral vanishes, and we may write the 
resulting relation in the form 

(51) I Pdx + Qdy = I Pdx + Qdy + I Pdx + 

J(C) J(C ) J(C") 

where each of the line integrals is to be taken in the sense desig 
nated above. 

Let us now return to the region A bounded by a single contour 
C, and let P and Q be two functions which satisfy the equation 
dP/dy = dQ/dx, and which, together with their first derivatives, are 
continuous except at a finite number 
of points of A, at which at least one of 
the functions P or Q is discontinuous. 
We shall suppose for definiteness that 
there are three points of discontinuity 
a, b, c in A. Let us surround each of 
these points by a small circle, and then 
join each of these circles to the contour 
C by a cross cut (Fig. 36). Then the 
integral j Pdx -\-Qdy taken from a fixed 
point (x , T/O) to a variable point (.r, y) 
along a curve which does not cross any 

of these cuts has a definite value at every point. For the contour C, 
the circles and the cuts form a single contour which may be described 
at one stroke, just as in the case discussed above. We shall call 
such a path direct, and shall denote the value of the line integral 
taken along it from M (x , y ) to M(x, y) by F(x, y}. 

We shall call the path composed of the straight line from M to 
a point a , whose distance from a is infinitesimal, the circumference 
of the circle of radius aa about a, and the straight line a M , a loop- 
circuit. The line integral fPdx -f Qdy taken along a loop-circuit 




FIG. 36 



320 MULTIPLE INTEGRALS [VII, 153 

reduces to the line integral taken along the circumference of the 
circle. This latter integral is not zero, in general, if one of the 
functions P or Q is infinite at the point a, but it is independent of 
the radius of the circle. It is a certain constant A, the double 
sign corresponding to the two senses in which the circumference 
may be described. Similarly, we shall denote by B and C the 
values of the integral taken along loop-circuits drawn about the two 
singular points b and c, respectively. 

Any path whatever joining M and M may now be reduced to a 
combination of loop-circuits followed by a direct path from M to M. 
For example, the path M mdefM may be reduced to a combination 
of the paths M mdM , M deM , M efM , and M fM. The path 
M^mdM^ may then be reduced to a loop-circuit about the singular 
point a, and similarly for the other two. Finally, the path M fM 
is equivalent to a direct path. It follows that, whatever be the path 
of integration, the value of the line integral will be of the form 



(52) F(x, y) = F(x, y) + m A + n B + pC , 

where m, n, and p may be any positive or negative integers. The 
quantities A, B, C are called the periods of the line integral. That 
integral is evidently a function of the variables x and y which 
admits of an infinite number of different determinations, and the 
origin of this indetermination is apparent. 

Remark. The function F(x, y} is a definitely defined function 
in the whole region A when the cuts aa, b/3, cy have been traced. 
But it should be noticed that the difference F(m) F(m ) between 
the values of the function at two points m and m which lie on 
opposite sides of a cut does not necessarily vanish. For we have 

/"" /" r*o 

A- /: + /+/ , 

i/a; J,,i Jmf 

which may be written 



But j"^ is zero ; hence 

J^-F<X)= A. 

It follows that the difference F(ni) F(m ) is constant and equal 
to A all along aa. The analogous proposition holds for each of 
the cuts. 



VII, 154] TOTAL DIFFERENTIALS 321 

Example. The line integral 



/ 

i/fll 



" 



" xdy ijdx 



~f~ y 



has a single critical point, the origin. In order to find the corre 
sponding period, let us integrate along the circle x 2 + y 2 = p 2 . 
Along this circle we have 

x = p cos o, y = p sin w, xdy ydx = p^dto, 

whence the period is equal to / "dw = 2?r. It is easy to verify 
this, for the integrand is the total differential of arc tan y/x. 

154. Common roots of two equations. Let X and Y be two functions of the 
variables x and y which, together with their first partial derivatives, are con 
tinuous in a region A bounded by a single closed contour C. Then the expres 
sion (XdY YdX)/(X* + Y 2 ) satisfies the condition of integrability, for it is 
the derivative of arc tan Y/X. Hence the line integral 



(53) 



L 



X 2 + 



taken along the contour C in the positive sense vanishes provided the coeffi 
cients of dx and dy in the integrand remain continuous inside (7, i.e. if the two 
curves X = 0, Y = have no common point inside that contour. But if these 
two curves have a certain number of common points a, 6, c, inside C, the value 
of the integral will be equal to the sum of the values of the same integral taken 
along the circumferences of small circles described about the points a, b, c, as 
centers. Let (a, 0) be the coordinates of one of the common points. We shall 
suppose that the functional determinant D(X, Y)/D(x, y) is not zero, i.e. that 
the two curves X and Y = are not tangent at the point. Then it is pos 
sible to draw about the point (a, /3) as center a circle c whose radius is so small 
that the point (JT, Y) describes a small plane region about the point (0, 0) 
which is bounded by a contour 7 and which corresponds point for point to the 
circle c ( 25 and 127). 

As the point (x, y) describes the circumference of the circle c in the positive 
sense, the point (X, Y) describes the contour y in the positive or in the negative 
sense, according as the sign of the functional determinant inside the circle c is 
positive or negative. But the definite integral along the circumference of c is 
equal to the change in arc tan Y/X in one revolution, that is, 2x. Similar 
reasoning for all of the roots shows that 





where P denotes the number of points common to the two curves at which 
-^(^, Y)/D(x, y) is positive, and N the number of common points at which the 
determinant is negative. 



322 MULTIPLE INTEGRALS [VII, 155 

The definite integral on the left is also equal to the variation in arc tan Y/X 
in going around c, that is, to the index of the function Y/X as the point (x, y) 
describes the contour C. If the functions X and Y are polynomials, and if the 
contour C is composed of a finite number of arcs of unicursal curves, we are led 
to calculate the index of one or more rational functions, which involves only 
elementary operations ( 77). Moreover, whatever be the functions X and Y, 
we can always evaluate the definite integral (54) approximately, with an error 
less than 7t, which is all that is necessary, since the right-hand side is always a 
multiple of 2it. 

The formula (54) does not give the exact number of points common to the 
two curves unless the functional determinant has a constant sign inside of C. 
Picard s recent work has completed the results of this investigation.* 

155. Generalization of the preceding. The results of the preceding paragraphs 
may be extended without essential alteration to line integrals in space. Let P, 
Q, and -B be three functions which, together with their first partial derivatives, 
are continuous in a region (E) of space bounded by a single closed surface S. 
Let us seek first to determine the conditions under which the line integral 



(55) 

(v j/ , z ) 

depends only upon the extremities (x , yo , Zo) and (x, y, z) of the path of inte 
gration. This amounts to inquiring under what conditions the same integral 
vanishes when taken along any closed path T. But by Stokes theorem ( 136) 
the above line integral is equal to the surface integral 



CC/ 

j I ( 
JJ \ 



t> p \j /^ R f>Q\^ , / 5P SR \ 7 

-- }dx dy + ( -- - I dy dz + ( --- }dz dx 
dx dy/ T \a* dz/ ^\cz ex/ 



extended over a surface S which is bounded by the contour T. In order that 
this surface integral should be zero, it is evidently necessary and sufficient that 
the equations 

8P_BQ ^Q = ^, <3jR = eP 

dy dx dz dy dx dz 

should be satisfied. If these conditions are satisfied, U is a function of the vari 
ables x, y, and z whose total differential is P dx + Q dy + R dz, and which is single 
valued in the region (E). In order to find the value of U at any point, the path 
of integration may be chosen arbitrarily. 

If the functions P, Q, and E satisfy the equations (56), but at least one of 
them becomes infinite at all the points of one or more curves in (E), results 
analogous to those of 153 may be derived. 

If, for example, one of the functions P, Q, R becomes infinite at all the points 
of a closed curve 7, the integral U will admit a period equal to the value of the 
line integral taken along a closed contour which pierces once and only once a 
surface <r bounded by 7. 

We may also consider questions relating to surface integrals which are exactly 
analogous to the questions proposed above for line integrals. Let A, B, and C 
be three functions which, together with their first partial derivatives, are 

* TraM d Analyse, Vol. II. 



VII, 155] TOTAL DIFFERENTIALS 323 

continuous in a region (E) of space bounded by a single closed surface S. Let 2 
be a surface inside of (E) bounded by a contour r of any form whatever. Then 
the surface integral 

(57) I = C f A dy dz + B dz dx + C dx dy 

depends in general upon the surface S as well as upon the contour r. In order 
that the integral should depend only upon F, it is evidently necessary and suffi 
cient that its value when taken over any closed surface in (E) should vanish. 
Green s theorem ( 149) gives at once the conditions under which this is true. 
For we know that the given double integral extended over any closed surface is 
equal to the triple integral 

dA dB dC\, 
1 1 \dxdydz 

dy dz / 

extended throughout the region bounded by the surface. In order that this latter 
integral should vanish for any region inside (E), it is evidently necessary that the 
functions A, B, and C should satisfy the equation 

dx + Ity + ~dz~ ~ 
This condition is also sufficient. 

Stokes theorem affords an easy verification of this fact. For if A, J5, and C 
are three functions which satisfy the equation (58), it is always possible to deter 
mine in an infinite number of ways three other functions P, Q, and R such that 

dy dz ~ dz dx dx dy 

In the first place, if these equations admit solutions, they admit an infinite 
number, for they remain unchanged if P, Q, and R be replaced by 

dx dy dz 

respectively, where X is an arbitrary function of x, y, and z. Again, setting 
R = 0, the first two of equations (59) give 

P = f *B(x, y, z) dz + <f>(x, y), Q = - C A(x, y, z) dz + $(x, y) , 

where <f>(x. y) and \f/(x, y) are arbitrary functions of x and y. Substituting these 
values in the last of equations (59), we find 

dA d B\ , d\b d<f> _,. 
h - ) dz H = C7(j y, z) , 

dx dy dx dy 

~ 

or, making use of (58), 

= C(x, y, z ) 

dx dy 

One of the functions or ^ may still be chosen at random. 

The functions P, Q, and R having been determined, the surface integral, by 
Stokes theorem, is equal to the line integral f (r) Pdx + Qdy + Rdz, which 
evidently depends only upon the contour F. 



324 MULTIPLE INTEGRALS [VII, Exs. 

EXERCISES 

1. Find the value of the triple integral 

( x -y) 2 + 3az- 4a*]dxdydz 

extended throughout the region of space defined by the inequalities 

x 2 + 7/ 2 -az<0, x 2 + 2/ 2 + z 2 -2a 2 <0. 

[Licence, Montpellier, 1895.] 

2. Find the area of the surface 



2 X 2 + & 2 y2 

and the volume of the solid bounded by the same surface. 

3. Investigate the properties of the function 

F(X, F, Z) = C dx C dy C /(z, y, z)dz 

x o J o z o 

considered as a function of JT, F, and Z. Generalize the results of 125. 

4. Find the volume of the portion of the solid bounded by the surface 

(z 2 + y 2 + z 2 ) 8 = 3a s xyz 
which lies in the first octant. 

5. Reduce to a simple integral the multiple integral 

C C C a a a 

I I x, x y * x " F(x\ + x 2 + + x n ) dx\ dx% dx n 
J J J 

extended throughout the domain D defined by the inequalities 

< Xi , < Zj , , ^ x n , x\ + x 2 + + x n ^ a . 
[Proceed as in 148.] 

6. Reduce to a simple integral the multiple integral 



extended throughout the domain D defined by the inequalities 



7*. Derive the formula 

n 

iri 



f f f--- Cdx 1 dx---dx n = - 
JJJ J 



VII. Exs.] EXERCISES 325 

where the multiple integral is extended throughout the domain D denned by the 
inequality 



8*. Derive the formula 

r27T 




C "de C n F(a cos 6 + b sin cos $ + c sin sin 0) sin 6 d<j> - 2n C F(uR) du , 

/0 Jo I 



where a, &, and c are three arbitrary constants, and where B = Va 2 + b 2 + c 2 . 

[POISSON.] 

[First observe that the given double integral is equal to a certain surface inte 
gral taken over the surface of the sphere x 2 + y 2 + z 2 = 1. Then take the plane 
ax + by + cz = &s the plane of xy in a new system of coordinates.] 

9*. Let p = F(6, <f>) be the equation in polar coordinates of a closed surface. 
Show that the volume of the solid bounded by the surface is equal to the double 
integral 



(a) J j ip cos y do- 



extended over the whole surface, where da- represents the element of area, and 7 
the angle which the radius vector makes with the exterior normal. 

10*. Let us consider an ellipsoid whose equation is 



and let us define the positions of any point on its surface by the elliptic coordi 
nates v and />, that is, by the roots which the above equation would have if /* 
were regarded as unknown (cf. 147). The application of the formulae (29) to 
the volume of this ellipsoid leads to the equation 



" 



r" --- dv m _ 

Jo 



Likewise, the formula (a) gives 
>& 



M 

i/O Jb 



- p 2 ) dv 



V(62-p 2 )( C 2 -p 2 )(, 2 -6 2 )(c 2 -, 2 ) 2 



11. Determine the functions P(z, y) and Q(x, y) which, together with their 
partial derivatives, are continuous, and for which the line integral 

x + a,y + /3)dx + Q(z + a, y + p)dy 

taken along any closed contour whatever is independent of the constants a and 

/3 and depends only upon the contour itself. 

[Licence, Paris, July, 1900.] 



326 MULTIPLE INTEGRALS [VII, Exs. 

12*. Consider the point transformation defined by the equations 




As the point (z , y , z ) describes a surface S , the point (x, y, z) describes a sur 
face S. Let a, /3, y be the direction angles of the normal to S ; a , , -/ the 
direction angles of the corresponding normal to the surface S ; and d<r and d<r 
the corresponding surface elements of the two surfaces. Prove the formula 

= M ^ y) coscr- + ^ y) cos + ^ 
r 



13*. Derive the formula (16) on page 304 directly. 

[The volume V may be expressed by the surface integral 

V = I z cos 7 d<r , 

J(S) 

and we may then make use of the identity 



D(f, *, *) d ( D(/, ) ) 8 j J>(/, 0) ) a 

Y 



D(x ,y ,z ) dx T D(y ,z ) dy D(z ,x ) 3z ( D(x , y ) 
which is easily verified.] 



it, 



CHAPTER VIII 
INFINITE SERIES 

I. SERIES OF REAL CONSTANT TERMS 
GENERAL PROPERTIES TESTS FOR CONVERGENCE 

156. Definitions and general principles. Sequences. The elementary 
properties of series are discussed in all texts on College Algebra 
and on Elementary Calculus. We shall review rapidly the principal 
points of these elementary discussions. 

First of all, let us consider an infinite sequence of quantities 

(1) S Q , Si, S Z , , S n , 

in which each quantity has a definite place, the order of precedence 
being fixed. Such a sequence is said to be convergent if s n approaches 
a limit as the index n becomes infinite. Every sequence which is 
not convergent is said to be divergent. This may happen in either 
of two ways : s n may finally become and remain larger than any 
preassigned quantity, or s n may approach no limit even though it 
does not become infinite. 

In order that a sequence should be convergent, it is necessary and 
sufficient that, corresponding to any preassigned positive number e, a 
positive integer n should exist such that the difference s n+p s n is 
less than e in absolute value for any positive integer p. 

In the first place, the condition is necessary. For if s n approaches 
a limit s as n becomes infinite, a number n always exists for which 
each of the differences s s n , s s n+l , , s s n+p , is less than 
e/2 in absolute value. It follows that the absolute value of s n+p s n 
will be less than 2 e/2 = c for any value of p. 

In order to prove the converse, we shall introduce a very impor 
tant idea due to Cauchy. Suppose that the absolute value of each 
of the terms of the sequence (1) is less than a positive number N. 
Then all the numbers between N and -f- N may be separated into 
two classes as follows. We shall say that a number belongs to the 
class A if there exist an infinite number of terms of the sequence (1) 

327 



328 INFINITE SERIES [VIII, 156 

which are greater than the given number. A number belongs to 
the class R if there are only a finite number of terms of the 
sequence (1) which are greater than the given number. It is 
evident that every number between N and + N belongs to one 
of the two classes, and that every number of the class A is less 
than any number of the class B. Let S be the upper limit of the 
numbers of the class A, which is obviously the same as the lower 
limit of the numbers of the class B. Cauchy called this number the 
greatest limit (la plus grande des limites) of the terms of the 
sequence (1).* This number S should be carefully distinguished 
from the upper limit of the terms of the sequence (1) ( 68). For 
instance, for the sequence 

11 1 

11 2 3 " n 

the upper limit of the terms of the sequence is 1, while the greatest 
limit is 0. 

The name given by Cauchy is readily justified. There always 
exist an infinite number of terms of the sequence (1) which lie 
between S and S + e, however small e be chosen. Let us then 
consider a decreasing sequence of positive numbers t l} c 2 , , 
e n , , where the general term ^ approaches zero. To each num 
ber ,- of the sequence let us assign a number . of the sequence (1) 
which lies between S e, and S + e^ We shall thus obtain a 
suite of numbers a 1? a 2 , , a n , belonging to the sequence (1) 
which approach S as their limit. On the other hand, it is clear 
from the very definition of S that no partial sequence of the kind just 
mentioned can be picked out of the sequence (1) which approaches 
a limit greater than S. Whenever the sequence is convergent its 
limit is evidently the number S itself. 

Let us now suppose that the difference s n+p s n of two terms of 
the sequence (1) can be made smaller than any positive number c 
for any value of p by a proper choice of n. Then all the terms of 
the sequence past s n lie between s lt e and s n + e. Let S be the 
greatest limit of the terms of the sequence. By the reasoning just 
given it is possible to pick a partial sequence out of the sequence (1) 
which approaches 5 as its limit. Since each term of the partial 
sequence, after a certain one, lies between s n e and s n + c, it is 

* Resumes analytiques de Turin, 1833 (Collected Works, 2d series, Vol. X, p. 49). 
The definition may be extended to any assemblage of numbers which has an upper 
limit. 



VIII, Ufi7] CONSTANT TERMS 329 

clear that the absolute value of S s n is at most equal to e. Now 
let s m be any term of the sequence (1) whose index m is greater 
than n. Then we may write 



and the value of the right-hand side is surely less than 2c. Since e 
is an arbitrarily preassigned positive number, it follows that the 
general term s m approaches S as its limit as the index m increases 
indefinitely. 

Note. If S is the greatest limit of the terms of the sequence (1), 
every number greater than S belongs to the class B, and every num 
ber less than S belongs to the class A. The number S itself may 
belong to either class. 

157. Passage from sequences to series. Given any infinite sequence 



the series formed from the terms of this sequence, 

(2) U + ! + U 2 H ----- \- U n -\ ---- , 

is said to be convergent if the sequence of the successive sums 

S = ?/ , S l = U + 1( 1 , , S n = U + ! + + U n , 

is convergent. Let 5 be the limit of the latter sequence, i.e. the 
limit which the sum S n approaches as n increases indefinitely: 

v=n 

S = lim S = lim ?/. 



Then S is called the sum of the preceding series, and this relation is 
indicated by writing the symbolic equation 

S = MO + MI H ----- h u n H 

A series which is not convergent is said to be divergent. 

It is evident that the problem of determining whether the series 
is convergent or divergent is equivalent to the problem of determin 
ing whether the sequence of the successive sums S 0) S lf 5 2 , is 
convergent or divergent. Conversely, the sequence 

S 0) s li S 2> " ) s n) 

will be convergent or divergent according as the series 

SQ + Oi - s ) + (s *i) H ----- 1- (* a.-i) H ---- 



330 INFINITE SERIES [VIII, 157 

is convergent or divergent. For the sum S n of the first n + 1 terms 
of this series is precisely equal to the general term s n of the given 
sequence. We shall apply this remark frequently. 
The series (2) converges or diverges with the series 

(3) S + Vn + * + *++ 

obtained by omitting the first p terms of (2). For, if S n (n > p) 
denote the sum of the first n + 1 terms of the series (2), and 2 n _ p 
the sum of the n p + 1 first terms of the series (3), i.e. 



the difference S n 2 n - P = u o + u \ H ----- h Mp_i is independent of w. 
Hence the sum 2 n - P approaches a limit if S n approaches a limit, 
and conversely. It follows that in determining whether the series 
converges or diverges we may neglect as many of the terms at the 
beginning of a series as we wish. 

Let S be the sum of a convergent series, S n the sum of the first 
n + 1 terms, and R n the sum of the series obtained by omitting the 
first Ti + 1 terms, 

RH = U n + l + + 2 H ----- h U n + p -\ ---- . 

It is evident that we shall always have 



It is not possible, in general, to find the sum S of a convergent 
series. If we take the sum 5 of the first n -+- 1 terms as an approxi 
mate value of S, the error made is equal to R n . Since S n approaches 
S as n becomes infinite, the error R n approaches zero, and hence the 
number of terms may always be taken so large at least theoret 
ically that the error made in replacing S by S n is less than any 
preassigned number. In order to have an idea of the degree of 
approximation obtained, it is sufficient to know an upper limit 
of R n . It is evident that the only series which lend themselves 
readily to numerical calculation in practice are those for which 
the remainder R n approaches zero rather rapidly. 

A number of properties result directly from the definition of con 
vergence. We shall content ourselves with stating a few of them. 

1) If each of the terms of a given series be multiplied by a constant 
k different from zero, the new series obtained will converge or diverge 
with the given series; if the given series converges to a sum S, the sum 
of the second series is kS. 



VIII, 158] CONSTANT TERMS 331 

2) If there be given two convergent series 

wo + MI + w 2 H h M H H , 

i o + vi + wH 1- v n H , 

whose sums are S and S , respectively, the new series obtained by 
adding the given series term by term, namely, 

Oo + * o) + (MX + Vj) + + (M + ) + , 

converges, and its sum is S -f- S . The analogous theorem holds for 
the term-by -term addition of p convergent series. 

3) The convergence or divergence of a series is not affected if the 
values of a finite number of the terms be changed. For such a change 
would merely increase or decrease all of the sums S n after a certain 
one by a constant amount. 

4) The test for convergence of any infinite sequence, applied to 
series, gives Cauchy s general test for convergence : * 

In order that a series be convergent it is necessary and sufficient 
that, corresponding to any preassigned positive number e, an integer 
n should exist, such that the sum of any number of terms what 
ever, starting with u ll+l , is less than c in absolute value. For 

S n + p S n = U n + l + U n + 2 H \~ ?+ 

In particular, the general term u n+l = S n+l S n must approach 
zero as n becomes infinite. 

Cauchy s test is absolutely general, but it is often difficult to 
apply it in practice. It is essentially a development of the very 
notion of a limit. We shall proceed to recall the practical rules most 
frequently used for testing series for convergence and divergence. 
None of these rules can be applied in all cases, but together they 
suffice for the treatment of the majority of cases which actually arise. 

158. Series of positive terms. We shall commence by investigating 
a very important class of series, those whose terms are all posi 
tive. In such a series the sum S n increases with n. Hence in 
order that the series converge it is sufficient that the sum S n should 
remain less than some fixed number for all values of n. The most 
general test for the convergence of such a series is based upon com 
parisons of the given series with others previously studied. The 
following propositions are fundamental for this process : 

* Exercices de Mathtmatlques, 1827. (Collected Works, Vol. VII, 2d series, p. 267.) 



332 INFINITE SERIES [VIII, 159 

1) If each of the terms of a given series of positive terms is less 
than or at most equal to the corresponding term of a known convergent 
series of positive terms, the given series is convergent. For the sum 
S n of the first n terms of the given series is evidently less than the 
sum S of the second series. Hence S n approaches a limit S which 
is less than S . 

2) If each of the terms of a given series of positive terms is greater 
than or equal to the corresponding term of a known divergent series 
of positive terms, the given series diverges. For the sum of the first 
n terms of the given series is not less than the sum of the first 
n terms of the second series, and hence it increases indefinitely 
with n. 

We may compare two series also by means of the following 
lemma. Let 

(U) MO + ! + MJ H h u n H , 

(V) v + Vl + Va + ... + , + ... 

be two series of positive terms. If the series (7) converges, and if, 
after a certain term, we always have v n+l /v n 5: u n+l /u n , the series (V) 
also converges. If the series (7) diverges, and if, after a certain 
term, we always have u n+ i/u n ^v n+} /v n , the series (F) also diverges. 

In order to prove the first statement, let us suppose that 
v n+i/ v n^ u n+i/ u n whenever n > 2). Since the convergence of a 
series is not affected by multiplying each term by the same con 
stant, and since the ratio of two consecutive terms also remains 
unchanged, we may suppose that v p < u p , and it is evident that we 
should have v p + l ^u p + 1 , v p+2 ^u p + 2 , etc. Hence the series (F) 
must converge. The proof of the second statement is similar. 

Given a series of positive terms which is known to converge or 
to diverge, we may make use of either set of propositions in order 
to determine in a given case whether a second series of positive 
terms converges or diverges. For we may compare the terms of 
the two series themselves, or we may compare the ratios of two 
consecutive terms. 

159. Cauchy s test and d Alembert s test. The simplest series which 
can be used for purposes of comparison is a geometrical progression 
whose ratio is r. It converges if r < 1, and diverges if r ^ 1. The 
comparison of a given series of positive terms with a geometrical 
progression leads to the following test, which is due to Cauchy: 



VIII, 159] CONSTANT TERMS 333 

If the nth root \u n of the general term u n of a series of positive 
terms after a certain term is constantlyjess than a fixed number less 
than unity, the series converges. If ~\/u n after a certain term is con 
stantly greater than unity, the series diverges. 

For in the first case ~\/u n <k<l, whence u n <k n . Hence each 
of the terms of the series after a certain one is less than the corre 
sponding term of a certain geometrical progression whose ratio is 
less than unity. In the second case, on the other hand, ~\/u n >l ) 
whence w n >l. Hence in this case the general term does not 
approach zero. 

This test is applicable whenever V^ approaches a limit. In 
fact, the following proposition may be stated : 

If -Vu n approaches a limit I as n becomes infinite, the series will 
converge if I is less than unity, and it will diverge if I is greater than 
unity. 

A doubt remains if I = 1, except when ~^fu n remains greater than 
unity as it approaches unity, in which case the series surely diverges. 

Comparing the ratio of two consecutive terms of a given series 
of positive terms with the ratio of two consecutive terms of a 
geometrical progression, we obtain d Alembert s test: 

If in a given series of positive terms the ratio of any term to the 
preceding after a certain term remains less than a fixed number 
less than unity, the series converges. If that ratio after a certain 
term remains greater than unity, the series diverges. 

From this theorem we may deduce the following corollary : 

If the ratio u n+l /u n approaches a limit I as n becomes infinite, the 
series converges if I < 1, and diverges ifl>l. 

The only doubtful case is that in which 1 = 1; even then, ifu n+l /u n 
remains greater than unity as it approaches unity, the series is divergent. 

General commentary. Cauchy s test is more general than d Alembert s. For 
suppose that the terms of a given series, after a certain one, are each less than 
the corresponding terms of a decreasing geometrical progression, i.e. that the 
general term u n is less than Ar n for all values of n greater than a fixed integer p, 
where A is a certain constant and r is less than unity. Hence Vu n < rvl 1 /", and 
the second member of this inequality approaches unity as n becomes infinite. 
Hence, denoting by A; a fixed number between r and 1, we shall have after a cer 
tain term \/M n < k. Hence Cauchy s test is applicable in any such case. But it 
may happen that the ratio u n + \/Un assumes values greater than unity, however 
far out in the series we may go. For example, consider the series 

1 + r | sin a | + r 2 1 sin 2a \ + + r n \ sin na \ -{ , 



334 INFINITE SERIES [VIII, 159 



where r < 1 and where a is an arbitrary constant. In this case v u n = r v | sin net \ < r, 

whereas the ratio 

sin(n + l) 



may assume, in general, an infinite number of values greater than unity as n 
increases indefinitely. 

Nevertheless, it is advantageous to retain d Alembert s test, for it is more 
convenient in many cases. For instance, for the series 

x x 2 x 3 x n 

A H~ ~ i ~ " I ~ ~ ~ ~T~ T 



1 1.2 1.2.3 1.2---W, 

the ratio of any term to the preceding is x/(n + 1), which approaches zero as n 
becomes infinite ; whereas some consideration is necessary to determine inde 
pendently what happens to Vun = x/\/l . 2 n as n becomes infinite. 

After we have shown by the application of one of the preceding tests that each 
of the terms of a given series is less than the corresponding term of a decreasing 
geometrical progression A, Ar, Ar 2 , , Ar n , , it is easy to find an upper 
limit of the error made when the sum of the first m terms is taken in place of 
the sum of the series. For this error is certainly less than the sum of the 

geometrical progression 

Ar m 
Ar m + Ar m + l 



When each of the two expressions -\iun and Un + i/u H approaches a limit, the 
two limits are necessarily the same. For, let us consider the auxiliary series 

. (4) + uix + u 2 z 2 H ----- h u n x" H ---- , 

where x is positive. In this series the ratio of any term to the preceding 
approaches the limit Ix, where I is the limit of the ratio u n + \/u n . Hence the 
series (4) converges when x < 1/Z, and diverges when x > I /I. Denoting the 
limit of -\/Un by * , the expression -\/u n x n also approaches a limit Z x, and 
the series (4) converges if x < 1/i , and diverges if x > 1/i . In order that the 
two tests should not give contradictory results, it is evidently necessary that I 
and I should be equal. If, for instance, I were greater than T, the series (4) would 
be convergent, by Cauchy s test, for any number x between 1/Z and 1/i , whereas 
the same series, for the same value of x, would be divergent by d Alembert s test. 
Still more generally, if Un + \/u n approaches a limit Z, >Xu n approaches the same 
limit.* For suppose that, after a certain term, each of the ratios 

U n + \ U,, + 2 U,, + ,, 



lies between I e and I + e, where e is a positive number which may be taken 
as small as we please by taking n sufficiently large. Then we shall have 



or 



*Cauchy, Cours d Analyse. 



Vlll, 160] CONSTANT TERMS 335 

As the number p increases indefinitely, while n remains fixed, the two terms on 
the extreme right and left of this double inequality approach I -f e and I e, 
respectively. Hence for all values of m greater than a suitably chosen number 
we shall have 



and, since e is an arbitrarily assigned number, it follows that Vu^ approaches 
the number I as its limit. 

It should be noted that the converse is not true. Consider, for example, the 

sequence 

1, a, a6, a" 2 6, a 2 6 2 , , a n b n ~ l , a"6 n , -, 

where a and b are two different numbers. The ratio of any term to the preced 
ing is alternately a and 6, whereas the expression -s/u^ approaches the limit Va6 
as n becomes infinite. 

The preceding proposition may be employed to determine the limits of cer 
tain expressions which occur in undetermined forms. Thus it is evident that 
the expression v 1 . 2 n increases indefinitely with n, since the ratio n \/(n 1)1 
increases indefinitely with n. In a similar manner it may be shown that each of 
the expressions v n and v^ogn approaches the limit unity as n becomes infinite. 

160. Application of the greatest limit. Cauchy formulated the preceding test 
in a more general manner. Let a n be the general term of a series of positive 
terms. Consider the sequence 

1 I I 

(5) on, a;, ajj, -, a*, . 

If the terms of this sequence have no upper limit, the general term a n will not 
approach zero, and the given series will be divergent. If all the terms of the 
sequence (5) are less than a fixed number, let w be the greatest limit of the terms 
of the sequence. 

The series Sa n is convergent if u is less than unity, and divergent if u is greater 
than unity. 

In order to prove the first part of the theorem, let 1 a be a number between 
w and 1. Then, by the definition of the greatest limit, there exist but a finite 
number of terms of the sequence (5) which are greater than 1 a. It follows 
that a positive integer p may be found such that 3/a n < 1 a for all values of n 
greater than p. Hence the series 2a re converges. On the other hand, if > 1, 
let 1 + a be a number between 1 and w. Then there are an infinite number of 
terms of the sequence (5) which are greater than 1 + a, and hence there are an 
infinite number of values of n for which a n is greater than unity. It follows that 
the series Sa n is divergent in this case. The case in which w = 1 remains in doubt. 

161. Cauchy s theorem. In case u n+l /u n and \u n both approach 
unity without remaining constantly greater than unity, neither 
d Alembert s test nor Cauchy s test enables us to decide whether 
the series is convergent or divergent. We must then take as a 
comparison series some series which has the same characteristic 



336 INFINITE SERIES [VIII, 161 

but which is known to be convergent or divergent. The following 
proposition, which Cauchy discovered in studying definite integrals, 
often enables us to decide whether a given series is convergent or 
divergent when the preceding rules fail. 

Let <f>(x) be a function which is positive for values of x greater 
than a certain number a, and which constantly decreases as x 
increases past x = a, approaching zero as x increases indefinitely. 
Then the x axis is an asymptote to the curve y = <f>(%), and the 
definite integral 



1 



<f>(x~)dx 



may or may not approach a finite limit as I increases indefinitely. 
The series 

(6) 4>( a ) + 4,(a +!)+ + <(> + ) + 

converges if the preceding integral approaches a limit, and diverges if 
it does not. 

For, let us consider the set of rectangles whose bases are each 
unity and whose altitudes are <f>(a), <f>(a + 1), -, <f>(a + n), respec 
tively. Since each of these rectangles extends beyond the curve 
y </>(#), the sum of their areas is evidently greater than the area 
between the x axis, the curve y = <j>(x), and the two ordinates x = a, 
x = a + n, that is, 



Xa + 



On the other hand, if we consider the rectangles constructed 
inside the curve, with a common base equal to unity and with the 
altitudes <j>(a +1), <j>(a + 2), , <j>(a -f- n), respectively, the sum of 
the areas of these rectangles is evidently less than the area under 
the curve, and we may write 

Xa + n 
+(x)dx. 

Hence, if the integral fj <j>(z) dx approaches a limit L as I increases 
indefinitely, the sum <f>(a) -\ ---- + <f>(a + n) always remains less than 
<f(a) -f L. It follows that the sum in question approaches a limit ; 
hence the series (6) is convergent. On the other hand, if the inte 
gral f a+n <j>(x)dx increases beyond all limit as n increases indefinitely, 
the same is true of the sum 

<a -f <t>a + 1 + + < a + ", 



VHi, 161] CONSTANT TERMS 337 

as is seen from the first of the above inequalities. Hence in this 
case the series (6) diverges. 

Let us consider, for example, the function <(ce) = l/x*, where /u, 
is positive and a = 1. This function satisfies all the requirements 
of the theorem, and the integral // [1/a^] dx approaches a limit as 
I increases indefinitely if and only if /u, is greater than unity. It 
follows that the series 

111 _! + . 

1* T 2* 3* n* 

converges if p. is greater than unity, and diverges if /x ^ 1. 

A.gain, consider the function <(#) = I/ [x(logo;) 1 ], where log a; 
denotes the natural logarithm, p. is a positive number, and a = 2. 
Then, if /* = 1, we shall have 



/ 

c/2 



^-[(log7 i ) 1 -^-(log2) 1 - 



The second member approaches a limit if /t > 1, and increases 
indefinitely with n if /u, < 1. In the particular case when /j. = 1 it 
is easy to show in a similar manner that the integral increases 
beyond all limit. Hence the series 



"t" r f\ o\u I I /i _\U. I 



2 (log 2)* 3 (log 3)* w(logn) 4 

converges if /t > 1, and diverges if /u. 51- 

More generally the series whose general term is 

1 

n log 7i log 2 n log 8 ?i log"" 1 n(log p ri)* 

converges if /t > 1, and diverges if /x ^ 1. In this expression log 2 n 
denotes log log n, log 8 n denotes log log log n, etc. It is understood, 
of course, that the integer n is given only values so large that 
log n, log 2 n, log 8 n, , log p n are positive. The missing terms in 
the series considered are then to be supplied by zeros. The 
theorem may be proved easily in a manner similar to the demon 
strations given above. If, for instance, /t = 1, the function 

1 



x log x log 2 x (log p xY 

is the derivative of (log a:) 1 ~ i /(l /A), and this latter function 
approaches a finite limit if and only if //. > 1. 



338 INFINITE SERIES [vill, 162 

Cauchy s theorem admits of applications of another sort. Let us suppose 
that the function </>(x) satisfies the conditions imposed above, and let us con 
sider the sum 

<j>(n) + 4>(n + 1) + + <f>(n + p) , 

where n andp are two integers which are to be allowed to become infinite. If the 
series whose general term is <p(n) is convergent, the preceding sum approaches 
zero as a limit, since it is the difference between the two sums S n + p + i and S n , 
each of which approaches the sum of the series. But if this series is divergent, 
no conclusion can be drawn. Keturning to the geometrical interpretation given 
above, we find the double inequality 



f 0(x)dx<0(n) + 0(n + l) + +<(n + p)<</>(n).+ f 

t/n t/n 



* P 



Since <f>(ri) approaches zero as n becomes infinite, it is evident that the limit of 
the sum in question is the same as that of the definite integral / n " +p 0(x)dx, 
and this depends upon the manner in which n and p become infinite. 
For example, the limit of the sum 

1 1 1 

n n + 1 n + p 

is the same as that of the definite integral f" +P [1/x] dx = log(l + p/n). It is 
clear that this integral approaches a limit if and only if the ratio p/n approaches a 
limit. If a is the limit of this ratio, the preceding sum approaches log(l + a) 
as its limit, as we have already seen in 49. 
Finally, the limit of the sum 



Vn Vn + 1 Vn + p 

is the same as that of the definite integral 

I+P j 



f 

\J n 



= 2 ( Vn + p Vn). 

Vx 



In order that this expression should approach a limit, it is necessary that the 
ratio p/Vn should approach a limit a. Then the preceding expression may be 
written in the form 



2- 



Vn + p + Vn 

1 
and it is evident that the limit of this expression is a. 

162. Logarithmic criteria. Taking the series 

11 1 

j^ 2 M n^ 

as a comparison series, Cauchy deduced a new test for convergence 
which is entirely analogous to that which involves -\/u n . 



vm, 162] CONSTANT TERMS 339 

If after a certain term the expression log(l/w n )/logn- is always 
greater than a fixed number which is greater than unity, the series 
converges. If after a certain term log(l/w n )/logn is always less 
than unity, the series diverges. 

If log(l/w n )/log n approaches a limit I as n increases indefinitely, 
the series converges if I > 1, and diverges if Ki. The case in 
which I = 1 remains in doubt. 

In order to prove the first part of the theorem, we will remark 
that the inequality 

log > k log n 

u n 

is equivalent to the inequality 

- > n k or u n < -T ; 

u n n k 

since k > 1, the series surely converges. 
Likewise, if 

log < log w, 

U n 

we shall have u n > 1/n, whence the series surely diverges. 

This test enables us to determine whether a given series con 
verges or diverges whenever the terms of the series, after a certain 
one, are each less, respectively, than the corresponding terms of 
the series 



where A is a constant factor and p. > 1. For, if 

A 

U n < 

n 
we shall have log ?/ + //. log n < log A or 

, 1 

lo S^ log A 

. - > /A ; 

log n log n 

and the right-hand side approaches the limit /A as n increases 
indefinitely. If K denotes a number between unity and p., we 
shall have, after a certain term, 



log n 



340 INFINITE SERIES [VIII, 163 

Similarly, taking the series 






_ _ 

n(\og n)^ ~r n log w(log 2 ri)* 

as comparison series, we obtain an infinite suite of tests for con 
vergence which may be obtained mechanically from the preceding 
by replacing the expression \og(l/u n )/logn by log[l/(/m n )]/log 2 ?i, 
then by 

log- 

nu n log n 



log 8 n 

and so forth, in the statement of the preceding tests.* These tests 
apply in more and more general cases. Indeed, it is easy to show 
that if the convergence or divergence of a series can be established 
by means of any one of them, the same will be true of any of those 
which follow. It may happen that no matter how far we proceed 
with these trial tests, no one of them will enable us to determine 
whether the series converges or diverges. Du Bois-Reymond f and 
Pringsheim have in fact actually given examples of both convergent 
and divergent series for which none of these logarithmic tests deter 
mines whether the series converge or diverge. This result is of great 
theoretical importance, but convergent series of this type evidently 
converge very slowly, and it scarcely appears possible that they 
should ever have any practical application whatever in problems 
which involve numerical calculation. 

163. Raabe s or Duhamel s test. Retaining the same comparison 
series, but comparing the ratios of two consecutive terms instead 
of comparing the terms themselves, we are led to new tests which 
are, to be sure, less general than the preceding, but which are 
often easier to apply in practice. For example, consider the series 
of positive terms 

(7) ?/ + m + w a H h u n -i , 

* See Bertrand, Traitt de Calcul differential et integral, Vol. I, p. 238; Journal 
de Liouville, 1st series, Vol. VII, p. 35. 

t Ueber Convergenz von Reihen . . . (Crelle s Journal, Vol. LXXVI, p. 85, 1873). 

J Allgemeine Theorie der Divergenz . . . (Mathematische Annalen, Vol. XXXV, 
1890). 

In an example of a certain convergent series due to du Bois-Reymond it would 
be necessary, according to the author, to take a number of terms equal to the volume 
of the earth expressed in cubic millimeters in order to obtain merely half the sum of 
the series. 



VIII, 163] CONSTANT TERMS 341 

in which the ratio u n + l /u a approaches unity, remaining constantly 
less than unity. Then we may write 



!+ 

where a n approaches zero as n becomes infinite. The comparison of 
this ratio with [n/(n +1)] M leads to the following rule, discovered 
first by Raabe* and then by Duhamel.f 

If after a certain term the product na n is always greater than a 
fixed number which is greater than unity, the series converges. If 
after a certain term the same product is always less than unity, the 
series diverges. 

The second part of the theorem follows immediately. For, since 
na n < 1 after a certain term, it follows that 



1 + ac H n+l 

and the ratio u n + l /u n is greater than the ratio of two consecutive 
terms of the harmonic series. Hence the series diverges. 

In order to prove the first part, let us suppose that after a certain 
term we always have na n >k>l. Let p. be a number which lies 
between 1 and k, 1 < p. < k. Then the series surely converges if 
after a certain term the ratio u n + l /u n is less than the ratio 
[_n/(n + 1)] 1 of two consecutive terms of the series whose general 
term is n~*. The necessary condition that this should be true 
is that 



(8) 



or, developing (1 + 1/n) 1 by Taylor s theorem limited to the term 
in 1/n 2 , 

l+- + -i< l+a n , 

W 7T 

where X n always remains less than a fixed number as n becomes 
infinite. Simplifying this inequality, we may write it in the form 



< na n . 
n 



* Zeitschrift fur Mathematik und Physik, Vol. X, 1832. 
t Journal de Liouville, Vol. IV, 1838. 



342 INFINITE SERIES [VIII, 103 

The left-hand side of this inequality approaches /u. as its limit as n 
becomes infinite. Hence, after a sufficiently large value of n, the 
left-hand side will be less than na n , which proves the inequality (8). 
It follows that the series is convergent. 

If the product na n approaches a limit I as n becomes infinite, we 
may apply the preceding rule. The series is convergent if 1>1, 
and divergent if I < 1. A doubt exists if I = 1, except when na n 
approaches unity remaining constantly less than unity : in that case 
the series diverges. 

If the product na n approaches unity as its limit, we may compare the ratio 
w + i/^n with the ratio of two consecutive terms of the series 



which converges if /*>!> and diverges if M^l- The ratio of two consecutive 
terms of the given series may be written in the form 



u " 1 ft, 

I + - + - 

n n 

where ft, approaches zero as n becomes infinite. If after a certain term the 
product ft, logn is always greater than a fixed number which is greater than unity, 
the series converges. If after a certain term the same product is always less than 
unity, the series diverges. 

In order to prove the first part of the theorem, let us suppose that ft, log n > k > 1. 
Let /A be a number between 1 and k. Then the series will surely converge if after 
a certain term we have 

(9\ u " +l < ~ n - [ logn T 

u n n + 1 Llog(n + l)J 
which may be written in the form 



logn 
or, applying Taylor s theorem to the right-hand side, 

1 + - + 



n \ nl ( log n 

where X B always remains less than a fixed number as n becomes infinite. 
Simplifying this inequality, it becomes 



V 



VIII, lt>;] CONSTANT TERMS 343 

The product (n + 1) log (1 + 1/n) approaches unity as n becomes infinite, for it 
may be written, by Taylor s theorem, in the form 

(10) 

where e approaches zero. The right-hand side of the above inequality therefore 
approaches fj. as its limit, and the truth of the inequality is established for suffi 
ciently large values of n, since the left-hand side is greater than k, which is itself 
greater than /^. 

The second part of the theorem may be proved by comparing the ratio 
ttn + i/w,, with the ratio of two consecutive terms of the series whose general 
term is l/(nlogn). For the inequality 

+ 1 > __ lg n 



u n n + 1 log(n + 1) 
which is to be proved, may be written in the form 



1 + - 



n n \ n/ 1_ log n 

<(* + !) log/1 + --). 

V */ 



The right-hand side approaches unity through values which are greater than 
unity, as is seen from the equation (10). The truth of the inequality is there 
fore established for sufficiently large values of n, for the left-hand side cannot 
exceed unity. 

From the above proposition it may be shown, as a corollary, that if the prod 
uct (8 n log n approaches a limit I as n becomes infinite, the series converges if I > 1, 
and diverges if l<l. The case in which I = 1 remains in doubt, unless /3 n logn 
is always less than unity. In that case the series surely diverges. 

If p n log n approaches unity through values which are greater than unity, we 
may write, in like manner, 



n n log n 

where > approaches zero as n becomes infinite. It would then be possible to 
prove theorems exactly analogous to the above by considering the product 
7,,log 2 n, and so forth. 

Corollary. If in a series of positive terms the ratio of any term to the pre 
ceding can be written in the form 

. r H n 
l --- 1 



u n n n 



i + n 



where /j. is a positive number, r a constant, and H n a quantity whose absolute 
value remains less than a fixed number as n increases indefinitely, the series con 
verges If r is greater than unity, and diverges in all other cases. 



344 INFINITE SERIES [VIII, 164 

For if we set 



we shall have 

r - 



na n = 



l- r - + 

n 



and hence lim na n = r. It follows that the series converges if r > 1, and diverges 
if r < 1. The only case which remains in doubt is that in which r = 1. In order 
to decide this case, let us set 



n n 
From this we find 



log n n + 1 __ log n 
-** 



, log n = 



and the right-hand side approaches zero as n becomes infinite, no matter how 
small the number /* may be. Hence the series diverges. 

Suppose, for example, that u n + \/u n is a rational function of n which ap 
proaches unity as n increases indefinitely : 



~ 2 -\ ---- 

Then, performing the division indicated and stopping with the term in 1/n 2 , we 
may write 

Un + l - 1 _L CTl ~ bl -L ^(^ 

- i -r -r ., > 

u n n n z 

where 0(n) is a rational function of n which approaches a limit as n becomes 
infinite. By the preceding theorem, the necessary and sufficient condition that 
the series should converge is that 

bi > ai + 1 . 

This theorem is due to Gauss, who proved it directly.* It was one of the first 
general tests for convergence. 

164. Absolute convergence. We shall now proceed to study series 
whose terms may be either positive or negative. If after a certain 
term all the terms have the same sign, the discussion reduces to 
the previous case. Hence we may restrict ourselves to series 
which contain an infinite number of positive terms and an infinite 

* (Collected Works, Vol. Ill, p. 138.) Disquisitiones generates circa seriem infinitam 

a.B 
1+ = + , 

l.y 



VIII, 164] CONSTANT TERMS 345 

number of negative terms. We shall prove first of all the fol 
lowing fundamental theorem : 

Any series whatever is convergent if the series formed of the abso 
lute values of the terms of the given series converges. 

Let 

(11) U Q + M! H ----- h u H ---- 

be a series of positive and negative terms, and let 

(12) l\+ L\ + .--+ / + 

be the series of the absolute values of the terms of the given series, 
where U n \ u n . If the series (12) converges, the series (11) like 
wise converges. This is a consequence of the general theorem of 
157. For we have 



and the right-hand side may be made less than any preassigned num 
ber by choosing n sufficiently large, for any subsequent choice of p. 
Hence the same is true for the left-hand side, and the series (11) 
surely converges. 

The theorem may also be proved as follows : Let us write 

u n = (u n + U n ) - U n , 

and then consider the auxiliary series whose general term is u n + U n , 
(13) (u + U -) + (u i + U l ) + ...+ (u n + C7 n ) + . . . . 

Let S n , S n , and S J denote the sums of the first n terms of the series 
(11), (12), and (13), respectively. Then we shall have 



The series (12) converges by hypothesis. Hence the series (13) 
also converges, since none of its terms is negative and its general 
term cannot exceed 2U n . It follows that each of the sums S n and 
S J, and hence also the sum S n , approaches a limit as n increases 
indefinitely. Hence the given series (11) converges. It is evident 
that the given series may be thought of as arising from the subtrac 
tion of two convergent series of positive terms. 

Any series is said to be absolutely convergent if the series of the 
absolute values of its terms converges. In such a series the order of 
the terms may be changed in any way whatever without altering the 



34G INFINITE SERIES [VIII, KM 

sum of the series. Let us first consider a convergent series of posi 
tive terms, 

(14) a +aH ----- \-a n -\ ---- , 
whose sum is S, and let 

(15) b + b l + .-. + b n + ..- 

be a series whose terms are the same as those of the first series 
arranged in a different order, i.e. each term of the series (14) is to 
be found somewhere in the series (15), and each term of the series 
(15) occurs in the series (14). 

Let S m be the sum of the first m terms of the series (15). Since 
all these terms occur in the series (14), it is evident that n may be 
chosen so .large that the first m terms of the series (15) are to be 
found among the first n terms of the series (14). Hence we shall have 

S m < S n < S, 

which shows that the series (15) converges and that its sum S does 
not exceed S. In a similar manner it is clear that S 5 S . Hence 
S = S. The same argument shows that if one of the above series 
(14) and (15) diverges, the other does also. 

The terms of a convergent series of positive terms may also be 
grouped together in any manner, that is, we may form a series each 
of whose terms is equal to the sum of a certain mimber of terms of 
the given series without altering the sum of the series.* Let us first 
suppose that consecutive terms are grouped together, and let 

(16) A, + A l + Ai + >"+A m + .... 

be the new series obtained, where, for example, 



Then the sum S m of the first m terms of the series (16) is equal to 
the sum 5 V of the first N terms of the given series, where N > m. 
As m becomes infinite, N also becomes infinite, and hence S m also 
approaches the limit S. 

Combining the two preceding operations, it becomes clear that any 
convergent series of positive terms may be replaced by another series 
each of whose terms is the sum of a certain number of terms of the 
given series taken in any order whatever, without altering the sum of 

* It is often said that parentheses may be inserted in a convergent series of positive 
terms in any manner whatever without altering the sum of the series. TRANS. 



Vlll, 165] CONSTANT TERMS 347 

the series. It is only necessary that each term of the given series 
should occur in one and in only one of the groups which form the 
terms of the second series. 

Any absolutely convergent series may be regarded as the differ 
ence of two convergent series of positive terms ; hence the preceding 
operations are permissible in any such series. It is evident that an 
absolutely convergent series may be treated from the point of view 
of numerical calculation as if it were a sum of a finite number of 
terms. 

165. Conditionally convergent series. A series whose terms do not all 
have the same sign may be convergent without being absolutely con 
vergent. This fact is brought out clearly by the following theorem 
on alternating series, which we shall merely state, assuming that it 
is already familiar to the student.* 

A series whose terms are alternately positive and negative converges 
if the absolute value of each term is less than that of the preceding, 
and if, in addition, the absolute value of the terms of the series 
diminishes indefinitely as the number of terms increases indefinitely. 

For example, the series 

i-| + |-* + - + (-l)-; + - 

converges. We saw in 49 that its sum is log 2. The series 
of the absolute values of the terms of this series is precisely the 
harmonic series, which diverges. A series which converges but 
which does not converge absolutely is called a conditionally conver 
gent series. The investigations of Cauchy, Lejeune-Dirichlet, and 
Riemann have shown clearly the necessity of distinguishing between 
absolutely convergent series and conditionally convergent series. 
For instance, in a conditionally convergent series it is not always 
allowable to change the order of the terms nor to group the terms 
together in parentheses in an arbitrary manner. These operations 
may alter the sum of such a series, or may change a convergent 
series into a divergent series, or vice versa. For example, let us 
again consider the convergent series 

i- 1 + 1 - 1 - 1 _!_ 

2,3 4 ^ r 2n+l 2n + 2 

* It is pointed out in 1(K> that this theorem is a special case of the theorem proved 
there. TRANS. 



348 INFINITE SERIES [VIII, 166 

whose sum is evidently equal to the limit of the expression 

1 1 



!, + 1 2n + 2, 

as m becomes infinite. Let us write the terms of this series in another 
order, putting two negative terms after each positive term, as follows : 

~2~4 + 3~ 6~~8 "*" "*" 2n + 1 ~~ 4r* -f 2 ~ 4 + 4 "*" 

It is easy to show from a consideration of the sums S 3n , S Stl+l , and 
S 3n+2 that the new series converges. Its sum is the limit of the 
expression 

yV i _!_ _ 

= o \ 2n + 1 4:7i + 2 4w + 4 
as m becomes infinite. From the identity 



2w + 1 4ra + 2 4n + 4 2 \2n + 1 2n + 2, 

it is evident that the sum of the second series is half the sum of 
the given series. 

In general, given a series which is convergent but not absolutely convergent, 
it is possible to arrange the terms in such a way that the new series converges 
toward any preassigned number A whatever. Let S p denote the sum of the 
first p positive terms of the series, and S g the sum of the absolute values of the 
first q negative terms, taken in such a way that the p positive terms and the q 
negative terms constitute the first p + q terms of the series. Then the sum of 
the first p + q terms is evidently S p S q . As the two numbers p and q increase 
indefinitely, each of the sums S p and S q must increase indefinitely, for otherwise 
the series would diverge, or else converge absolutely. On the other hand, since 
the series is supposed to converge, the general term must approach zero. 

We may now form a new series whose sum is A in the following manner : 
Let us take positive terms from the given series in the order in which they occur 
in it until their sum exceeds A. Let us then add to these, in the order in which 
they occur in the given series, negative terms until the total sum is less than A. 
Again, beginning with the positive terms where we left off, let us add positive 
terms until the total sum is greater than A. We should then return to the 
negative terms, and so on. It is clear that the sum of the first n terms of the 
new series thus obtained is alternately greater than and then less than A, and 
that it differs from A by a quantity which approaches zero as its limit. 

166. Abel s test. The following test, due to Abel, enables us to establish the 
convergence of certain series for which the preceding tests fail. The proof is 
based upon the lemma stated and proved in 75. 

Let 

MO + ui + h + 



VIII, 166] CONSTANT TERMS 349 

be a series which converges or which is indeterminate (that is, for which the sum 
of the first n terms is always less than a fixed number A in absolute value). 
Again, let 

fQ ) 1 ) e n , 

be a monotonically decreasing sequence of positive numbers which approach 
zero as n becomes infinite. Then the series 

(17) foU + eiMi + + e n Wn + . . . 

converges under the hypotheses made above. 

For by the hypotheses made above it follows that 



for any value of n and p. Hence, by the lemma just referred to, we may write 

|u, i + ie,, + i + + U n+p e n+p \ < 2Ae n + i. 

Sine? e n+ i approaches zero as n becomes infinite, n may be chosen so large that 
the absolute value of the sum 



will be less than any preassigned positive number for all values of p. The 
series (17) therefore converges by the general theorem of 157. 
When the series u + MI + + u n + reduces to the series 

1 - 1 + 1 - 1 + 1 -!-, 

whose terms are alternately + 1 and 1, the theorem of this article reduces to 
the theorem stated in 165 with regard to alternating series. 
As an example under the general theorem consider the series 

sin 6 + sin 2 + sin 3 6 + + sin n + , 

which is convergent or indeterminate. For if sin 6 = 0, every term of the series 
is zero, while if sin ^ 0, the sum of the first n terms, by a formula of Trigo 
nometry, is equal to the expression 

. nft 
sin 

2 . /n + 1 

sin ( 

.9 \ 2 
sin - 
2 

which is less than | I/sin (6/2) \ in absolute value. It follows that the series 
sin 6 sin 2 sin n 6 

I i T I -f- -j- 

converges for all values of 6. It may be shown in a similar manner that the 

series 

cos 6 cos 2 6 cos n 6 



12 n 

converges for all values of except 2krt. 



h 



350 INFINITE SERIES [VIII, 167 

Corollary. Restricting ourselves to convergent series, we may state a more 
general theorem. Let 

MO + MI H ----- f- u n + 

be a convergent series, and let 



be any monotonically increasing or decreasing sequence of positive numbers 
which approach a limit k different from zero as n increases indefinitely. Then 
the series 

(18) e o + eii H ----- h H ---- 

also converges. 

For definiteness let us suppose that the e s always increase. Then we may 
write 

e = k a , ei = k a-i , , e n = k a n , , 

where the numbers a , a\ , , a n , form a sequence of decreasing positive 
numbers which approach zero as n becomes infinite. It follows that the two 
series 

ku + kui + + ku n + , 



both converge, and therefore the series (18) also converges. 

II. SERIES OF COMPLEX TERMS MULTIPLE SERIES 

167. Definitions. In this section we shall deal with certain gen 
eralizations of the idea of an infinite series. 
Let 

(19) u + u, + i, z + + u n + 

be a series whose terms are imaginary quantities: 



Such a series is said to be convergent if the two series formed of 
the real parts of the successive terms and of the coefficients of the 
imaginary parts, respectively, both converge: 

(20) a + a, + a 2 -f . + a n + = S , 

(21) &. + &1+6.+- -+^ + --- = ^". 

Let S and S" be the sums of the series (20) and (21), respectively. 
Then the quantity S = S + is" is called the sum of the series (19). 
It is evident that S is, as before, the limit of the sum S n of the first 
n terms of the given series as n becomes infinite. It is evident 
that a series of complex terms is essentially only a combination of 
two series of real terms. 



VIII, 168] COMPLEX TERMS MULTIPLE SERIES 351 

When the series of absolute values of the terms of the series (19) 

converges, each of the series (20) and (21) evidently converges abso 



lutely, for \a n < vX + l and IVI = 

In this case the series (19) is said to be absolutely convergent. The 
sum of such a series is not altered by a change in the order of the 
terms, nor by grouping the terms together in any way. 

Conversely, if each of the series (20) and (21) converges absolutely, 
the series (22) converges absolutely, for y a* + b\ ;> | a n | + | b n \ . 

Corresponding to every test for the convergence of a series of 
positive terms there exists a test for the absolute convergence of 
any series whatever, real or imaginary. Thus, if the absolute value 
of the ratio of two consecutive terms of a series \u n + l /u n \, after a cer 
tain term, is less than a fixed number less than unity, the series con 
verges absolutely. For, let 7, = | u, \ . Then, since | u n+l /u n \ < k < 1 
after a certain term, we shall have also 

U rl _ , 



which shows that the series of absolute values 

U + U l + + U n + 

converges. If \u n+l /u n [ approaches a limit I as n becomes infinite, 
the series converges if I < 1, and diverges if I > 1. The first half is 
self-evident. In the second case the general term u n does not 
approach zero, and consequently the series (20) and (21) cannot 
both be convergent. The case I = 1 remains in doubt. 

More generally, if to be the greatest limit of Vt7 n as n becomes infinite, the 
series (19) converges if w<l, and diverges if w>l. For in the latter case the 
modulus of the general term does not approach zero (see 161). The case in 
which w = 1 remains in doubt the series may be absolutely convergent, simply 
convergent, or divergent. 

168. Multiplication of series. Let 

(23) u + Ul + u z + + "+-, 

(24) r + Vl + v, + + v n + 

be any two series whatever. Let us multiply terms of the first 
series by terms of the second in all possible ways, and then group 



352 INFINITE SERIES [VIII, 168 

together all the products u t Vj for which the sum i+j of the sub 
scripts is the same ; we obtain in this way a new series 

(25) \ U V + ( H VI + UlV ) + ( M * 2 + u * v i + W 2 W ) ^ 

( +( u <>v n + u l v n _ l -{ hw B Vo)H 

If each of the series (23) and (24) is absolutely convergent, the 
series (25) converges, and its sum is the product of the sums of the 
two given series. This theorem, which is due to Cauchy, was gener 
alized by Mertens,* who showed that it still holds if only one of the 
series (23) and (24) is absolutely convergent and the other is merely 
convergent. 

Let us suppose for definiteness that the series (23) converges 
absolutely, and let w n be the general term of the series (25): 

W n = 0^n + MlW-l H 1- U n V . 

The proposition will be proved if we can show that each of the 
differences 

^0 + W l H 1" W Sn ~ ( U + % -{ 1- ) (l + Vl H h V n ) , 

^o + w l -\ + w 2n+l - (u + u l -\ + n + i)0o + *i H + v n + 1 ) 

approaches zero as n becomes infinite. Since the proof is the same 
in each case, we shall consider the first difference only. Arranging 
it according to the u s, it becomes 

S = U (v n + l -\ h V 2n ) + HI (W n + 1 + + V 2n _,) H h M.,-1 t . + i 

+ w n+ i(v H -f v_i)+ u n + a (v -\ h v n _ 2 )-\ \-u 2n v . 

Since the series (23) converges absolutely, the sum U + U l H f- U n 

is less than a fixed positive number A for all values of n. Like 
wise, since the series (24) converges, the absolute value of the sum 
v o + Vi + + v n is less than a fixed positive number B. Moreover, 
corresponding to any preassigned positive number e a number m 
exists such that 



A + B 
c 



for any value of p whatever, provided that n > m. Having so chosen 
n that all these inequalities are satisfied, an upper limit of the quan 
tity 1 8 is given by replacing u , u l} u z , ., u 2n by U , U 1} U 2 , , L\ nJ 



* Crelle s Journal, Vol. LXXIX. 



VIII, 169] COMPLEX TERMS MULTIPLE SERIES 353 

respectively, v n + l + v n + 2 -\ ----- \- v n + p by (./(A + B), and finally each 
of the expressions v + v t + + v n _ l , v + + v n _ 2 , , v by B. 
This gives 

1 8 1 < u + Ul + " + Un l 



or 



A + B A + B 

whence, finally, 1 8 1 < e. Hence the difference 8 actually does approach 
zero as n becomes infinite. 

169. Double series. Consider a rectangular network which is lim 
ited upward and to the left, but which extends indefinitely down 
ward and to the right. The network will contain an infinite number 
of vertical columns, which we shall number from left to right from 
to + oo . It will also contain an infinite number of horizontal 
rows, which we shall number from the top downward from to + oo . 
Let us now suppose that to each of the rectangles of the network a 
certain quantity is assigned and written in the corresponding rec 
tangle. Let a ik be the quantity which lies in the ith row and in the 
kih column. Then we shall have an array of the form 



(26) 



We shall first suppose that each of the elements of this array is real 
and positive. 

Now let an infinite sequence of curves C lt C 2 , , C n , be drawn 
across this array as follows : 1) Any one of them forms with the two 
straight lines which bound the array a closed curve which entirely 
surrounds the preceding one ; 2) The distance from any fixed point 
to any point of the curve C B , which is otherwise entirely arbitrary, 
becomes infinite with n. Let S t be the sum of the elements of the 
array which lie entirely inside the closed curve composed of C,- and 



354 INFINITE SERIES [VIII, 169 

the two straight lines which bound the array. If S n approaches a 
limit S as n becomes infinite, we shall say that the double series 

\-~r- +-r> 

(27) 

converges, and that its sum is S. In order to justify this definition, 
it is necessary to show that the limit 5 is independent of the form 
of the curves C. Let C{, C" z , , C m , be another set of curves 
which recede indefinitely, and let S{ be the sum of the elements 
inside the closed curve formed by C- and the two boundaries. If m 
be assigned any fixed value, n can always be so chosen that the 
curve C n lies entirely outside of C m . Hence S m < S n , and therefore 
S m ^ S, for any value of m. Since S m increases steadily with m, it 
must approach a limit S < 5 as m becomes infinite. In the same 
way it follows that S < S . Hence S = S. 

For example, the curve C. may be chosen as the two lines which 
form with the boundaries of the array a square whose side increases 
indefinitely with i, or as a straight line equally inclined to the two 
boundaries. The corresponding sums are, respectively, the following : 

00 + fan) + 11 + Ol)H ----- K a + a n\ H ----- h + -!, n H ----- H 0n) > 



If either of these sums approaches a limit as n becomes infinite, the 
other will also, and the two limits are equal. 

The array may also be added by rows or by columns. For, sup 
pose that the double series (27) converges, and let its sum be 5. It 
is evident that the sum of any finite number of elements of the series 
cannot exceed 5. It follows that each of the series formed of the 
elements in a single row 

(28) % + *! + + <*. + , i = 0, 1, 2, , 

converges, for the sum of the first n + 1 terms a ;o + a n + + a, n 
cannot exceed S and increases steadily with n. Let o-, be the sum of 
the series formed of the elements in the ith row. Then the new series 

(29) o- + o-, + - + o-, + 

surely converges. For, let us consider the sum of the terms of the 
array 2 tt for which i^p, k^r. This sum cannot exceed S, and 
increases steadily with r for any fixed value of p; hence it 
approaches a limit as r becomes infinite, and that limit is equal to 

(30) <r n + o-, -f- + tr 



VIII, 1(50] COMPLEX TERMS MULTIPLE SERIES 355 

for any fixed value of p. It follows that <r + o^ + + v p cannot 
exceed 5 and increases steadily with p. Consequently the series (29) 
converges, and its sum 2 is less than or equal to S. Conversely, if 
each of the series (28) converges, and the series (29) converges to a 
sum 2, it is evident that the sum of any finite number of elements 
of the array (26) cannot exceed 2. Hence S 5= 2, and consequently 
2 = s. 

The argument just given for the series formed from the elements 
in individual rows evidently holds equally well for the series formed 
from the elements in individual columns. The sum of a convergent 
double series whose elements are all positive may be evaluated by 
rows, by columns, or by means of curves of any form which recede 
indefinitely. In particular, if the series converges when added by rows, 
it will surely converge when added by columns, and the sum will be the 
same. A number of theorems proved for simple series of positive 
terms may be extended to double series of positive elements. For 
example : if each of the elements of a double series of positive elements 
is less, respectively, than the corresponding elements of a knoivn con 
vergent double series, the first series is also convergent; and so forth. 

A double series of positive terms which is not convergent is said 
to be divergent. The sum of the elements of the corresponding 
array which lie inside any closed curve increases beyond all limit 
as the curve recedes indefinitely in every direction. 

Let us now consider an array whose elements are not all positive. 
It is evident that it is unnecessary to consider the cases in which 
all the elements are negative, or in which only a finite number of 
elements are either positive or negative, since each of these cases 
reduces immediately to the preceding case. We shall therefore sup 
pose that there are an infinite number of positive elements and an 
infinite number of negative elements in the array. Let a lk be the 
general term of this array T. If the array 7\ of positive elements, 
each of which is the absolute value | a ik \ of the corresponding element 
in T, converges, the array T is said to be absolutely convergent. Such 
an array has all of the essential properties of a convergent array of 
positive elements. 

In order to prove this, let us consider two auxiliary arrays T 
and T", defined as follows. The array T is formed from the array T 
by replacing each negative element by a zero, retaining the positive 
elements as they stand. Likewise, the array T" is obtained from 
the array T by replacing each positive element by a zero and chang 
ing the sign of each negative element. Each of the arrays T and T" 



356 INFINITE SERIES [VIII, 169 

converges whenever the array 7\ converges, for each element of T , 
for example, is less than the corresponding element of 7^. The sum 
of the terms of the series T which lie inside any closed curve is 
equal to the difference between the sum of the terms of T which 
lie inside the same curve and the sum of the terms of T" which 
lie inside it. Since the two latter sums each approach limits as 
the curve recedes indefinitely in all directions, the first sum also 
approaches a limit, and that limit is independent of the form of 
the boundary curve. This limit is called the sum of the array T. 
The argument given above for arrays of positive elements shows 
that the same sum will be obtained by evaluating the array T by 
rows or by columns. It is now clear that an array whose elements 
are indiscriminately positive and negative, if it converges absolutely, 
may be treated as if it were a convergent array of positive terms. 
But it is essential that the series 7\ of positive terms be shown to 
be convergent. 

If the array TI diverges, at least one of the arrays T and T" diverges. If 
only one of them, T for example, diverges, the other T" being convergent, the 
sum of the elements of the array T which lie inside a closed curve C becomes 
infinite as the curve recedes indefinitely in all directions, irrespective of the 
form of the curve. If both arrays 7" and T" diverge, the above reasoning 
shows only one thing, that the sum of the elements of the array T inside 
a closed curve C is equal to the difference between two sums, each of which 
increases indefinitely as the curve C recedes indefinitely in all directions. It 
may happen that the sum of the elements of T inside C approach different 
limits according to the form of the curves C and the manner in which they 
recede, that is to say, according to the relative rate at which the number of 
positive terms and the number of negative terms in the sum are made to increase. 
The sum may even become infinite or approach no limit whatever for certain 
methods of recession. As a particular case, the sum obtained on evaluating by 
rows may be entirely different from that obtained on evaluating by columns if 
the array is not absolutely convergent. 

The following example is due to Arndt.* Let us consider the array 



(31) 



1 /1\ 1 /2\ 1 /2\ 1 /3 



p 



2\2/ SW 3\3/ 4\4/ A P I P + 

l^V-^-V 1/ 2 V-1/ 3 V I/^liV-- 1 X n X 

2V2/ 3V3/ 3V3/ 4\4/ P\ P / P 

I /l\" 1 /2\ n 1 /2\" 1 /3\" 



2\2/ 3\3/ 3V3/ 4\4/ p\ p / p + 



* Grunert s Archiv, Vol. XI, p. 319. 



VIII, 169] COMPLEX TERMS MULTIPLE SERIES 357 

which contains an infinite number of positire and an infinite number of negative 
elements. Each of the series formed from the elements in a single row or from 
those in a single column converges. The sum of the series formed from the 
terms in the nth row is evidently 



2\2 

Hence, evaluating the array (31) by rows, the result obtained is equal to the 
sum of the convergent series 

2a + 28 + " + 2Ml + " 

which is 1/2. On the other hand, the series formed from the elements in the 
(p l)th column, that is, 



converges, and its sum is 

jj-1 p -1 1 1 

p p+ 1 p(p +1) p + 1 p 

Hence, evaluating the array (31) by columns, the result obtained is equal to the 
sum of the convergent series 



3 

which is 1/2. 

This example shows clearly that a double series should not be used in a 
calculation unless it is absolutely convergent. 

We shall also meet with double series whose elements are complex 
quantities. If the elements of the array (26) are complex, two other 
arrays 7" and T" may be formed where each element of T is the 
real part of the corresponding element of T and each element of T" 
is the coefficient of i in the corresponding element of T. If the 
array 7\ of absolute values of the elements of T, each of whose 
elements is the absolute value of the corresponding element of T, 
converges, each of the arrays T and T" converges absolutely, and 
the given array T is said to be absolutely convergent. The sum of 
the elements of the array which lie inside a variable closed curve 
approaches a limit as the curve recedes indefinitely in all directions. 
This limit is independent of the form of the variable curve, and it 
is called the sum of the given array. The sum of any absolutely 
convergent array may also be evaluated by rows or by columns. 



358 INFINITE SERIES [VIII, 170 

170. An absolutely convergent double series may be replaced by a simple 
series formed from the same elements. It will be sufficient to show that the 
rectangles of the network (26) can be numbered in such a way that each rec 
tangle has a definite number, without exception, different from that of any other 
rectangle. In other words, we need merely show that the sequence of natural 
numbers 

(32) 0, 1, 2, -.., n, -.-, 

and the assemblage of all pairs of positive integers (i, fc), where i^O, k>0, can 
be paired off in such a way that one and only one number of the sequence (32) 
will correspond to any given pair (i, k), and conversely, no number n corresponds 
to more than one of the pairs (t, k). Let us write the pairs (i, k) in order as 
follows : 

(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), -.., 

where, in general, all those pairs for which i + k = n are written down after 
those for which i + k < n have all been written down, the order in which those 
of any one set are written being the same as that of the values of i for the various 
pairs beginning with (n, 0) and going to (0, n). It is evident that any pair (i, k) 
will be preceded by only a, finite number of other pairs. Hence each pair will 
have a distinct number when the sequence just written down is counted off 
according to the natural numbers. 

Suppose that the elements of the absolutely convergent double series SSaa- are 
written down in the order just determined. Then we shall have an ordinary series 

(33) doo + io + oi + 2o + ii + floa + + ao + a-i,i + 

whose terms coincide with the elements of the given double series. This simple 
series evidently converges absolutely, and its sum is equal to the sum of the given 
double series. It is clear that the method we have employed is not the only pos 
sible method of transforming the given double series into a simple series, since 
the order of the terms of the series (33) can be altered at pleasure. Conversely, 
any absolutely convergent simple series can be transformed into a double series 
in an infinite variety of ways, and that process constitutes a powerful instrument 
in the proof of certain identities.* 

It is evident that the concept of double series is not essentially different from 
that of simple series. In studying absolutely convergent series we found that 
the order of the terms could be altered at will, and that any finite number of 
terms could be replaced by their sum without altering the sum of the series. 
An attempt to generalize this property leads very naturally to the introduction 
of double series. 

171. Multiple series. The notion of double series may be generalized. 
In the first place we may consider a series of elements a mn with two 
subscripts ra and n, each of which may vary from oo to + oo . 
The elements of such a series may be arranged in the rectangles of 
a rectangular network which extends indefinitely in all directions ; 

*Tanuery, Introduction a hi theorie desf auctions d une variable, p. G7. 



VIII, 172] COMPLEX TERMS MULTIPLE SERIES 359 

it is evident that it may be divided into four double series of the 
type we have just studied. 

A more important generalization is the following. Let us consider 
a series of elements of the type ,,. .., mp , where the subscripts 
tK. 1} m 2 , , m p may take on any values from to -f oo , or from GO 
to + oo, but may be restricted by certain inequalities. Although no 
such convenient geometrical form as that used above is available 
when the number of subscripts exceeds three, a slight consideration 
shows that the theorems proved for double series admit of immediate 
generalization to multiple series of any order p. Let us first sup 
pose that all the elements MI , ,,,..., mp are real and positive. Let S l 
be the sum of a certain number of elements of the given series, S 2 
the sum of <S\ and a certain number of terms previously neglected, 
S a the sum of S 2 and further terms, and so on, the successive sums 
S 1} S 2 , -, S n , being formed in such a way that any particular 
element of the given series occurs in all the sums past a certain one. 
If S n approaches a limit S as n becomes infinite, the given series 
is said to be convergent, and S is called its sum. As in the case of 
double series, this limit is independent of the way in which the 
successive sums are formed. 

If the elements of the given multiple series have different signs 
or are complex quantities, the series will still surely converge if the 
series of absolute values of the terms of the given series converges. 

172. Generalization of Cauchy s theorem. The following theorem, 
which is a generalization of Cauchy s theorem ( 161), enables us to 
determine in many cases whether a given multiple series is conver 
gent or divergent. Let/(.x, y) be a function of the two variables x 
and y which is positive for all points (x, y) outside a certain closed 
curve T, and which steadily diminishes in value as the point (x, y) 
recedes from the origin.* Let us consider the value of the double 
integralJJ"/^, y} dx dij extended over the ring-shaped region between 
T and a variable curve C outside T, which we shall allow to recede 
indefinitely in all directions ; and let us compare it with the double 
series 2/(wi, n), where the subscripts m and n may assume any posi 
tive or negative integral values for which the point (m, n) lies out 
side the fixed curve T. Then the double series converges if the double 
integral approaches a limit, and conversely. 

* All that is necessary for the present proof is that/to, l/i)>/(a; 2) 7/ 2 ) whenever 
x\>X2 ond y\>y^ outside T. It is easy to adapt the proof to still more general 
hypotheses. TRANS. 



360 INFINITE SERIES [VIH,173 

The lines x = 0, x = 1, x = 2, and y = 0, y = 1, y = 2, 
divide the region between F and C into squares or portions of squares. 
Selecting from the double series the term which corresponds to that 
corner of each of these squares which is farthest from the origin, it 
is evident that the sum 2/(w, n) of these terms will be less than the 
value of the double integral ///(#, y) dx dy extended over the region 
between F and C. If the double integral approaches a limit as C 
recedes indefinitely in all directions, it follows that the sum of any 
number of terms of the series whatever is always less than a fixed 
number ; hence the series converges. Similarly, if the double series 
converges, the value of the double integral taken over any finite 
region is always less than a fixed number ; hence the integral 
approaches a limit. The theorem may be extended to multiple 
series of any order p, with suitable hypotheses ; in that case the 
integral of comparison is a multiple integral of order p. 

As an example consider the double series whose general term is 
l/(?n 2 + n 2 ) M , where the subscripts m and n may assume all integral 
values from oo to + o except the values m = n 0. This series 
converges for p. > 1, and diverges for /x^l. For the double integral 

dx dy 



extended over the region of the plane outside any circle whose 
center is the origin has a definite value if ^ > 1 and becomes 
infinite if p.<l (133). 

More generally the multiple series whose general term is 

1 

(m\ + m\ + -.. + miy 

where the set of values m l = m z = = m p = is excluded, con 
verges if 2n > j9.* 

III. SERIES OF VARIABLE TERMS UNIFORM CONVERGENCE 

173. Definition of uniform convergence. A series of the form 
(35) u (x) + KJ (*)+... + u n (x) + -, 

whose terms are continuous functions of a variable x in an inter 
val (a, i), and which converges for every value of x belonging to 
that interval, does not necessarily represent a continuous function, 

*More general theorems are to be found in Jordan s C ours d Analyse, Vol. I, p. 163. 



VIII, 173] VARIABLE TERMS 361 

as we might be tempted to believe. In order to prove the fact we 
need only consider the series studied in 4 : 



1 + x 2 (1 + a; 2 ) 2 (1 + a; 2 )" 

which satisfies the above conditions, but whose sum is discontinuous 
for x 0. Since a large number of the functions which occur in 
mathematics are defined by series, it has been found necessary to 
study the properties of functions given in the form of a series. The 
first question which arises is precisely that of determining whether 
or not the sum of a given series is a continuous function of the 
variable. Although no general solution of this problem is known, 
its study has led to the development of the very important notion 
of uniform, convergence. 

A series of the type (35), each of whose terms is a function of x 
which is defined in an interval (a, 6), is said to be uniformly con 
vergent in that interval if it converges for every value of x between 
a and b, and if, corresponding to any arbitrarily preassigned positive 
number c, a positive integer N, independent of x, can be found such 
that the absolute value of the remainder R n of the given series 

R n = U n + l () + MH + S (*) H ----- H ^ n+p (x) -\ ---- 



is less than e for every value of n^N and for every value of x 
which lies in the interval (a, &). 

The latter condition is essential in this definition. For any pre 
assigned value of x for which the series converges it is apparent 
from the very definition of convergence that, corresponding to any 
positive number e, a number N can be found which will satisfy 
the condition in question. But, in order that the series should con 
verge uniformly, it is necessary further that the same number N 
should satisfy this condition, no matter what value of x be selected 
in the interval (a, b). The following examples show that such is not 
always the case. Thus in the series considered just above we have 



The series in question is not uniformly convergent in the inter 
val (0, 1). For, in order that it should be, it would be necessary 
(though not sufficient) that a number N exist, such that 

1 



362 INFINITE SERIES [VIII, 17.5 

for all values of x in the interval (0, 1), or, what amounts to the 
same thing, that 



Whatever be the values of N and e, there always exist, however, 
positive values of x which do not satisfy this inequality, since the 
right-hand side is greater than unity. 

Again, consider the series denned by the equations 



= S - S 



n _ lf = 



The sum of the first n terms of this series is evidently S n (#), which 
approaches zero as n increases indefinitely. The series is therefore 
convergent, and the remainder R n (cc) is equal to nxe~" x *. In order 
that the series should be uniformly convergent in the interval (0, 1), 
it would be necessary and sufficient that, corresponding to any arbi 
trarily preassigned positive number e, a positive integer N exist such 
that for all values of n > N 



But, if x be replaced by 1/w, the left-hand side of this inequality is 
equal to e~ l/n , which is greater than 1/e whenever n > 1. Since e 
may be chosen less than 1/e, it follows that the given series is not 
uniformly convergent. 

The importance of uniformly convergent series rests upon the 
following property: 

The sum of a series whose terms are continuous functions of a 
variable x in an interval (a, i) and which converges uniformly in that 
interval, is itself a continuous function of x in the same interval. 

Let X Q be a value of x between a and b, and let x -f h be a value 
in the neighborhood of X Q which also lies between a and b. Let n 
be chosen so large that the remainder 

fl() = u n + l (x) + u n+z (x) H ---- 

is less than e/3 in absolute value for all values of x in the interval 
(a, b), where e is an arbitrarily preassigned positive number. Let/(cc) 
be the sum of the given convergent series. Then we may write 



where <}>(x) denotes the sum of the first n -f 1 terms, 

K*) = M uO*0 + M, 00 H + (). 



VIII, 17.5] VARIABLE TERMS 

Subtracting the two equalities 



363 



f(x + It) = <f>(x u + A) + R tt (x + A), 
we find 

f(x + h) -/(*) = [>(x + A) - <(*)] + /^(^u + A) - TJ.^o). 
The number was so chosen that we have 

O O 

On the other hand, since each of the terms of the series is a continu 
ous function of x, <(o:) is itself a continuous function of x. Hence 
a positive number 77 may be found such that 



whenever | h is less than rj. It follows that we shall have, a fortiori, 



whenever \h is less than rj. This shows that f(x) is continuous 
for x = x . 

Note. It would seem at first very difficult to determine whether 
or not a given series is uniformly convergent in a given interval. 
The following theorem enables us to show in many cases that a 
given series converges uniformly. 

Let 

(36) MO (a:) + MI (x) + - . - + (*) + 

be a series each of whose terms is a continuous function of x in an 
interval (a, b), and let 

(37) Jf. + Jtfi + ... + *; + ... 

be a convergent series whose terms are positive constants. Then, 
tf I u n | ^ M n for a H values of x in the interval (a, b) and for all 
values of n, the first series (36) converges uniformly in the interval 
considered. 

For it is evident that we shall have 



364 INFINITE SERIES [VIII, 174 

for all values of x between a and b. If N be chosen so large that 
the remainder R n of the second series is less than e for all values 
of n greater than N, we shall also have 



whenever n is greater than N, for all values of x in the interval (a, b). 
For example, the series 

M + Mi sin x + Mjj sin 2x -\ ----- \- M a smnx -\ ---- , 

where M , M l} M z , have the same meaning as above, converges 
uniformly in any interval whatever. 

174. Integration and differentiation of series. 

Any series of continuous functions which converges uniformly in an 
interval (a, b) may be integrated term by term, provided the limits of 
integration are finite and lie in the interval (a, b*). 

Let x and x v be any two values of x which lie between a and b, 
and let N be a positive integer such that | R B (ar)| < e for all values 
of x in the interval (a, b) whenever n > N. Let f(x) be the sum of 
the series 



and let us set 

/*i. r x i r x i r x \ r x * 

D n \ f(x) dx I u dx I Uidx ----- I u n dx = I R n dx. 

Jx t Jr tA Jx Jx 



The absolute value of D n is less than c x^ x 1 whenever n^ 
Hence D n approaches zero as n increases indefinitely, and we have 
the equation 

r*i r*i r x * r* 1 

I f(x}dx=\ u (x)dx+l Ui(x)dx-\ ----- hi u H (x)dx -\ ---- . 

J*t Jft J* ^ x * 

Considering X Q as fixed and Xj as variable, we obtain a series 

I u Q (x)dx-\ ---- +lu n (x )dx-\ ---- 

J*t J*o 

which converges uniformly in the interval (a, b) and represents a 
continuous function whose derivative is f(x). 



VIII, 174] VARIABLE TERMS 365 

Conversely, any convergent series may be differentiated term by term 
if the resulting series converges uniformly.* 

For, let 

f(x) = M (a;) + Ul (x) + + u n (x) -\ ---- 

be a series which converges in the interval (a, ft). Let us suppose 
that the series whose terms are the derivatives of the terms of the 
given series, respectively, converges uniformly in the same interval, 
and let <(z) denote the sum of the new series 



Integrating this series term by term between two limits x and x, 
each of which lies between a and b, we find 



/ 

Jx K 



= [u (a:) - M O (* O )] + [MJ (a;) - 

*SJC 

or 



This shows that <f>(x) is the derivative of /(x) . 
Examples. 1) The integral 



dx 
x 



cannot be expressed by means of a finite number of elementary 
functions. Let us write it as follows : 

/e x C dx C e x 1 C e x 1 
dx = I (-/ dx = log x -f I dx. 
x J x J x J x 

The last integral may be developed in a series which holds for all 
values of x. For we have 



and this series converges uniformly in the interval from R to + R, 
no matter how large R be taken, since the absolute value of any 

* It is assumed in the proof also that each term of the new series is a continuous 
function. The theorem is true, however, in general. TRANS. 



366 INFINITE SERIES [VIII, 174 

term of the series is less than the corresponding term of the con 
vergent series 



It follows that the series obtained by term-by-term integration 

OC OC^ 3C n 

F ^ =l+ I + 2Y72 + + nl.2...n + 

converges for any value of x and represents a function whose deriva 
tive is (tf !)/. 

2) The perimeter of an ellipse whose major axis is 2a and whose eccentricity 
is e is equal, by 1 12, to the definite integral 



= 4c f 

/o 



The product e 2 sin 2 <j> lies between and e 2 (< 1). Hence the radical is equal to the 
sum of the series given by the binomial theorem 



Vl e 2 sin 2 <f> = 1 -- e 2 sin 2 -- e 4 sin*^> 





2 . 4 . 6 2n 

The series on the right converges uniformly, for the absolute value of each of 
its terms is less than the corresponding term of the convergent series obtained 
by setting sin = 1. Hence the series may be integrated term by term; and 
since, by 116, 

C\ . .,. 1.8.6...(2n-l) it 
sin^"0a* -- 1 
Jo 2 . 4 . 6 2n 2 

we shall have 

f Vl-e 2 sinV<Z0 = -Sl- - e 2 - e* - eft ---- 
Jo 2 ( 4 64 256 

3.6...(2n-3)-| 2 
2.4.6--.2n J V 

If the eccentricity e is small, a very good approximation to the exact value of the 
integral is obtained by computing a few terms. 
Similarly, we may develop the integral 



f Vl- e 2 sin 2 0drf> 
Jo 



in a series for any value of the upper limit <f>. 

Finally, the development of Legendre s complete integral of the first kind 
leads to the formula 



Vlll, 174] VARIABLE TERMS 367 

The definition of uniform convergence may be extended to series 
whose terms are functions of several independent variables. For 
example, let 

u (x, y) + % (x, y) H h u n (x, y}-\ 

be a series whose terms are functions of two independent variables x 
and ij, and let us suppose that this series converges whenever the 
point (x, y} lies in a region R bounded by a closed contour C. 
The series is said to be uniformly convergent in the region R if, 
corresponding to every positive number e, an integer N can be found 
such that the absolute value of the remainder R n is less than e 
whenever n is equal to or greater than N, for every point (x, y) 
inside the contour C. It can be shown as above that the sum of 
such a series is a continuous function of the two variables x and 
y in this region, provided the terms of the series are all continu 
ous in R. 

The theorem on term-by-term integration also may be generalized. 
If each of the terms of the series is continuous in R and if f(x, y) 
denotes the sum of the series, we shall have 

I I f( x > y}dxdy =11 u 9 (x, y) dx dy + I I u l (x, y)dxdy -\ 

+ 11 u n (x,y}dxdy-\ , 

where each of the double integrals is extended over the whole inte 
rior of any contour inside of the region R. 

Again, let us consider a double series whose elements are functions 
of one or more variables and which converges absolutely for all sets 
of values of those variables inside of a certain domain D. Let the 
elements of the series be arranged in the ordinary rectangular array, 
and let R c denote the sum of the double series outside any closed 
curve C drawn in the plane of the array. Then the given double 
series is said to converge uniformly in the domain D if correspond 
ing to any preassigned number c, a closed curve K, not dependent 
on the values of the variables, can be drawn such that | R c < e for 
any curve C whatever lying outside of K and for any set of values 
of the variables inside the domain D. 

It is evident that the preceding definitions and theorems may be 
extended without difficulty to a multiple series of any order whose 
elements are functions of any number of variables. 



368 INFINITE SERIES [VIII, 175 

Note. If a series does not converge uniformly, it is not always allowable to 
integrate it term by term. For example, let us set 

S H (x) = nxe-*?, S (x) = 0, u a (x) = S n - S n ^ . n = 1, 2, . 

The series whose general term is u n (x) converges, and its sum is zero, since S n (x) 
approaches zero as n becomes infinite. Hence we may write 

f(x) = = 1*1 (z) + MS (x) + + u, (x) + , 

whence J f(x) dx = 0. On the other hand, if we integrate the series term by 
term between the limits zero and unity, we obtain a new series for which the 
sum of the first n terms is 



which approaches 1/2 as its limit as n becomes infinite. 

175. Application to differentiation under the integral sign. The proof 
of the formula for differentiation under the integral sign given in 
97 is based essentially upon the supposition that the limits x 
and X are finite. If X is infinite, the formula does not always hold. 
Let us consider, for example, the integral 

f + " s max 
F(a) = I dx , a > . 

i/O X 

This integral does not depend on a, for if we make the substitu 
tion y = ax it becomes 



y 

If we tried to apply the ordinary formula for differentiation to F(a~), 
we should find 

F (o-) = I cos ax dx 
Jo 

This is surely incorrect, for the left-hand side is zero, while the 
right-hand side has no definite value. 

Sufficient conditions may be found for the application of the 
ordinary formula for differentiation, even when one of the limits 
is infinite, by connecting the subject with the study of series. Let 

us first consider the integral 

p + 

f(x)dx, 

which we shall suppose to have a determinate value ( 90). Let 
a u a 2 , , a n , be an infinite increasing sequence of numbers, all 



Vni, 175] VARIABLE TERMS 369 

greater than a , where a n becomes infinite with n. If we set 

/" 2 r a +i 

Ja^ Ja n 

the series 

converges and its sum is j*" f(x) dx, for the sum S n of the first n terms 
is equal to f a nn f(x) dx. 

It should be noticed that the converse is not always true. 
If, for example, we set 

f(x) = cosx, a 0, ! = TT, -, a n = mr, -, 
we shall have 

U n = I cos x dx = . 

U nit 

Hence the series converges, whereas the integral f o cosxdx ap 
proaches no limit whatever as I becomes infinite. 

Now let f(x, a) be a function of the two variables x and a which 
is continuous whenever x is equal to or greater than a and a lies 
in an interval (a > i)- If the integral J a f(x, a) dx approaches a 
limit as I becomes infinite, for any value" of a, that limit is a 
function of a, 



r +x 

(a)= f(x,a)dx, 

i/a. 



which may be replaced, as we have just shown, by the sum of a 
convergent series whose terms are continuous functions of a : 



U (a) = I \f(x, a) dx , U, (a) = / /(ar, a) rfx , 

A, ^t 

This function F(a) is continuous whenever the series converges uni 
formly. By analogy we shall say that the integral f* m f(x t a) dx 

converges uniformly in the interval (a , aj if, corresponding to any 
preassigned positive quantity e, a number N independent of a can 
be found such that | f t + "f(x, a)dx < e whenever I > N, for any value 
of a which lies in the interval (a , a^* If the integral converges 

* See W. F. OSGOOD, Annals of Mathematics, 2d series, Vol. Ill (1902), p. 129. 
TRANS. 



370 INFINITE SERIES [vm,i75 

uniformly, the series will also. For if a n be taken greater than N, 
we shall have 



\R.\ = 



r + * 

I f(x, cr) dx 

Ja n 



hence the function F(a) is continuous in this case throughout the 
interval (a , a^. 

Let us now suppose that the derivative df/da is a continuous 
function of x and a when x ^ a and a < a < a^ , that the integral 



da 

has a finite value for every value of a in the interval (a , a^, and 
that the integral converges uniformly in that interval. The integral 
in question may be replaced by the sum of the series 



dx = F (a-) + F! (tr) + + F n (a) 
where 



The new series converges uniformly, and its terms are equal to the 
corresponding terms of the preceding series. Hence, by the theorem 
proved above for the differentiation of series, we may write 



In other words, the formula for differentiation under the integral sign 
still holds, provided that the integral on the right converges uniformly. 
The formula for integration under the integral sign ( 123) also 
may be extended to the case in which one of the limits becomes 
infinite. Let f(x, a) be a continuous function of the two variables 
x and a, for x > a , a < a < a t . If the integral // V(*, ) dx is uni 
formly convergent in the interval (a , a^, we shall have 

r + * r^ f, /> + 

(A) / dx I f(x,a}da=\ da f(x,a)dx. 

A| ^ J J 

To prove this, let us first select a number I > a ; then we shall 
have 

C l r a * r a i r l 

(B) / dx \ f(x, a)da= I da f(x, a)dx. 

^ u a J*f J Ja 



VIII, 176] 



VARIABLE TERMS 371 



As I increases indefinitely the right-hand side of this equation 
approaches the double integral 



. + ao 

f(x, a)dx, 



for the difference between these two double integrals is equal to 

a i r +x 

da I /(cc, <x)dx. 

Jl 



Suppose N chosen so large that the absolute value of the integral 
/ + :0 /(x, a)dx is less than c whenever / is greater than N, for any 
value of a in the interval (a , a^. Then the absolute value of the 
difference in question will be less than c a 1 a \ , and therefore it 
will approach zero as I increases indefinitely. Hence the left-hand 
side of the equation (B) also approaches a limit as I becomes infi 
nite, and this limit is represented by the symbol 

X+0 /! 

dx I f(x, a) da. 
J*t 

This gives the formula (A) which was to be proved.* 

176. Examples. 1) Let us return to the integral of 91 : 

/ + 

F(a}- I er"* - dx, 
Jo x 

where a is positive. The integral 

/ + - 
I er ax sin x dx , 

Jo 

* The formula for differentiation may be deduced easily from the formula (A). For, 
suppose that the two functions f(x, a) and f a (x, a) are continuous for a < a < en , 
x ;> a ; that the two integrals F(a) = f a + */(*, ) dx and *() = f a * *S ( x > ^ dx have 
finite values ; and that the latter converges uniformly in the interval (a , i). From 
the formula (A), if a lies in the interval (a > <*!) we have 

fduC "/<*t)d*= f dx C f u (x,u)du, 
J "o J "o Jm t Ja o 

where for distinctness a has heen replaced by u under the integral sign. But this 
formula may be written in the form 



f "*(!*) du = ( + */(*, a)dx- C + f(x, 

Ja o Ja o Ja o 

whence, taking the derivative of each side with respect to a, we find 



372 INFINITE SERIES [VIII, 176 

obtained by differentiating under the integral sign with respect to a, converges 
uniformly for all values of a greater than an arbitrary positive number k. For 
we have 

/ +<=0 ~ +00 1 

I e- ax smxdx<l e~ ax dx = er al . 
Ji Ji a 

and hence the absolute value of the integral on the left will be less than e for all 
values of a greater than k, if I > N, where N is chosen so large that ke ky > 1/e 
It follows that 

*- \ (X) I Q sin ctj& , 
Jo 

The indefinite integral was calculated in 119 and gives 

~e-<* x (cosx + a sinx)~| + 1 



F (a) = I" 



1 + a 2 Jo 1 + a 2 

whence we find 

F(a) = C - arc tan a , 

and the constant C may be determined by noting that the definite integral F(a) 
approaches zero as a becomes infinite. Hence C = x/2, and we finally find the 
formula 

r + 
,. sin x , 1 

e~ ax dx = arc tan . 



J + 
Sll 
e~ ax 
, 



This formula is established only for positive values of n-, but we saw in 91 that 
the left-hand side is the sum of an alternating series whose remainder R n is always 
less than 1/n. Hence the series converges uniformly, and the integral is a con 
tinuous function of a, even for a = 0. As a approaches zero we shall have in 
the limit 



f +ao 

Jo 
2) If in the formula 



(39) 

2 



rv-<fc=^ 

Jo 2 

of 134 we set x = yVa, where a is positive, we find 

(40) r + V 

t/o 



and it is easy to show that all the integrals derived from this one by successive 
differentiations with respect to the parameter a converge uniformly, provided 
that a is always greater than a certain positive constant A;. From the preceding 
formula we may deduce the values of a whole series of integrals : 



2 2 
(41) 



VIII, Exs.] EXERCISES 373 

By combining these an infinite number of other integrals may be evaluated. 
We have, for example, 



/^ -4- oo 

- I t-**W** 

Jo ! 2 



All the integrals on the right have been evaluated above, and we find 
/- + 1 \n (2/3) 2 Vn a-i 

1 ^0032^ = -^--^ + ... 

+ /_iy, ( 2 fl 2 " ^* 1.8.6.. .(8n-l)8 
7 1.2.3.--2n 2 2" 

or, simplifying, 

/ + *> 1 Iff _ 5! 

(42) I e~ a v*cos2pydy = ~ %/- e . 

^o 2 \a 



EXERCISES 
1. Derive the formula 

; [Z" (logZ)><] = 1 + Si lOgZ + ^- (logz) 2 + h " (logz)" , 

1 . 2 n uz" 1.2 1 . 2 n 

where S p denotes the sum of the products of the first n natural numbers taken p 

at a time. r ,, n 

[MURPHY.] 

[Start with the formula 



1.2 
and differentiate n times with respect to z.] 

2. Calculate the value of the definite integral 



(log) + ..1 

1 . 2 n 



j 



by means of the formula for differentiation under the integral sign. 
3. Derive the formula 
/ 



r + * -+ ^ 

=1 e &dx = 
Jo 



[First show that dl/da = - 21.] 



374 INFINITE SERIES rviil, 

4. Derive the formula 

J_ a _ *? da i ,. 
e = = VTre- 2 * 
Va 

by making use of the preceding exercise. 

5. From the relation 



derive the formula 



CHAPTER IX 

POWER SERIES TRIGONOMETRIC SERIES 

In this chapter we shall study two particularly important classes 
of series power series and trigonometric series. Although we shall 
speak of real variables only, the arguments used in the study of 
power series are applicable without change to the case where the 
variables are complex quantities, by simply substituting the expres 
sion modulus or absolute value (of a complex variable) for the expres 
sion absolute value (of a real variable).* 

I. POWER SERIES OF A SINGLE VARIABLE 

177. Interval of convergence. Let us first consider a series of the form 
(1) A + AiX + A 2 X* + -f A H X + -, 

where the coefficients A , A 1} A 2 , are all positive, and where 
the independent variable A is assigned only positive values. It is 
evident that each of the terms increases with A . Hence, if the 
series converges for any particular value of A, say X l} it converges 
a fortiori for any value of A less than X l . Conversely, if the series 
diverges for the value A 2 , it surely diverges for any value of A 
greater than A 2 . We shall distinguish the following cases. 

1) The series (1) may converge for any value of A whatever. 
Such is the case, for example, for the series 

Y A 2 A" 

1 " f I + lT2 + > " + l72T^ + "" 

2) The series (1) may diverge for any value of A except A = 
The following series, for example, has this property : 

1 + A + 1 . 2A 2 H +1.2.3-.- nX n H . 

3) Finally, let us suppose that the series converges for certain 
values of A and diverges for other values. Let A x be a value of A 
for which it converges, and let A 2 be a value for which it diverges. 

* See Vol. II, 2GG-275. TKAKS. 
375 



376 SPECIAL SERIES [IX, 177 

From the remark made above, it follows that X l is less than X 2 . The 
series converges if A <A j, and it diverges if AT>A 2 . The only 
uncertainty is about the values of A between X l and X 2 . But all 
the values of A for which the series converges are less than X 2 , and 
hence they have an upper limit, which we shall call R. Since all the 
values of X for which the series diverges are greater than any value 
of A for which it converges, the number R is also the lower limit of 
the values of X for which the series diverges. Hence the series (1) 
diverges for all values of X greater than R, and converges for all values 
of X less than R. It may either converge or diverge when X = R. 
For example, the series 

converges if A < 1, and diverges if A ^ 1. In this case R =1. 

This third case may be said to include the other two by suppos 
ing that R may be zero or may become infinite. 

Let us now consider a power series, i.e. a series of the form 

(2) a -f a 1 x + a z x 2 + + a n x a + , 

where the coefficients a, and the variable x may have any real values 
whatever. From now on we shall set A { = |a,-|, X = \x\. Then the 
series (1) is the series of absolute values of the terms of the series (2). 
Let R be the number defined above for the series (1). Then the 
series (2) evidently converges absolutely for any value of x between 
- R and + R, by the very definition of the number R. It remains 
to be shown that the series (2) diverges for any value of x whose 
absolute value exceeds R. This follows immediately from a funda 
mental theorem due to Abel : * 

If the series (2) converges for any particular value x , it converges 
absolutely for any values of x whose absolute value is less than \x \. 

In order to prove this theorem, let us suppose that the series (2) 
converges for x = x , and let M be a positive number greater than 
the absolute value of any term of the series for that value of x. 
Then we shall have, for any value of n, 

A lt x n <M, 
and we may write 

/ \ _ / 

<M /X 



, m in (m 1) 

* Recherche sur la sene 1 H x -\ 

1 . 2t 



IX, 177] POWER SERIES 377 

It follows that the series (1) converges whenever X<|cc |, which 
proves the theorem. 

In other words, if the series (2) converges for x x , the series (1) 
of absolute values converges whenever X is less than | x 1 . Hence 
|cc cannot exceed R, for R was supposed to be the upper limit of 
the values of X for which the series (1) converges. 

To sum up, given a power series (2) whose coefficients may have 
either sign, there exists a positive number R which has the follow 
ing properties : The series (2) converges absolutely for any value of x 
between R and -f R, and diverges for any value of x whose absolute 
value exceeds R. The interval ( R, + R) is called the interval of 
convergence. This interval extends from oo to + cc in the case in 
which R is conceived to have become infinite, and reduces to the 
origin if R = 0. The latter case will be neglected in what follows. 

The preceding demonstration gives us no information about what 
happens when x = R or x = R. The series (2) may be absolutely 
convergent, simply convergent, or divergent. For example, R = 1 
for each of the three series 



for the ratio of any term to the preceding approaches x as its limit 
in each case. The first series diverges for aj = 1. The second 
series diverges for x = 1, and converges for x = 1. The third con 
verges absolutely for x = 1. 

Note. The statement of Abel s theorem may be made more general, 
for it is sufficient for the argument that the absolute value of any 
term of the series 

+ a l X + ---- 1- a n X H ---- 

be less than a fixed number. Whenever this condition is satisfied, 
the series (2) converges absolutely for any value of x whose absolute 
value is less than \x . 

The number R is connected in a very simple way with the number to defined 
in 160, which is the greatest limit of the Sequence 



For if we consider the analogous sequence 



378 SPECIAL SERIES [IX, 178 

it is evident that the greatest limit of the terms of the new sequence is <aX. The 
sequence (1) therefore converges if X < 1/w, and diverges if X > 1/w ; hence 



178. Continuity of a power series. Let f(x) be the sum of a power 
series which converges in the interval from R to + R, 



(3) f(x) = a 

and let R be a positive number less than R. We shall first show 
that the series (3) converges uniformly in the interval from R 
to + R . For, if the absolute value of x is less than R , the 

remainder R n 

R n = a n+l x n + l + + a n+p x n + P + 

of the series (3) is less in absolute value than the remainder 

4.+I*** 1 +-*.+;**** 

of the corresponding series (1). But the series (1) converges for 
X = R , since R < R. Consequently a number N may be found 
such that the latter remainder will be less than any preassigned 
positive number c whenever n ^ N. Hence R n \ < c whenever n > N 
provided that \x\ < R . 

It follows that the sum f(x} of the given series is a continuous 
function of x for all values of x between R and + R. For, let x 
be any number whose absolute value is less than R. It is evident 
that a number R 1 may be found which is less than R and greater 
than \x \. Then the series converges uniformly in the interval 
(_ R , + R ~), as we have just seen, and hence the sum f(x) of the 
series is continuous for the value x , since x belongs to the interval 
in question. 

This proof does not apply to the end points + R and R of the 
interval of convergence. The function f(x} remains continuous, 
however, provided that the series converges for those values. 
Indeed, Abel showed that if the series (3) converges for x = R, its 
sum for x = R is the limit which the sum /(#) of the series approaches 
as x approaches R through values less than J?.f 

Let S be the sum of the convergent series 

S = + a l R + a 2 R* -\ ---- + a n R" H ---- , 



* This theorem was proved by Cauchy in his Cours d Analyse. It was rediscovered 
by Hadamard in his thesis. 

t As stated above, these theorems can be immediately generalized to the case of 
series of imaginary terms. In this case, however, care is necessary in formulating 
the generalization. See Vol. II, 266. TRANS. 



IX, m] POWER SERIES 379 

and let n be a positive integer such that any one of the sums 



is less than a preassigned positive number e. If we set x = R0, and 
then let increase from to 1, a; will increase from to R, and we 
shall have 



/(z) =/(0J?) = a + a l 0R + a 2 2 2 + + a n $R" 4. .... 
If n be chosen as above, we may write 

S -f(x) = ai R(l - 0) + 2 # 2 (1 - 2 ) + + a n R n (l - 0") 
(4)- +a n+l R n + l + --- + a n+p R n+ ? + -. 



and the absolute value of the sum of the series in the second line can 
not exceed e. On the other hand, the numbers O n + l , B + 2 , -, H+P 
form a decreasing sequence. Hence, by Abel s lemma proved in 75, 
we shall have 

\a n + l 6 n + l R n + l -i ---- + a n+p O n+p R n+p < 6 n + l t < e. 

It follows that the absolute value of the sum of the series in the 
third line cannot exceed e. Finally, the first line of the right-hand 
side of the equation (4) is a polynomial of degree n in which 
vanishes when 0=1. Therefore another positive number rj may be 
found such that the absolute value of this polynomial is less than c 
whenever lies between 1 r\ and unity. Hence for all such values 
of we shall have 

\S~f(x)\<3e. 

But e is an arbitrarily preassigned positive number. Hence f(x) 
approaches S as its limit as x approaches R. 

In a similar manner it may be shown that if the series (3) con 
verges for x = R, the sum of the series for x = R is equal to 
the limit which /(x) approaches as x approaches R through values 
greater than R. Indeed, if we replace x by x, this case reduces 
to the preceding. 

An application. This theorem enables us to complete the results of 168 
regarding the multiplication of series. Let 

(5) S =UQ + UI + U Z -\ ---- + u n H ---- , 

(6) S = v + v l + t> 2 H ----- h v + 

be two convergent series, neither of which converges absolutely. The series 

(7) u o + (ot>i 4- Uiu ) H ---- + ( u o H ----- 1- no) H ---- 



380 SPECIAL SERIES [IX, 179 

may converge or diverge. If it converges, its sum S is equal to the product of 
the sums of the two given series, i.e. S = <S<S . For, let us consider the three 
power series 



f(x) = 

tf>(x) = v + vix + + v n x n H ---- , 

-\ ----- h 



Each of these series converges, by hypothesis, when z = 1. Hence each of them 
converges absolutely for any value of x between 1 and + 1. For any such 
value of x Cauchy s theorem regarding the multiplication of series applies and 
gives us the equation 

(8) /(*)0(z) = *(z). 

By Abel s theorem, as x approaches unity the three functions /(x), #(x), \ft(x) 
approach S, S , and 2, respectively. Since the two sides of the equation (8) 
meanwhile remain equal, we shall have, in the limit, S = SS . 

The theorem remains true for series whose terms are imaginary, and the proof 
follows precisely the same lines. 

179. Successive derivatives of a power series. If a power series 
f(x) = a + a v x + a 2 x 2 -\ ----- h *" H ---- 



which converges in the interval ( R, -f R) be differentiated term 
by term, the resulting power series 

converges in the same interval. In order to prove this, it will be 
sufficient to show that the series of absolute values of the terms of 
the new series, 

A v + 2A 9 X-\ h nA n X n ~ l -\ , 

where A t = |a,.| and X = \x\, converges for X<R and diverges for 
X>R. 

For the first part let us suppose that X < R, and let R be a num 
ber between X and R, X < R < R. Then the auxiliary series 

h 4- ( Y 4- 4- I}* 1 4 

converges, for the ratio of any term to the preceding approaches 
X/R , which is less than unity. Multiplying the successive terms 
of this series, respectively, by the factors 



IX, 179] POWER SERIES 381 

each of which is less than a certain fixed number, since R <R, we 
obtain a new series 



which also evidently converges. 

The proof of the second part is similar to the above. If the series 

A 1 + 2A 2 X l + + nA n Xr + > 
where X t is greater than R, were convergent, the series 
A l X l + 2A t X* + .-- + nA n X n l + 

+ 

would converge also, and consequently the series 2^ n X" would con 
verge, since each of its terms is less than the corresponding term of 
the preceding series. Then R would not be the upper limit of the 
values of X for which the series (1) converges. 

The sum /j (a:) of the series (9) is therefore a continuous function 
of the variable x inside the same interval. Since this series con 
verges uniformly in any interval ( R , -f- R ), where R < R ) f 1 (x) 
is the derivative of f(x) throughout such an interval, by 174. 
Since R may be chosen as near R as we please, we may assert that 
the function f(x) possesses a derivative for any value of x between 
R and -f R, and that that derivative is represented by the series 
obtained by differentiating the given series term by term : * 

(10) /(*) = ! + 2a 2 x + + na n x*- 1 + .... 

Repeating the above reasoning for the series (10), we see that f(x) 
has a second derivative, 

/"(a) = 2a 2 + 6a 3 x + . . . + w (n - 1) a n x n ~ 2 + -, 

and so forth. The function f(x) possesses an unlimited sequence of 
derivatives for any value of x inside the interval (/, + R), and 
these derivatives are represented by the series obtained by differen 
tiating the given series successively term by term : 

(11) f<*\x) = 1.2---na n + 2.3---n(n + l~)a n + l x-i ---- . 
If we set x = in these formulae, we find 



or, in general, 



* Although the corresponding theorem is true for series of imaginary terms, the 
proof follows somewhat different lines. See Vol. II, 266. TRANS. 



382 SPECIAL SERIES [IX, 179 



The development of /(#) thus obtained is identical with the develop 
ment given by Maclaurin s formula : 

/(*) =/(0) + f / (<>) + f^/"(<>) + + iTf^/^O) + -. 

The coefficients a 0) a l9 , a n , are equal, except for certain 
numerical factors, to the values of the function f(x) and its succes 
sive derivatives for x = 0. It follows that no function can have two 
distinct developments in power series. 

Similarly, if a power series be integrated term by term, a new 
power series is obtained which has an arbitrary constant term and 
which converges in the same interval as the given series, the given 
series being the derivative of the new series. If we integrate again, 
we obtain a third series whose first two terms are arbitrary ; and so 
forth. 

Examples. 1) The geometrical progression 

1 X + X* X s -\ ----- \- (l) n X n -\ ---- , 

whose ratio is x, converges for every value of x between 1 and 
+ 1, and its sum is 1/(1 + x). Integrating it term by term between 
the limits and x, where x < 1, we obtain again the development 
of log (1 + x*) found in 49 : 



This formula holds also for x = 1, for the series on the right con 
verges when x = 1. 

2) For any value of x between 1 and + 1 we may write 
^-^ = 1 - x 2 + * 4 - x 6 + ... + (- 1 )" x 2 " + 

Integrating this series term by term between the limits and x, 
where Ice I < 1, we find 



Since the new series converges for x = 1, it follows that 
TT 111 1 



IX, 17 . ] 



row EH SERIES 



383 



3) Let F(x) be the sum of the convergent series 



m m(m 1) 



+ 



i(m !) (in, 



where m is any number whatever and | a: < 1. Then we shall have 

[m 1 
! + * + 



(m -1). (m p +1) 
< 1.2.. -1 J 



Let us multiply each side by (1+ a;) and then collect the terms in 
like powers of x. Using the identity 

(m 1) (m p +1) (m !) -(m />) _ m(?rc 1) (m p -f-1) 
1.2.--OJ-1) 1.2-.-.P 1.2--.J9 

which is easily verified, we find the formula 



, 



P , 



or 



From this result we find, successively, 

F (x) _ m 
~F(x)~l+x 

log [/ ()] = m log (1 + a-) + log 



or 



To determine the constant C we need merely notice that F(0) = l. 
Hence C = 1. This gives the development of (1 + a*) " found in 50 : 

Y_1. . m(m-l)...(m-.p+l) , 



4) Replacing a; by x 2 and /, by 1/2 in the last formula above, 
we find 



,1 

=1+ - 



1.3 

- - 
2.4 



, 

H ----- 1 



1.3. 5- ..(27t- 

v ^ 
2 . 4 . 6 2n 



This formula holds for any value of x between 1 and +1. Inte 
grating both sides between the limits and x, where | x < 1, we 
obtain the following development for the arcsine : 

x , 1 a- 3 1.3a; 5 , , 1 . 3 . 5 - - (2n -1) x z " + l 
arc sm x = - -f 7: -^ + ^ T -=- H ----- 



. g 
2.4.6---2n 



384 



SPECIAL SERIES 



[IX, 180 



180. Extension of Taylor s series. Let/(x) be the sum of a power 
series which converges in the interval ( R, + R~), a- a point inside 
that interval, and x + h another point of the same interval such 
that |a; | + | h\ < R. The series whose sum is f(x + A), 

o + ai( + A) + a 2 (x + A) 2 H \- a n (x + A)" -] , 

may be replaced by the double series obtained by developing each 
of the powers of (x + A) and writing the terms in the same power 
of h upon the same line : 



(12) 



-f- 2a 2 x h 



n 

n(n 1) 
1.2 



This double series converges absolutely. For if each of its terms 
be replaced by its absolute value, a new double series of positive 
terms is obtained : 



(13) 



If we add the elements in any one column, we obtain a series 



(- 


^n\ X 


n i _ 


+ 


n A n x 


n ~ l \h\ + 


1 


i(n 1) 


n-2| ^|2_1_. . 


T 


1.2 ff 





which converges, since we have supposed that | x \ + 1 h \ < R. Hence 
the array (12) may be summed by rows or by columns. Taking 
the sums of the columns, we obtain f(x + A). Taking the sums 
of the rows, the resulting series is arranged according to powers of 
A, and the coefficients of A, h 2 , are f (x ),f"(x )/2l, , respec 
tively. Hence we may write 



(14) 



A) =/(.r ) + 



.L . 



71 



if we assume that | A [ < R | x \ . 

This formula surely holds inside the interval from x R +\x \ 
to x + R \x , but it may happen that the series on the right 
converges in a larger interval. As an example consider the function 



IX, 180] POWER SERIES 386 

(1 + x~) m , where m is not a positive integer. The development 
according to powers of x holds for all values of x between 1 and 
+ 1. Let x be a value of x which lies in that interval. Then we 
may write 

(i + x} m = (i + x + x - XY = (i + <r (i + *) m , 

where 



z = 



X X 



l+X 



We may now develop (1 + z) m according to powers of 2, and this 
new development will hold whenever \z\ < 1, i.e. for all values of x 
between 1 and 1 + 2x . If x is positive, the new interval will be 
larger than the former interval (1, +!) Hence the new formula 
enables us to calculate the values of the function for values of the 
variable which lie outside the original interval. Further investiga 
tion of this remark leads to an extremely important notion, that 
of analytic extension. We shall consider this subject in the second 
volume. 

Note. It is evident that the theorems proved for series arranged 
according to positive powers of a variable x may be extended immedi 
ately to series arranged according to positive powers of x a, or, 
more generally still, to series arranged according to positive powers 
of any continuous function <(x) whatever. We need only consider 
them as composite functions, <f>(x) being the auxiliary function. 
Thus a series arranged according to positive powers of I/a; con 
verges for all values of x which exceed a certain positive constant in 
absolute value, and it represents a continuous function of x for all 
such values of the variable. The function Va/ 2 a, for example, may 
be written in the form x(\ a/a: 2 )*. The expression (1 a/ar 2 )* 
may be developed according to powers of I/a; 2 for all values of x 
which exceed V a in absolute value. This gives the formula 

1 . 2 . 3 - (2 P - 3) a" 



2.4.6---2p a- 2 - 1 

which constitutes a valid development of Va; 2 a whenever x > Va. 
When x < Va, the same series converges and represents the func 
tion Va: 2 a. This formula may be used advantageously to obtain 
a development for the square root of an integer whenever the first 
perfect square which exceeds that integer is known. 



386 SPECIAL SERIES [IX, 181 

181. Dominant functions. The theorems proved above establish a 
close analogy between polynomials and power series. Let ( r, + r) 
be the least of the intervals of convergence of several given power 
series /i (x), / 2 (x), -,/ (#) When |cc|<r, each of these series 
converges absolutely, and they may be added or multiplied together 
by the ordinary rules for polynomials. In general, any integral poly 
nomial in / 1 (x),/ 2 (a;), ,/() may be developed in a convergent 
power series in the same interval. 

For purposes of generalization we shall now define certain expres 
sions which will be useful in what follows. Let f(x~) be a power 

series 

f(x) = a + aj_x + a 2 x 2 -) h a n x" H , 

and let <f>(x) be another power series with positive coefficients 
<f,(x) = a + a^x + a 2 x 2 H f- a n x n -\ 

which converges in a suitable interval. Then the function <(a;) is 
said to dominate * the function f(x) if each of the coefficients a n is 
greater than the absolute value of the corresponding coefficient of 

/(*): 

\< 1 0\<C*0, 

Poincare has proposed the notation 

f(x) < <(*) 

to express the relation which exists between the two functions f(x) 
and <f>(x~). 

The utility of these dominant functions is based upon the fol 
lowing fact, which is an immediate consequence of the definition. 
Let P(a , a\, , ) be a polynomial in the first n -f- 1 coefficients 
of f(x) whose coefficients are all real and positive. If the quanti 
ties a Q , !, -, a n be replaced by the corresponding coefficients of 
<(#), it is clear that we shall have 

|P(Oo, a 1} -, a n ~)\<P(a , a 1} , a n ). 

For instance, if the function <f>(x) dominates the function /(#), 
the series which represents [<(:r)] 2 will dominate [/(ic)] 2 , and so 
on. In general, [<(#)]" w iU dominate [/()]". Similarly, if <f> and 
^>! are dominant functions for / and / t , respectively, the product <<j 
will dominate the product jff\ ; and so forth. 

*This expression will be used as a translation of the phrase " <p(x) est majorante 
pour la fonction /(a;)." Likewise, "dominant functions " will be used for " fonctions 
majorantes." TRANS. 



IX, 181] 



POWER SERIES 387 



Given a power series/(x) which converges in an interval ( R, + R), 
the problem of determining a dominant function is of course indeter 
minate. But it is convenient in what follows to make the domi 
nant function as simple as possible. Let r be any number less than 
R and arbitrarily near R. Since the given series converges for x = r, 
the absolute value of its terms will have an upper limit, which we 
shall call M. Then we may write, for any value of n, 



or 
Hence the series 



x Mx n M 

M + M- 



r r" x 

r 

whose general term is M(x n /r n ), dominates the given function /(x). 
This is the dominant function most frequently used. If the series 
/(x) contains no constant term, the function 

M 



may be taken as a dominant function. 

It is evident that r may be assigned any value less than R, and 
that M decreases, in general, with r. But M can never be less than 
A . If A is not zero, a number p less than R can always be found 
such that the function -4 /(1 x/p) dominates the function /(x). 
For, let the series 

M + M - + M ^ + + M ^ + 
r r 2 r" 

where M > A , be a first dominant function. If p be a number less 
than rA /M and n > 1, we shall have 



whence |a n p n | < A . On the other hand, |a c | = A . Hence the series 






dominates the function f(x). We shall make use of this fact pres 
ently. More generally still, any number whatever which is greater 
than or equal to A may be used in place of M. 






388 SPECIAL SERIES [IX, 182 

It may be shown in a similar manner that if a = 0, the function 



is a dominant function, where /t is any positive number whatever. 

Note. The knowledge of a geometrical progression which dominates the func 
tion f(x) also enables us to estimate the error made in replacing the function 
f(x) by the sum of the first n + 1 terms of the series. If the series M/(l x/r) 
dominates /(z), it is evident that the remainder 



of the given series is less in absolute value than the corresponding remainder 



of the dominant series. It follows that the error in question will be less than 

(T 1 
jfW 



182. Substitution of one series in another. Let 

(15) * =/(>/) = o + iy 4- + a n y + 

be a series arranged according to powers of a variable y which con 
verges whenever \y\<. R. Again let 

/ 1 A \ -JL ^ \ TitA I t A w. i 

be another series, which converges in the interval ( r, + r). If 
y> y 2 ) y 8 ) in the series (15) be replaced by their developments in 
series arranged according to powers of x from (16), a double series 

+ 

I ,. 7, ., l O 7, 7, ~. 

(17) 



a 2 (b\ 



is obtained. We shall now investigate the conditions under which 
this double series converges absolutely. In the first place, it is 
necessary that the series written in the first row, 



IX, 182] POWER SERIES 389 

should converge absolutely, i.e. that \b \ should be less than R.* This 
condition is also sufficient. For if it is satisfied, the function <(>(x) 
will be dominated by an expression of the form m/(l x/p), where 
m is any positive number greater than \b \ and where p < r. We 
may therefore suppose that m is less than R. Let R be another 
positive number which lies between m and R. Then the function 
f(y) is dominated by an expression of the form 



y R R 2 

R 

If y be replaced by wi/(l x/p) in this last series, and the powers 
of y be developed according to increasing powers of x by the binomial 
theorem, a new double series 



(18) 



M ^ 



m x 



M j^ + - + nMt- 



is obtained, each of whose coefficients is positive and greater than 
the absolute value of the corresponding coefficients in the array (17), 
since each of the coefficients in (17) is formed from the coefficients 
a , a*!, a 2 , , b 0) bi, b 2 , -by means of additions and multiplications 
only. The double series (17) therefore converges absolutely pro 
vided the double series (18) converges absolutely. If x be replaced 
by its absolute value in the series (18), a necessary condition for abso 
lute convergence is that each of the series formed of the terms in any 
one column should converge, i.e. that \x\ < p. If this condition be 
satisfied, the sum of the terms in the (n + l)th column is equal to 




Then a further necessary condition is that we should have 



or 

(19) 



* The case in which the series (15) converges for y = R (see 177) will be neglected 
in what follows. TRANS. 



390 SPECIAL SERIES [IX, 182 

Since this latter condition includes the former, \x\ < p, it follows 
that it is a necessary and sufficient condition for the absolute con 
vergence of the double series (18). The double series (17) will 
therefore converge absolutely for values of x which satisfy the 
inequality (19). It is to be noticed that the series <j>(x) converges 
for all these values of x, and that the corresponding value of y is 
less than R in absolute value. For the inequalities 

i. t x, . m. \x\ m 

1 ( ! 



necessitate the inequality $(x)\<.R . Taking the sum of the series 
(17) by columns, we find 



that is, f[$(x)~\- On the other hand, adding by rows, we obtain a 
series arranged according to powers of x. Hence we may write 



where the coefficients c , c i} c 2 , are given by the formulae 

CQ = <ZQ -f- fti w + ft 2 t>o ~r ~r <^ n "o i > 
(21) 



which are easily verified. 

The formula (20) has been established only for values of x which 
satisfy the inequality (19), but the latter merely gives an under 
limit of the size of the interval in which the formula holds. It may 
be valid in a much larger interval. This raises a question whose 
solution requires a knowledge of functions of a complex variable. 
We shall return to it later. 

Special cases. 1) Since the number R which occurs in (19) may 
be taken as near R as we please, the formula (20) holds whenever x 
satisfies the inequality \x < p(l m/R). Hence, if the series (15) 
converges for any value of y whatever, R may be thought of as infinite, 
p may be taken as near r as we please, and the formula (20) applies 
whenever |a;| < r, that is, in the same interval in which the series 
(16) converges. In particular, if the series (16) converges for all 
values of x, and (15) converges for all values of y, the formula (20) 
is valid for all values of x. 



IX, 182] 



POWER SERIES 391 



2) When the constant term b of the series (16) is zero, the func 
tion <(z) is dominated by an expression of the form 



l- x - 

p 

where p < r and where m is any positive number whatever. An 
argument similar to that used in the general case shows that the 
formula (20) holds in this case whenever x satisfies the inequality 

(22) z|< P , 

v r R + m 

where R is as near to R as we please. The corresponding interval 
of validity is larger than that given by the inequality (19). 

This special case often arises in practice. The inequality 
\b \ < R is evidently satisfied, and the coefficients c n depend upon 



Examples. 1) Cauchy gave a method for obtaining the binomial theorem from 
the development of log(l + x). Setting 

y = 
we may write 



whence, substituting the first expansion in the second, 



If the right-hand side be arranged according to powers of x, it is evident that 
the coefficient of x" will be a polynomial of degree n in ju, which we shall call 
P n (n)- This polynomial must vanish when ^ = 0, 1, 2, , n 1, and must 
reduce to unity when ^ = n. These facts completely determine P n in the form 

.M p _M(M-l)---(M-n+l) 

P " - ~""" 



2) Setting z = (1 + z) 1 /*, where x lies between - 1 and + 1, we may write 



where 



392 SPECIAL SERIES [IX, 183 

The first expansion is valid for all values of y, and the second is valid whenever 
\x\< 1. Hence the formula obtained by substituting the second expansion in 
the first holds for any value of x between 1 and + 1. The first two terms of 
this formula are 



T __ . 

It follows that (1 + x) 1 /* approaches e through values less than e as z approaches 
zero through positive values. 

183. Division of power series. Let us first consider the reciprocal 

f( x \ _ _ _ 
A } l+b lX + b 2 x* + ... 

of a power series which begins with unity and which converges in 
the interval ( r, -f- ?). Setting 

y = b v x + b 2 x 2 H ---- , 
we may write 

/(*) = r+~y =1 ~ y + y2 - y * + 

whence, substituting the first development in the second, we obtain 
an expansion for f(x) in power series, 

(25) f(x) = 1 - b lX + (b\ - 6.) * 2 + -, 

which holds inside a certain interval. In a similar manner a devel 
opment may be obtained for the reciprocal of any power series 
whose constant term is different from zero. 

Let us now try to develop the quotient of two convergent power 
series 



$(x) b ti + biX + b 3 x* -\ ---- 
If b is not zero, this quotient may be written in the form 



= ( a +a x + a 2 x* + -- ) X 



Then by the case just treated the left-hand side of this equation is the 
product of two convergent power series. Hence it may be written 
in the form of a power series which converges near the origin : 



-f 



Clearing of fractions and equating the coefficients of like powers 
of x, we find the formulae 



IX, 184] POWER SERIES 393 

(27) a n = l c H + biC H _ l +--- + b H c 9 , n = 0, 1, 2, -, 

from which the coefficients c , c lt , c n may be calculated succes 
sively. It will be noticed that these coefficients are the same as 
those we should obtain by performing the division indicated by the 
ordinary rule for the division of polynomials arranged according to 
increasing powers of x. . 

If b = 0, the result is different. Let us suppose for generality 
that ij/(x~) = x k fa (x), where A; is a positive integer and \l/\(x) is a 
power series whose constant term is not zero. Then we may write 



and by the above we shall have also 

It follows that the given quotient is expressible in the form 
\ / "77 \ = * *tI < * ,~ c k -j- Cj. + j a; -p , 

where the right-hand side is the sum of a rational fraction which 
becomes infinite for x = and a power series which converges near 
the origin. 

Note. In order to calculate the successive powers of a power series, it is con 
venient to proceed as follows. Assuming the identity 

(a + a\x + + a n x n + ) " = c + c t x + + c n x n + , 

let us take the logarithmic derivative of each side and then clear of fractions. 
This leads to the new identity 

+ na n x n - 1 + -)(c + Cix + 4- c n x n + ) 



(29) 

a, ( x" 

The coefficients of the various powers of x are easily calculated. Equat 
ing coefficients of like powers, we find a sequence of formulae from which 
CQ, Ci, , c n , may be found successively if c be known. It is evident that 



184. Development of 1/Vl 2xz -|- z 2 . Let us develop 1/Vl - 2xz + z 2 
according to powers of z. Setting y = 2xz z 2 , we shall have, when \y\< 1, 



or 

(30) 1 = 1 + 2xz ~ 28 + - (2xz - z 2 ) 2 + 

Vl^2xzTP 2 8 



394 SPECIAL SERIES [ix,i86 

Collecting the terms which are divisible by the same power of z, we obtain an 
expansion of the form 

(31) l -- = p + p lZ + p zZ 2 + ... + p nZ n + ..., 

VI - ?xz + z 2 
where 



2 * 

and where, in general, P n is a polynomial of the nth degree in x. These poly 
nomials may be determined successively by means of a recurrent formula. Dif 
ferentiating the equation (31) with respect to z, we find 



(1 - 2xz + z 2 )* 
or, by the equation (31), 

(x - z)(P + PIZ + h P n * n + ) = (1 - 2xz + z 2 )(Pi + 2P 2 z H ) . 

Equating the coefficients of z", we obtain the desired recurrent formula 

This equation is identical with the relation between three consecutive Legendre 
polynomials ( 88), and moreover P = X , P! = X\ , P 2 = X 2 . Hence P n = X n 
for all values of n, and the formula (31) may be written 

Vl - 2xz + z 2 ~ 
where X n is the Legendre polynomial of the nth order 

-V id i\_n 

-&-n = \\&* 1)1 

2 . 4 . 6 2n dx n 
We shall find later the interval in which this formula holds. 

II. POWER SERIES IN SEVERAL VARIABLES 

185. General principles. The properties of power series of a single 
variable may be extended easily to power series in several independ 
ent variables. Let us first consider a double series 2 t a mn x m y n , where 
the integers m and n vary from zero to + oo and where the coeffi 
cients a mn may have either sign. If no element of this series exceeds 
a certain positive constant in absolute value for a set of values 
x x o> y = 2/0) the series converges absolutely for all values of x and 
y which satisfy the inequalities \x\ < \x \, \y\ < \y \. 

For, suppose that the inequality 

M 

I a mn x Vo \<M or | a mu I < , , m . , 

I x o \ \yo\ 



IX, 185] DOUBLE POWER SERIES 395 

is satisfied for all sets of values of m and n. Then the absolute value 
of the general element of the double series 2a mn .r m y is less than the 
corresponding element of the double series 2M\x/x \ m \y/y \ n . But 
the latter series converges whenever |#|<|a: |, |y|<|y |> an d its 
sum is 

M 



2/o 

as we see by taking the sums of the elements by columns and then 
adding these sums. 

Let r and p be two positive numbers for which the double series 
2 a mn \r m p n converges, and let R denote the rectangle formed by the 
four straight lines x = r, x = r, y = p, y p. For every point 
inside this rectangle or upon one of its sides no element of the 
double series 

exceeds the corresponding element of the series ^\a mn \r m p n in abso 
lute value. Hence the series (33) converges absolutely and uni 
formly inside of R, and it therefore defines a continuous function 
of the two variables x and y inside that region. 

It may be shown, as for series in a single variable, that the 
double series obtained by any number of term-by-term differen 
tiations converges absolutely and uniformly inside the rectangle 
bounded by the lines x = r c, x = r + c, y = p e , y = p + e , 
where c and e are any positive numbers less than r and p, respec 
tively. These series represent the various partial derivatives of 
F(x,y). For example, the sum of the series 2 t ma mn x m - l y n is equal 
to cF/dx. For if the elements of the two series be arranged accord 
ing to increasing powers of x, each element of the second series is 
equal to the derivative of the corresponding element of the first. 
Likewise, the partial derivative d m+n F/dx m dy n is equal to the sum 
of a double series whose constant factor is a ran l . 2 m . 1 . 2 n. 
Hence the coefficients a mn are equal to the values of the correspond 
ing derivatives of the function F(x, y) at the point x = y 0, except 
for certain numerical factors, and the formula (33) may be written 
in the form 



F(x, y)= 
\ >yj 



/8 m + "F\ 



396 SPECIAL SERIES [IX, 186 

It follows, incidentally, that no function of two variables can have 
two distinct developments in power series. 

If the elements of the double series be collected according to 
their degrees in x and y, a simple series is obtained : 

(35) F(x, y) = < + fa + <f> 2 + + < + , 

where </> is a homogeneous polynomial of the nth degree in x and 
y which may be written, symbolically, 



The preceding development therefore coincides with that given by 
Taylor s series ( 51). 

Let (a- , y ~) be a point inside the rectangle R, and (x + h, y + k) 
be a neighboring point such that | x \ + 1 h \ < r, \y \ + \k\ < p. Then 
for any point inside the rectangle formed by the lines 

* = ar [r-|a: |], y = y [p - |y |], 

the function F(x, y) may be developed in a power series arranged 
according to positive powers of x x v and y y : 

F\ 



ZdT m f)i/ n / x=x 
1.2 ,! 1. ;*..*"* 



For if each element of the double series 



be replaced by its development in powers of h and k, the new multi 
ple series will converge absolutely under the hypotheses. Arrang 
ing the elements of this new series according to powers of h and k, 
we obtain the formula (36). 

The reader will be able to show without difficulty that all the 
preceding arguments and theorems hold without essential altera 
tion for power series in any number of variables whatever. 

186. Dominant functions. Given a power series f(x, y, z, ) in n 
variables, we shall say that another series in n variables </>(, y, z, ) 
dominates the first series if each coefficient of <j>(x, y,z, ) is positive 
and greater than the absolute value of the corresponding coefficient 
of f(x, y, z, ). The argument in 185 depends essentially upon 



IX, 186] DOUBLE POWER SERIES 397 

the use of a dominant function. For if the series 2,\a mn x "y"\ con 
verges for x = r, y = p, the function 

M /x\ m /vY* 

d>(x, y} = = 3/2 ( - ) ( - > 

/ x\ I y\ \ r / \P/ 



where M is greater than any coefficient in the series 2,\a mn r m p"\, 
dominates the series 2,a mn x m y n . The function 






is another dominant function. For the coefficient of x m y n in 1/^(3*, y) 
is equal to the coefficient of the corresponding term in the expan 
sion of M(x/r -+- y/p} m+n , and therefore it is at least equal to the 
coefficient of x m y n in <f>(x, y). 
Similarly, a triple series 



which converges absolutely for x = r, y = r , z = r", where r, r , r" 
are three positive numbers, is dominated by an expression of the 

form 

M 

y> 

l_Ul_\/l_- 



and also by any one of the expressions 

M M 



x\[~. /?/ z \ ~1 

- ) !-(-, + ) 
r/\_ \r r"/ J 



If f(x, y, z) contains no constant term, any one of the preceding expres 
sions diminished by M may be selected as a dominant function. 

The theorem regarding the substitution of one power series in 
another ( 182) may be extended to power series in several variables. 

If each of the variables in a convergent power series in p variables 
y\i y-ii ) y p b & replaced by a convergent power series in q variables 
x \j x ii *> x q which has no constant term, the result of the substitu 
tion may be written in the form of a power series arranged according 
to powers of x l , x 2 , , x q , provided that the absolute value of each 
of these variables is less than a certain constant. 



398 SPECIAL SERIES [IX, 186 

Since the proof of the theorem is essentially the same for any 
number of variables, we shall restrict ourselves for definiteness to 
the following particular case. Let 



(37) 

be a power series which converges whenever y < r and z < r , and let 
(38) (y=-t>i*--b 2 x^ -b m * .., 

f Cj 00 ~~r~ ^ 2 **^ ~T~ " * * j* t <C ~J~ 

be two series without constant terms both of which converge if the 
absolute value of x does not exceed p. If y and z in the series (37) 
be replaced by their developments from (38), the term in y m z" becomes 
a new power series in x, and the double series (37) becomes a triple 
series, each of whose coefficients may be calculated from the coeffi 
cients a mn , b n , and c n by means of additions and multiplications 
only. It remains to be shown that this triple series converges abso 
lutely when the absolute value of x does not exceed a certain con 
stant, from which it would then follow that the series could be 
arranged according to increasing powers of x. In the first place, 
the function f(y, z) is dominated by the function 

(39) 



and both of the series (38) are dominated by an expression of the form 

N ^T^ (x\ n 

(40) N= > .V (-) , 

, x L*i \P/ 

1 71= 1 

P 

where M and N are two positive numbers. If y and z in the double 
series (39) be replaced by the function (40) and each of the products 
y m z n be developed in powers of x, each of the coefficients of the result 
ing triple series will be positive and greater than the absolute value 
of the corresponding coefficient in the triple series found above. It 
will therefore be sufficient to show that this new triple series con 
verges for sufficiently small positive values of x. Now the sum of 
the terms which arise from the expansion of any term y m z* of the 
series (39) is 

M- 

AT -" Vp/ 



IX, 187] REAL ANALYTIC FUNCTIONS 399 

which is the general term of the series obtained by multiplying the 
two series 

x \ m i x 

P 



y Mry _p_ 

L*\rl 1 _ x _ 



term by term, except for the constant factor M. Both of the latter 
series converge if x satisfies both of the inequalities 



x < - x < 



It follows that all the series considered will converge absolutely, 
and therefore that the original triple series may be arranged accord 
ing to positive powers of x, whenever the absolute value of x is less 
than the smaller of the two numbers pr/(r + JV) and pr /(r + N). 

Note. The theorem remains valid when the series (38) contain 
constant terms b and c , provided that \b \< r and c < r . For 
the expansion (37) may be replaced by a series arranged according 
to powers of y b and z c , by 185, which reduces the discus 
sion to the case just treated. 

III. IMPLICIT FUNCTIONS 
ANALYTIC CURVES AND SURFACES 

187. Implicit functions of a single variable. The existence of implicit 
functions has already been established (Chapter II, 20 et ff.) under 
certain conditions regarding continuity. When the left-hand sides 
of the given equations are power series, more thorough investigation 
is possible, as we shall proceed to show. 

Let F(x, y~) = be an equation whose left-hand side can be developed 
in a convergent power series arranged according to increasing powers 
of x x and y y Q , where the constant term is zero and the coeffi 
cient of y y is different from zero. Then the equation has one and 
only one root which approaches >/ n as x approaches X Q , and that root 
can be developed in a power series arranged according to powers of 
x - x . 

For simplicity let us suppose that x = y = 0, which amounts to 
moving the origin of coordinates. Transposing the term of the first 
degree in y, we may write the given equation in the form 

(41) y = f(x, y} = a^x + a 20 a; 2 + a n xy + a^y* H , 



400 SPECIAL SERIES [IX, 187 

where the terms not written down are of degrees greater than the 
second. We shall first show that this equation can be formally sat 
isfied by replacing y by a series of the form 

(42) y = Cl x + c 2 x* + -.- + c n x n + --- 

if the rules for operation on convergent series be applied to the series 
on the right. For, making the substitution and comparing the coeffi 
cients of x, we find the equations 

c i = a io> c 2 = a so -f anCj + a 02 cf, ; 

and, in general, c n can be expressed in terms of the preceding c s 
and the coefficients a ik , where i + k < n, by means of additions and 
multiplications only. Thus we may write 

(43) c n = P n (a 10 , a w> a n , , a 0n ), 

where P n is a polynomial each of whose coefficients is a positive 
integer. The validity of the operations performed will be estab 
lished if we can show that the series (42) determined in this way 
converges for all sufficiently small values of x. We shall do this by 
means of a device which is frequently used. Its conception is due 
to Cauchy, and it is based essentially upon the idea of dominant 
functions. Let 



be a function which dominates the f unction /(x, y), where # 00 = b ol = 
and where b mn is positive and at least equal to a mn \ . Let us then 
consider the auxiliary equation 

(41 ) Y = 4>(x, Y) = 2b mn x m Y n 

and try to find a solution of this equation of the form 

(42 ) Y=C,x+ C 2 x* + ... + c n z" + --.. 

The values of the coefficients C 15 C 2 , can be determined as above, 

and are 

C l = b lo , C 2 = b 20 + b n C l -{-b 02 Cl, -, 

and in general 

(43 ) C. = P n (b w ,b 20 , ,*) 

It is evident from a comparison of the formulae (43) and (43 ) 
that \c n \ < C n , since each of the coefficients of the polynomial P n is 
positive and \a mn \^b mn . Hence the series (42) surely converges 



EX, 187] REAL ANALYTIC FUNCTIONS 401 

whenever the series (42 ) converges. Now we may select for the 
dominant function <j>(x, Y) the function 



Y 

" M ~ M 7 



where M, r, and p are three positive numbers. Then the auxiliary 
equation (41 ) becomes, after clearing of fractions, 



p + M p + M -L _ * 
r 

This equation has a root which vanishes for x = 0, namely : 

F = 




The quantity under the radical may be written in the form 



where 

P 



Hence the root Y may be written 



] 



It follows that this root Y may be developed in a series which con 
verges in the interval ( a, -f a), and this development must coin 
cide with that which we should obtain by direct substitution, that 
is, with (42 ). Accordingly the series (42) converges, a fortiori, in 
the interval ( a, + a). This is, however, merely a lower limit of 
the true interval of convergence of the series (42), which may be 
very much larger. 

It is evident from the manner in which the coefficients c n were 
determined that the sum of the series (42) satisfies the equation (41). 
Let us write the equation F(x, y) in the form y f(x, y) = 0, and 
let y = P(x) be the root just found. Then if P(x) + be substi 
tuted for y in F(x, y), and the result be arranged according to 
powers of x and z, each term must be divisible by z, since the whole 
expression vanishes when z = for any value of x. We shall have 
then F[x, P(x) -f *] = sQ(#, ) , where Q(x, z) is a power series in x 



402 SPECIAL SERIES [IX, 188 

and z. Finally, if z be replaced by y P(x} in Q(x, z), we obtain 
the identity 



where the constant term of Q t must be unity, since the coefficient 
of y on the left-hand side is unity. Hence we may write 

(44) F(x, y) = [y - P(x)] (1 + ax + fa + ...). 

This decomposition of F(x, y) into a product of two factors is due 
to Weierstrass. It exhibits the root y = P(x), and also shows that 
there is no other root of the equation F(x, y) = which vanishes 
with x, since the second factor does not approach zero with x and y. 

Note. The preceding method for determining the coefficients c n is 
essentially the same as that given in 46. But it is now evident 
that the series obtained by carrying on the process indefinitely is 
convergent. 

188. The general theorem. Let us now consider a system of p equa 
tions in p -f q variables. 

i(i,*a * ; yi,y a j J y P ) = 0, 
2 (x 1} x 2 , - ,x 9 i yi,y 2 , -,y,)= 0, 

........... 

(*u ** >*; yi>y 2 , %)== o, 

where each of the functions F 1} F 2 , , F p vanishes when x i y k = 0, 
and is developable in power series near that point. We shall further 
suppose that the Jacobian D(F l} F 2 , -, F p )/D(y 1 , y 2 , , y p ) does 
not vanish for the set of values considered. Under these conditions 
there exists one and only one system of solutions of the equations (45) 
of the form 



where fa, fa, , < y , are power series in x l} x 2 , , a*, tvhich vanish 
when x l = a; 2 = = x q = 0. 

In order to simplify the notation, we shall restrict ourselves to 
the case of two equations between two dependent variables u and v 
and three independent variables x, y, and z : 

/^CN (F l = au -f bv + ex +dy + ez -\ ---- = 0, 

(4o) < 

I F 2 = a u + Vv + c x + d y + e z ^ ---- = 0. 

Since the determinant ab ba is not zero, by hypothesis, the two 
equations (46) may be replaced by two equations of the form 



IX, 188] REAL ANALYTIC FUNCTIONS 403 



mnpqr 
(v = ^b mnpqr 

where the left-hand sides contain no constant terms and no terms 
of the first degree in u and v. It is easy to show, as above, that 
these equations may be satisfied formally by replacing u and v by 
power series in x, //, and z : 

(48) u = 2c,. t ,a;V*S v = 2c ikl x { y k z 1 , 

where the coefficients c ikl and c\ kl may be calculated from a mnpqr and 
b mn ( r by means of additions and multiplications only. In order to 
show that these series converge, we need merely compare them with 
the analogous expansions obtained by solving the two auxiliary 
equations 



where M, r, and p are positive numbers whose meaning has been 
explained above. These two auxiliary equations reduce to a single 
equation of the second degree 

x + if + z 



,r. | 



_ 

2 P + 4Af 2p + 4M 1 _ x + y + z 

r 

which has a single root which vanishes for x = y = z = 0, namely : 



__ 

~ 



p 2 ___ p 2 

~ 



4(p + 2M) 4( P -f 2M) 



r 



where a = r [p/(p + 4M)] 2 . 

This root may be developed in a convergent power series when 
ever the absolute values of x, y, and z are all less than or equal to 
a/3. Hence the series (48) converges under the same conditions. 

Let x and v t be the solutions of (47) which are developable in 
series. If we set u = ?/ t + u , v = v^ -f v in (47) and arrange the 
result according to powers of x, y, z, u , v , each of the terms must 
be divisible by u or by v . Hence, returning to the original varia 
bles x, y, z, u, v, the given equations may be written in the form 

M -/ +(-iH =0, 






404 SPECIAL SERIES [IX, 189 

where /, <f>, f t , ^ are power series in x, y, z, u, and v. In this 
form the solutions u = u l} v = v-i are exhibited. It is evident also 
that no other solutions of (47 ) exist which vanish for x = y = z = 0. 
For any other set of solutions must cause ffa <&/\ to vanish, 
and a comparison of (47) with (47 ) shows that the constant term 
is unity in both / and <j> l} whereas the constant term is zero in 
both/! and <; hence the condition ffa </i = cannot be met by 
replacing u and v by functions which vanish when x y = z = 0. 

189. Lagrange s formula. Let us consider the equation 

(49) y = a + x$(y) , 

where <j>(y) is a function which is developable in a power series in y a, 

4>(y) = <t>(a) + (y - a) (a) + (y ~ a)2 0"(a) + , 

1 . l 

which converges whenever y a does not exceed a certain number. By the 
general theorem of 187, this equation has one and only one root which 
approaches a as x approaches zero, and this root is represented for sufficiently 
small values of x by a convergent power series 

y = a + aix -f a 2 x 2 + . 

In general, if f(y) is a function which is developable according to positive 
powers of y a, an expansion of f(y) according to powers of x may be obtained 
by replacing y by the development just found, 

(50) f(y) =f(a) + A t x + A 2 x* + + A,,x + . . , 

and this expansion holds for all values of x between certain limits. 
The purpose of Lagrange s formula is to determine the coefficients 

AI, A 2 , -, A n , 

in terms of a. It will be noticed that this problem does not differ essentially 
from the general problem. The coefficient A n is equal to the nth derivative of 
f(y) for y = 0, except for a constant factor n!, where y is defined by (49); and 
this derivative can be calculated by the usual rules. The calculation appears to 
be very complicated, but it may be substantially shortened by applying the fol 
lowing remarks of Laplace (cf. Ex. 8, Chapter II). The partial derivatives of 
the function y defined by (49), with respect to the variables x and a, are given 
by the formulae 



whence we find immediately 

i*.i\ d u 

(51) 

ox da 

where u =f(y). On the other hand, it is easy to show that the formula 



da 



! = -!><*) - 

acJ dx I W c>a 



IX, 189] REAL ANALYTIC FUNCTIONS 405 

is identically satisfied, where F(y) is an arbitrary function of y. For either 
side becomes 



on performing the indicated differentiations. We shall now prove the formula 



for any value of n. It holds, by (51), for n = 1. In order to prove it in gen 
eral, let us assume that it holds for a certain number n. Then we shall have 



dx + l da n ~ l dx 
But we also have, from (51) and (51 ), 

du 



l- [*]-[>* 

caJ ca L cxj ca L 



whence the preceding formula reduces to the form 
c n + 1 u B n 



which shows that the formula in question holds for all values of n. 

Now if we set x = 0, y reduces to a, u to /(a), and the nth derivative of u 
with respect to x is given by the formula 



da"- 1 
Hence the development of f(y) by Taylor s series becomes 

/(2/)=/(a) + z0(a)/ (a) . 
(52) 



This is the noted formula due to Lagrange. It gives an expression for the 
root y which approaches zero as x approaches zero. We shall find later the 
limits between which this formula is applicable. 

Note. It follows from the general theorem that the root y, considered as a 
function of x and a, may be represented as a double series arranged according 
to powers of x and a. This series can be obtained by replacing each of the 
coefficients A n by its development in powers of a. Hence the series (52) may 
be differentiated term by term with respect to a. 

Examples. 1) The equation 

(53) y = a + ?(y*~l) 



40G SPECIAL SERIES [IX, 190 

has one root which is equal to a when x = 0. Lagrange s formula gives the 
following development for that root : 



. 2 V 27 da 



(54) 
On the other hand, the equation (53) may be solved directly, and its roots are 



, _J_M" d- (q2-l)n 
1.2...nV27 do-- 



y = - - Vl 2ax + x 2 . 

The root which is equal to a when x = is that given by taking the sign ~~. 
Differentiating both sides of (54) with respect to a, we obtain a formula which 
differs from the formula (32) of 184 only in notation. 

2) Kepler s equation for the eccentric anomaly u,* 

(55) u = a + e sin u , 

which occurs in Astronomy, has a root u which is equal to a f or e = 0. Lagrange s 
formula gives the development of this root near e = in the form 

(56) u = a e " 



1.2 da 1.2..-n da-i 

Laplace was the first to show, by a profound process of reasoning, that this 
series converges whenever e is less than the limit 0.662743 

190. Inversion. Let us consider a series of the form 

(57) y = aix + a z x 2 + + a n x n + , 

where ai is different from zero and where the interval of convergence is( r, + r). 
If y be taken as the independent variable and x be thought of as a function of y, 
by the general theorem of 187 the equation (57) has one and only one root which 
approaches zero with y, and this root can be developed in a power series in y : 

(58) x = biy -f b 2 y 2 + b^y 3 + + b n y" + . 

The coefficients bi, b 2 , 63, may be determined successively by replacing x in 
(57) by this expansion and then equating the coefficients of like powers of y. 
The values thus found are 

bi = , & 2 = ~ b s = - 1 3 , . 

<*! a? af 

The value of the coefficient b n of the general term may be obtained from 
Lagrange s formula. For, setting 

^(X) = tti + a 2 X + + OnX n ~ l + , 

the equation (57) may be written in the form 

1 



*See p. 248, Ex. 19; and ZIWET, Elements of T?teoretical Mechanics, 2d ed., 
p. 356. TRANS. 



IX, l!ii] REAL ANALYTIC FUNCTIONS 407 

and the development of the root of this equation which approaches zero with y 
is given by Lagrange s formula in the form 

1 ?/" d-i / 1 \" 

+ + " 



where the subscript indicates that we are to set x = after performing the 
indicated differentiations. 

The problem just treated has sometimes been called the reversion of series. 

191. Analytic functions. In the future we shall say that a func 
tion of any number of variables x, y, z, is analytic if it can be 
developed, for values of the variables near the point x , y , z , , 
in a power series arranged according to increasing powers of 
x ~ x o> V 2/o j z ~ z o> " which converges for sufficiently small 
values of the differences x x , . The values which ar , y , z , 
may take on may be restricted by certain conditions, but we shall 
not go into the matter further here. The developments of the pres 
ent chapter make clear that such functions are, so to speak, inter 
related. Given one or more analytic functions, the operations of 
integration and differentiation, the algebraic operations of multipli 
cation, division, substitution, etc., lead to new analytic functions. 
Likewise, the solution of equations whose left-hand member is ana 
lytic leads to analytic functions. Since the very simplest functions, 
such as polynomials, the exponential function, the trigonometric 
functions, etc., are analytic, it is easy to see why the first functions 
studied by mathematicians were analytic. These functions are still 
predominant in the theory of functions of a complex variable and in 
the study of differential equations. Nevertheless, despite the funda 
mental importance of analytic functions, it must not be forgotten 
that they actually constitute merely a very particular group among 
the whole assemblage of continuous functions.* 

192. Plane curves. Let us consider an arc AB of a plane curve. 
We shall say that the curve is analytic along the arc AB if the 
coordinates of any point M which lies in the neighborhood of any 
fixed point M of that arc can be developed in power series arranged 
according to powers of a parameter t t , 



(59) T" -rw-^o + aiC to)+*2(t t o y + ... + a n (t 

y = f(f) = y + b l (t- f ) + b z (t - t o y + + b n (t - 



which converge for sufficiently small values of t t . 

* In the second volume an example of a non-analytic function will be given, all of 
whose derivatives exist throughout an interval (a, b). 



408 SPECIAL SERIES [IX, 192 

A point 3/o will be called an ordinary point if in the neighbor 
hood of that point one of the differences y y , x x can be 
represented as a convergent power series in powers of the other. 
If, for example, y y can be developed in a power series in 
x x 0) 

(60) y y ( > = ^(x x ) + c a (x x ) 2 H ----- 1- c n (x z ) n H ---- , 

for all values of x between x n h and -a- + h, the point (x , y ~) is 
an ordinary point. It is easy to replace the equation (60) by two 
equations of the form (59), for we need only set 



(61) 



If c v is different from zero, which is the case in general, the equa 
tion (60) may be solved for x x in a power series in y y which 
is valid whenever y y is sufficiently small. In this case each of 
the differences x x , y y can be represented as a convergent 
power series in powers of the other. This ceases to be true if c t is 
zero, that is to say, if the tangent to the curve is parallel to the 
x axis. In that case, as we shall see presently, x x may be devel 
oped in a series arranged according to fractional powers of y y . 
It is evident also that at a point where the tangent is parallel to 
the y axis x x can be developed in power series in y y oy but 
y t/o cannot be developed in power series in x x . 

If the coordinates (a-, ?/) of a point on the curve are given by the 
equations (59) near a point M , that point is an ordinary point if 
at least one of the coefficients a l} bi is different from zero.* If a l 
is not zero, for example, the first equation can be solved for t t 
in powers of x x , and the second equation becomes an expansion 
of y y in powers of x x when this solution is substituted for 
t-t . 

The appearance of a curve at an ordinary point is either the cus 
tomary appearance or else that of a point of inflection. Any point 
which is not an ordinary point is called a singular point. If all 
the points of an arc of an analytic curve are ordinary points, the 
arc is said to be regular. 



* This condition is sufficient, but not necessary. However, the equations of any 
curve, near an ordinary point M , may always be written in such a way that 04 and 
b l do not both vanish, by a suitable choice of the parameter. For this is actually 
accomplished in equations (61). See also second footnote, p. 409. TRANS. 



IX, 193] KEAL ANALYTIC FUNCTIONS 409 

If each of the coefficients a and b v is zero, but a 2 , for example, 
is different from zero, the first of equations (59) may be written in 
the form (x x u )* = (t ) [ 2 -f a 8 (t ) -\ ---- ]*, where the right- 
hand member is developable according to powers of t t . Hence 
t t is developable in powers of (x a^)*, and if t t in the 
second equation of (59) be replaced by that development, we obtain 
a development for y y in powers of (x z )* : 



In this case the point (z , y ~) is usually a cusp of the first kind.* 

The argument just given is general. If the development of 
x x in powers of t t begins with a term of degree n, y y 
can be developed according to powers of (x a- ). The appearance 
of a curve given by the equation (59) near a point (x , y ) is of 
one of four types : a point with none of these peculiarities, a point 
of inflection, a cusp of the first kind, or a cusp of the second kind.* 

193. Skew curves. A skew curve is said to be analytic along an arc 
AB if the coordinates x, y, z of a variable point M can be developed 
in power series arranged according to powers of a parameter t t 

f x = x + ai(t - *) H ----- h a n (t - t u ) n -i ---- , 

(62) J y = y + b, (t - * ) + + b n (t - t,} n + -.., 

(* SS K 9 +C l (t-tj + -~+ C n (t - t ) n + -, 

in the neighborhood of any fixed point M of the arc. A point 
M is said to be an ordinary point if two of the three differences 
x x oi y y< z s o can be developed in power series arranged 
according to powers of the third. 

It can be shown, as in the preceding paragraph, that the point 
M will surely be an ordinary point if not all three of the coefficients 
a l ,b l , c l vanish. Hence the value of the parameter t for a singular 
point must satisfy the equations f 

^ = dy_ Q dz_ 

dt dt U dt 

* For a cusp of the first kind the tangent lies between the two branches. For a 
cusp of the second kind both branches lie on the same side of the tangent. The 
point is an ordinary point, of course, if the coefficients of the fractional powers 
happen to be all zeros. TRANS. 

t These conditions are not sufficient to make the point 3f , which corresponds to 
a value t of the parameter, a singular point when a point M of the curve near Jf 
corresponds to several values of t which approach t as M approaches M . Such is 
the case, for example, at the origin on the curve defined by the equations x = t 3 , 



410 SPECIAL SERIES [IX, 194 

Let x , y , z be the coordinates of a point M on a skew curve T 
whose equations are given in the form 

(63) F(x,y,z)=0, F,(x, </,*)= 0, 

where the functions Fand F t are power series in x x , y y , z z . 
The point M will surely be an ordinary point if not all three of 
the functional determinants 

D(F, F,} (F, FQ D(F, FQ 

D(x, y) D(y, z) D(z, x) 

vanish simultaneously at the point x = x , y = ?/ , z = z . For if 
the determinant D(F, Fi)/D(x, y), for example, does not vanish at 
M , the equations (63) can be solved, by 188, for x X Q and y y Q 
as power series in z z . 

194. Surfaces. A surface S will be said to be analytic throughout 
a certain region if the coordinates x, y, z of any variable point M 
can be expressed as double power series in terms of two variable 
parameters t t and u H O 

ex x = a 10 (t f ) + oi( M - OH , 

(64) J y - y = b 10 (t - * ) + b 01 (u - w ) -\ , 

U - = c io(t <o) + c oi ( u ? o) H 1 

in the neighborhood of any fixed point M of that region, where 
the three series converge for sufficiently small values of t 1 and 
u u . A point M of the surface will be said to be an ordinary 
point if one of the three differences x x , y y , z z can be 
expressed as a power series in terms of the other two. Every point 
M for which not all three of the determinants 

D(y, g) D(z, x)^ D(x, y} 
D(t, u) D(t, u) D(t, u) 

vanish simultaneously is surely an ordinary point. If, for exam 
ple, the first of these determinants does not vanish, the last two of 
the equations (64) can be solved for t t and u u , and the first 
equation becomes an expansion of x x in terms of y y and 
z z upon replacing t t and u u by these values. 

Let the surface S be given by means of an unsolved equation 
F(x, y, z) = 0, and let x , y , z be the coordinates of a point M 
of the surface. If the function F(x, y, z) is a power series in 
x x o> y y< z z o) an( i ^ n t a ll three of the partial derivatives 
dF/dx , 8F/8y , 8F/dz vanish simultaneously, the point 3/ is surely 
an ordinary point, by 188. 



IX,195J TRIGONOMETRIC SERIES 411 

Note. The definition of an ordinary point on a curve or on a sur 
face is independent of the choice of axes. For, let 3/ (x , y , ) be an 
ordinary point on a surface S. Then the coordinates of any neigh 
boring point can* be written in the form (64), where not all three of 
the determinants D(y, z)/D(t, u), D(z, x)/D(t, u}, D(x, y)/D(t, it} 
vanish simultaneously for t = t , u = u . Let us now select any new 
axes whatever and let 

A = ai x + fay + yiz + 8 1} 
Y= a^x + {3 2 y -f y z z + S 2 , 
Z = a 3 x + fay + y 3 z + 8 3 

be the transformation which carries x, y, z into the new coordinates 
X, Y, Z, where the determinant A = D(X, Y, Z)/D(x, y, z) is differ 
ent from zero. Replacing x, y, z by their developments in series 
(64), we obtain three analogous developments for X, Y, Z ; and we 
cannot have 

1>(X, Y} = Z>( J^Z) = D(Z, X} = 
J)(t, if) D(t, u) ~ D(t, u) 

for t = , u = ii , since the transformation can be written in the form 

X = A 1 X + B 1 Y+ C\Z + D ly 

y = A 2 X + B 2 Y + C 2 Z + J9 2 , 
z = A 3 X + B 3 Y+C 3 Z + D t , 

and the three functional determinants involving .Y, Y, Z cannot 
vanish simultaneously unless the three involving x, y, z, also vanish 
simultaneously. 

IV. TRIGONOMETRIC SERIES MISCELLANEOUS SERIES 

195. Calculation of the coefficients. The series which we shall study 
in this section are entirely different from those studied above. 
Trigonometric series appear to have been first studied by D. Ber 
noulli, in connection with the problem of the stretched string. The 
process for determining the coefficients, which we are about to give, 
is due to Euler. 

Let /(a:) be a function defined in the interval (a, &). We shall 
first suppose that a and b have the values TT and + TT, respec 
tively, which is always allowable, since the substitution 

2-7TX (a -f 1} 7T 

x = r-*- 

o a 

* See footnote, p. 408. TRANS. 



412 SPECIAL SERIES [IX, 195 

reduces any case to the preceding. Then if the equation 

(tn 

(65) /(#) = -^ -f (i cos x + b : sin x) -\ \- (a m cos mx + & m sin mx) -\ 

holds for all values of x between TT and + TT, where the coefficients 
a o) a i> bi> ) a m> b m >--- are unknown constants, the following device 
enables us to determine those constants. We shall first write down 
for reference the following formulae, which were established above, 
for positive integral values of m and n : 



(66) 



sin mx dx = ; 



L 

L 
f 

\J 71 



/cos -mxdx = I dx = TT, if m 3= 0; 

7T V/-7T ^ 

r + " 

I sin mx sin nx dx 

\J 7T 

f + * cos (m n) x cos (m -f n)x ^ _ 
/ o " x 

C/-7T 



cos m rfa; = 0, if m ^ ; 

r 
cos wcc cos nx dx 

" cos(m ri)x + cos (m + ri)x 7 . 

^ * 5 rfx = , if m = n ; 



2 



r + 7r . r + " 1 cos 2mrc , . 

I sm 2 ma;rfx= I ^ ^ x =TT, ifw^O; 

I 



sin mx cos nx dx 



X 77 sin (m + n) x + sin (TO n)x __ 
2 

Integrating both sides of (65) between the limits TT and -f TT, 
the right-hand side being integrated term by term, we find 

/" -\- 7T /^ "t" T 

I /(x) dx = - I dx = 7ra , 

J-TT /- 

which gives the value of a . Performing the same operations upon 
the equation (65) after having multiplied both sides either by cos mx 



IX, 195] TRIGONOMETRIC SERIES 413 

or by sin mx, the only term on the right whose integral between TT 
and + TT is different from zero is the one in cos 2 mx or in sin 2 mx. 
Hence we find the formulae 

/ + " (* + * 

f(x) cos mx dx = 7nz m , I f(x) sin mx dx = 7rb m , 

TT \J TT 

respectively. The values of the coefficients may be assembled as 
follows : 



(67) 



a = - I /(a) da, a m = - I /(a) cos ma da, 

** t/ n *i^/ jr 

i r +)r 

b m = I f(a~) sin mar c?nr. 

T 7 " t/-n- 



The preceding calculation is merely formal, and therefore tenta 
tive. For we have assumed that the function f(x) can be developed 
in the form (65), and that that development converges uniformly 
between the limits TT and -f TT. Since there is nothing to prove, 
a priori, that these assumptions are justifiable, it is essential that 
we investigate whether the series thus obtained converges or not. 
Eeplacing the coefficients a { and b { by their values from (67) and 
simplifying, the sum of the first (m + 1) terms is seen to be 

1 f +7r fl 1 

S m+l = I /() -^ + cos(a a-) + cos2(rt x}-\ ----- \-COSm(a x) \da. 

But by a well-known trigonometric formula we have 

2m +1 
sin - a 

- + cos a -f cos 2a + + cos m a = - - - > 
2 . a 

2sm- 

whence 

. 2m + 1 
sin - (a a-) 



or, setting a = x -f- 
(68) S ^ 



The whole question is reduced to that of finding the limit of this 
sum as the integer m increases indefinitely. In order to study this 
question, we shall assume that the function f(x) satisfies the fol 
lowing conditions : 



414 SPECIAL SERIES [IX, 196 

1) The function f(x) shall be in general continuous between TT 
and + TT, except for a, finite number of values of x, for which its value 
may change suddenly in the following manner. Let c be a number 
between TT and + TT. For any value of c a number h can be found 
such that f(x) is continuous between c h and c and also between 
c and c + h. As c approaches zero, f(c -f- c) approaches a limit which 
we shall call f(c + 0). Likewise, f(c e) approaches a limit which 
we shall call f(c 0) as c approaches zero. If the function f(x) 
is continuous for x = c, we shall have /(c) f(c + 0) =/(c 0). If 
f(c + 0) = f(c 0), f(x) is discontinuous for x = c, and we shall agree 
to take the arithmetic mean of these values [/(c + 0) + f(c 0)]/2 
for /(c). It is evident that this definition of /(c) holds also at points 
where f(x) is continuous. We shall further suppose that /( TT + e) 
and /(TT e) approach limits, which we shall call /( TT + 0) and 
/(TT 0), respectively, as e approaches zero through positive values. 
The curve whose equation is y f(x) must be similar to that of 
Fig. 11 on page 160, if there are any discontinuities. We have 
already seen that the function /(x) is integrable in the interval from 
TT to + TT, and it is evident that the same is true for the product 
of f(x) by any function which is continuous in the same interval. 

2) It shall be possible to divide the interval ( TT, + TT) into a 
finite number of subintervals in such a way thaty(cr) is a monoton- 
ically increasing or a monotonically decreasing function in each of 
the subintervals. 

For brevity we shall say that the function f(x) satisfies Dirichlet s 
conditions in the interval ( TT, + TT). It is clear that a function 
which is continuous in the interval ( TT, + TT) and which has a 
finite number of maxima and minima in that interval, satisfies 
Dirichlet s conditions. 

196. The integral / h f (x) [sin nx/sin x] dx. The expression obtained 
for <S TO + 1 leads us to seek the limit of the definite integral 



C sin nx 

f(x) dx 

Jo since 



as n becomes infinite. The first rigorous discussion of this ques 
tion was given by Lejeune-Dirichlet.* The method which we shall 
employ is essentially the same as that given by Bonnet. t 

* Crelle s Journal, Vol. IV, 1829. 

t Mtmoires des savants etrangers publics par 1 Academic de Belgique, Vol. XXIII. 



IX, 196] TRIGONOMETRIC SERIES 415 

Let us first consider the integral 

r>h 

I . . sin nx , 
(69) ./ = I <K*)~ dx > 

c/O 

where h is a positive number less than TT, and <j>(x) is a function 
which satisfies Dirichlet s conditions in the interval (0, A). If <f>(x) 
is a constant C, it is easy to find the limit of /. For, setting y = nx, 
we may write 

sink 

I sin y 7 
J=C\ 2-dy, 

Jo y 

and the limit of J as n becomes infinite is CTr/2, by (39), 176. 

Next suppose that <f>(x) is a positive monotonically decreasing 
function in the interval (0, h). The integrand changes sign for 
all values of x of the form kir/n. Hence J may be written 

j= Uo - Ul + u 2 - u 3 + + (- 1)*%- + + (- I) m 0",, < < 1, 
where 



C " 
I 

Jk* 



sn nx . 
- dx 



and where the upper limit h is supposed to lie between mir/n and 
(ra -f T)7r/n. Each of the integrals u k is less than the preceding. 
For, if we set nx = &TT + y in u k , we find 

y + kw\ sin y 
n / y + A:TT * 

and it is evident, by the hypotheses regarding <(#), that this inte 
gral decreases as the subscript k increases. Hence we shall have 
the equations 

./ = UQ \Ml ^"i) y^ s ^4/ ) 

which show that / lies between ?/ and n u^ . It follows that / is 
a positive number less than ii , that is to say, less than the integral 



/ 

Jo 



smnx 
- -dx. 



x 

But this integral is itself less than the integral 



Jo y 

where A denotes the value of the definite integral //[(sin y)/y~\ dy. 



416 SPECIAL SERIES [IX, 196 

The same argument shows that the definite integral 



Xh 
, . ^ sm nx . 
*(x)- -dx, 
x 



where c is any positive number less than h, approaches zero as n 
becomes infinite. If c lies between (i l)?r/n and i-rr/n, it can be 
shown as above that the absolute value of J is less than 



C " . sin nx , C 

<p(x) dx + 

Jc X Jiz 

n 

and hence, a fortiori, less than 



snnx 




_ 

c \ n 



Hence the integral approaches zero as n becomes infinite.* 

This method gives us no information if c = 0. In order to dis 
cover the limit of the integral J, let c be a number between 
and h, such that <f>(x) is continuous from to c, and let us set 
<l>(x) = <f>(c) + \f/(x"). Then i^(a-) is positive and decreases in the 
interval (0, c) from the value <(+ 0) - <(c) when x = to the 
value zero when x = c. If we write / in the form 



/""sin nx C r sin nx C 

J = <(c) I dx + I $(x) - dx + 

JO X Jo X Jc 

and then subtract (7r/2)< T i(-f 0), we find 



sin nx 7 
-- dx 



(70) 

/e . ,-> . 

I sinnx I smnir 

I f(x) - -dx+ <#>(x) - 

JO -^ Jc x 

In order to prove that ./ approaches the limit (Tr/2) ^>(+ 0), it will 
be sufficient to show that a number m exists such that the absolute 

* This result may be obtained even more simply by the use of the second theorem 
of the mean for integrals (75). Since the function <f>(r) is a decreasing function, 
that formula gives 



. . sin nx , 0(c) r 1 <f>M 

() ax = -J- J sin wx dx = - -j* (cos nc - cosn|) , 

and the right-hand member evidently approaches zero. 



IX, 196] TRIGONOMETRIC SERIES 417 

value of each of the terms on the right is less than a preassigned 
positive number e/4 when n is greater than m. By the remark 
made above, the absolute value of the integral 



/ 

Jo 



sin nx 
ax 



is less than A$(+ 0) = A [<(+ 0) <(e)]. Since <(z) approaches 
<(+ 0) as x approaches zero, c may be taken so near to zero that 
A [>(+ 0) - (c)] and (7r/2)[>(+ 0) - (c)] are both less than e/4. 
The number c having been chosen in this way, the other two terms 
on the right-hand side of equation (70) both approach zero as n 
becomes infinite. Hence n may be chosen so large that the abso 
lute value of either of them is less than e/4. It follows that 

(71) lim/=|*(+0). 

(=QO A 

We shall now proceed to remove the various restrictions which 
have been placed upon <f>( x ) ^ n the preceding argument. If <f>(x) is 
a monotonically decreasing function, but is not always positive, the 
function \j/(x) <f>(x) + C is a positive monotonically decreasing func 
tion from to h if the constant C be suitably chosen. Then the 
formula (71) applies to \j/(x). Moreover we may write 



(* * sinnx C k sinnx C 

I (*) dx = <K*)- -dx-C 

JO Jo JO 



h sinnx 

dx, 



and the right-hand side approaches the limit (w/2) \j/(-{- 0) (?r/2) C, 
that is, (?r/2) <#(+ 0). 

If <(#) is a monotonically increasing function from to h, <f>(x~) 
is a monotonically decreasing function, and we shall have 



C h sin nx C 

I ^(*) - - dx = - I 

Jo * t7o 



sin nx 
- dx. 



Hence the integral approaches (7r/2)<(+ 0) in this case also. 

Finally, suppose that <(#) is any function which satisfies Dirich- 
let s conditions in the interval (0, A). Then the interval (0, A) 
may be divided into a finite number of subintervals (0, a), (a, b), 
(b, c), -, (7, A), in each of which <f>(x~) is a monotonically increasing 
or decreasing function. The integral from to a approaches the limit 
(7T/2) <(-)- 0). Each of the other integrals, which are of the type 



=/ 

Ja 



sinnx 
dx, 



418 SPECIAL SERIES [IX, 107 

approaches zero. For if <f>(x~) is a monotonically increasing function, 
for instance, from a to b, an auxiliary function \j/(x) can be formed 
in an infinite variety of ways, which increases monotonically from 
to b, is continuous from to a, and coincides with <f>(x) from a to b. 
Then each of the integrals 



C 
/ 

t/O 



sinnx sinnx 

<**, 



approaches $(+ 0) as n becomes infinite. Hence their difference, 
which is precisely H, approaches zero. It follows that the formula 
(71) holds for any function <f>(x) which satisfies Dirichlet s condi 
tions in the interval (0, h). 

Let us now consider the integral 



(72) f=f(x)dX) (X/K7T, 



where f(x) is a positive monotonically increasing function from 
to A. This integral may be written 



C v ff \ x 

= 1 f( x )~ 

Jo L ^ sm x 



n s 



-, 

dx, 



sill x J x 

and the function tf>(x) = f(x) x/sin x is a positive monotonically 
increasing function from to h. Since /(+ 0) = <(+ 0), it follows 
that 

(73) lim 7 = /(+ 0). 

This formula therefore holds if f(x) is a positive monotonically 
increasing function from to h. It can be shown by successive 
steps, as above, that the restrictions upon f(x) can all be removed, 
and that the formula holds for any function f(x) which satisfies 
Dirichlet s conditions in the interval (0, li). 

197. Fourier series. A trigonometric series whose coefficients are 
given by the formulae (67) is usually called a Fourier series. Indeed 
it was Fourier who first stated the theorem that any function arbi 
trarily defined in an interval of length 2?r may be represented by a 
series of that type. By an arbitrary function Fourier understood 
a function which could be represented graphically by several cur 
vilinear arcs of curves which are usually regarded as distinct curves. 
We shall render this rather vague notion precise by restricting our 
discussion to functions which satisfy Dirichlet s conditions. 



IX, 197] 



TRIGONOMETRIC SERIES 419 



In order to show that a function of this kind can be represented 
by a Fourier series in the interval ( TT, + TT), we must find the 
limit of the integral (68) as m becomes infinite. Let us divide 
this integral into two integrals whose limits of integration are 
and (TT r)/2, and (TT + x)/2 and 0, respectively, and let us 
make the substitution y = z in the second of these integrals. 
Then the formula (68) becomes 



sn 



1 C * s, ox sin (2m + 1) * , 

I f(x 2z) dz . 

TT Jo J v sm 



When x lies between TT and 4- TT, (TT a;)/2 and (TT + #)/2 both 
lie between and TT. Hence by the last article the right-hand 
side of the preceding formula approaches 



as m becomes infinite. It follows that the series (65) converges and 
that its sum is/(x) for every value of x between TT and + TT. 

Let us now suppose that x is equal to one of the limits of the 
interval, TT for example. Then S m + l may be written in the form 



1 C*., , X sin(2m4-1)// j 

= - I /( TT + 2y) -- i ; dy 

7rJ J smy 



The first integral on the right approaches the limit /( TT + 0)/2. 
Setting y = TT 2 in the second integral, it takes the form 



sm* 

which approaches /(TT 0)/2. Hence the sum of the trigonometric 
series is [/(TT 0) +/( TT + 0)] /2 when x = TT. It is evident 
that the sum of the series is the same when x = + TT. 

If, instead of laying off x as a length along a straight line, we 
lay it off as the length of an arc of a unit circle, counting in the 



420 SPECIAL SERIES [IX, 197 

positive direction from the point of intersection of the circle with 
the positive direction of some initial diameter, the sum of the series 
at any point whatever will be the arithmetic mean of the two limits 
approached by the sum of the series as each of the variable points 
ra and ra", taken on the circumference on opposite sides of ra, 
approaches m. If the two limits /( TT + 0) and f(ir 0) are 
different, the point of the circumference on the negative direction 
of the initial line will be a point of discontinuity. 

In conclusion, every function which is defined in the interval 
( TJ -f- "") and which satisfies Dirichlet s conditions in that inter 
val may be represented by a Fourier series in the same interval. 

More generally, let f(x) T^e a function which is defined in an 
interval (a, a + 2?r) of length 2?r, and which satisfies Dirichlet s 
conditions in that interval. It is evident that there exists one and 
only one function F(x) which has the period 2?r and coincides with 
/(#) in the interval (a, a + 2?r). This function is represented, for 
all values of x, by the sum of a trigonometric series whose coeffi 
cients a m and b m are given by the formulas (67): 

a m = ~ I F(r) cos mx dx, b m = - I F(x) sin mx dx . 
The coefficient a m , for example, may be written in the form 

1 r* i r- 2 *- 

a m I F(x) cos mx dx + - I F(x) cos mx dx, 

*V-tte vj_ n 

where a is supposed to lie between 2hir TT and 2hjr + TT. Since 
F(x} has the period 2?r and coincides with f(x) in the interval 
(a, a + 2?r), this value may be rewritten in the form 

( 1 / l2 " r + !r ^a: + 2T 

\ a m = ~ I f(y) cos mx dx + / / (.r) cos mx dx 

(74) I Ja Jm*+m 

- ,^a + 2jr 

= / f(x) cos mx dx . 

V. t/ ft 

Similarly, we should find 

(75) b m = ~ I f(x) sin mx dx . 

"ft J a 

Whenever a function f(x) is defined in any interval of length 2?r, 
the preceding formulae enable us to calculate the coefficients of its 
development in a Fourier series without reducing the given interval 
to the interval ( TT, + TT). 



IX, 198] TRIGONOMETRIC SERIES 421 

198. Examples. 1) Let us find a Fourier series whose sum is 1 for 
it < x < 0, and +1 f or < x < + it. The formulae (67) give the values 

1 r 1 c n 

a = I dx H I dx = , 

It J-ir Tt Jo 

1 r i r* 

a m = I cos mxdx -\ cos mx dx = , 

It J -n It JQ 

I f . j , ! T* A 2 cosmTT cos( mir) 
n m = ~ sm mx dx -\ sin mx ax = - > - - . 

71 J -T, Tt Jo mit 

If m is even, b m is zero. If m is odd, b m is 4/mir. Multiplying all the coeffi 
cients by 7T/4, we see that the sum of the series 



(76) v = -i- Sin3x + + 

1 3 2m + 1 

is 7T/4 for it < x < 0, and + ff/4 for < x < it. The point x = is a point 
of discontinuity, and the sum of the series is zero when x = 0, as it should be. 
More generally the sum of the series (76) is 7t/4 when sin x is positive, ?r/4 
when sin x is negative, and zero when sin x 0. 

The curve represented by the equation (76) is composed of an infinite number 
of segments of length n of the straight lines y = 7T/4 and an infinite num 
ber of isolated points (y = 0, x = kit) on the x axis. 

2) The coefficients of the Fourier development of x in the interval from to 
27T are 

1 r-* 
do = I xdx = 2?r, 

Tt Jo 

1 r 2 " rxsinmx-l 2 * 1 c^ 

dm = -- x cos mxdx = -{ -- I sin mx dx = , 

Tt Jo \_ 11171 Jo mit Jo 

1 r 2 " rxcos?/ixl 2;r 1 (**" 2 

o m = - x sin mx dx = \ -\ --- | cos mx dx = --- 

n Jo \_ mit Jo iwt Jo m 



Hence the formula 

x it sinx sin 2x sin3x 



__ 



22 



is valid for all values of x between and 2?r. If we set y equal to the series on 
the right, the resulting equation represents a curve composed of an infinite num 
ber of segments of straight lines parallel to y x/2 and an infinite number of 
isolated points. 

Note. If the function /(x) defined in the interval ( - Tt, + Tt) is even, that is 
to say, if /( x) =/(x), each of the coefficients b m is zero, since it is evident that 

nO n-n 

I f(x) sin mx dx I /(x) sin mx dx . 

J-n Jo 

Similarly, if /(x) is an odd function, that is, if /( x) = /(x), each of the 
coefficients a m is zero, including a . A function /(x) which is defined only in 



422 SPECIAL SERIES [IX, 199 

the interval from to it may be defined in the interval from it to by either 
of the equations 

/(-*)= /() or /( - z) = - f(x) 

if we choose to do so. Hence the given function /(x) may be represented either 
by a series of cosines or by a series of sines, in the interval from to ic. 

199. Expansion of a continuous function. Weierstrass theorem. Let /(z) be a 
function which is defined and continuous in the interval (a, 6). The following 
remarkable theorem was discovered by Weierstrass : Given any preassigned posi 
tive number e, a polynomial P(x) can always be found such that the difference 
f(x)P(x) is less than e in absolute value for all values of x in the interval (a, 6). 

Among the many proofs of this theorem, that due to Lebesgue is one of the 
simplest.* Let us first consider a special function \j/(x) which is continuous in 
the interval (1, +1) and which is defined as follows : ^(x) = for 1 < x < 0, 
V (x) = 2kx for < x < 1, where fc is a given constant. Then \f/(x) = (x -f | x |) k. 
Moreover for - 1 < x < + 1 we shall have 



and for the same values of x the radical can be developed in a uniformly con 
vergent series arranged according to powers of (1 - x 2 ). It follows that x|, and 
hence also i//(x), may be represented to any desired degree of approximation in 
the interval ( 1, + 1) by a polynomial. 

Let us now consider any function whatever, /(x), which is continuous in 
the interval (a, 6), and let us divide that interval into a suite of subintervals 
(<*oi ai), (a\ , a 2 ), , (a,,_i , a n ), where a = a < a x < a 2 < < a,,_i < a n = 6, 
in such a way that the oscillation of /(x) in any one of the subintervals is less 
than e/2. Let L be the broken line formed by connecting the points of the 
curve y =f(x) whose abscissae are a , a x , a 2 , , b. The ordinate of any point 
on L is evidently a continuous function 0(x), and the difference /(x) - <f>(x) is 
less than e/2 in absolute value. For in the interval (a M _!, a M ), for example, 
we shall have 

/(x) - 0(x) = [/(x) -/(<v_,)] (1 - 6) + [/(x) -/(a M )] 0, 

where x - a M _i = 0(a M - a^_i). Since the factor d is positive and less than 
unity, the absolute value of the difference/ <f> is less than e(l - 6 + e)/2 = e/2. 
The function </>(x) can be split up into a sum of n functions of the same type as 
V (x). For, let A Q , AI, A 2 , , A n be the successive vertices of L. Then </>(x) 
is equal to the continuous function ^ (x) which is represented throughout the 
interval (a, 6) by the straight line A A l extended, plus a function 0i(x) which 
is represented by a broken line A A{ A whose first side A Q A\ lies on the 
x axis and whose other sides are readily constructed from the sides of L. Again, 
the function fa (x) is equal to the sum of two functions ^ 2 and 2 , where f 2 is 
zero between a and cti, and is represented by the straight line A\A i extended 
between aj and 6, while 2 is represented by a broken line A(, Ai A 2 - A n whose 
first three vertices lie on the x axis. Finally, we shall obtain the equation 
= f i + </"2 + + </- , where ^ is a continuous function which vanishes 
between a and o,-_i and which is represented by a segment of a straight line 

* Bulletin des sciences mathtmatiques, p. 278, 1898. 



IX, 200] TRIGONOMETRIC SERIES 423 

between ai_i and b. If we then make the substitution X = mx + n, where m 
and n are suitably chosen numbers, the function \j/i(x) may be defined in the 
interval ( 1, + 1) by the equation 



and hence it can be represented by a polynomial with any desired degree of 
approximation. Since each of the functions f;(x) can be represented in the 
interval (a, 6) by a polynomial with an error less than c/2n, it is evident that the 
sum of these polynomials will differ from /(x) by less than e. 

It follows from the preceding theorem that any function f(x) which is contin 
uous in an interval (a, b) may be represented by an infinite series of polynomials 
which converges uniformly in that interval. For, let i , c 2 , , e , be a sequence 
of positive numbers, each of which is less than the preceding, where e n approaches 
zero as n becomes infinite. By the preceding theorem, corresponding to each of 
the e s a polynomial P(x) can be found such that the difference /(z) P,(x) is 
less than e,- in absolute value throughout the interval (a, b). Then the series 

^1 (3) + t P 2 (X) - Pi (X)] + + [P n (X) - P.-i ()] + 

converges, and its sum is/(x) for any value of x inside the interval (a, b). For 
the sum of the first n terms is equal to P B (x), and the difference /(x) <S M , which 
is less than e,, , approaches zero as n becomes infinite. Moreover the series con 
verges uniformly, since the absolute value of the difference /(x) S n will be less 
than any preassigned positive number for all values of n which exceed a certain 
fixed integer N, when x has any value whatever between a and b. 

200. A continuous function without a derivative. We shall conclude this chapter 
by giving an example due to Weierstrass of a continuous function which does 
not possess a derivative for any value of the variable whatever. Let 6 be a posi 
tive constant less than unity and let a be an odd integer. Then the function 
F(x) defined by the convergent infinite series 



-r ^ 

(78) F(x) = V b" cos (a" nx) 



is continuous for all values of x, since the series converges uniformly in any 
interval whatever. If the product ab is less than unity, the same statements 
hold for the series obtained by term-by-term differentiation. Hence the func 
tion F(x) possesses a derivative which is itself a continuous function. We shall 
now show that the state of affairs is essentially different if the product ab exceeds 
a certain limit. 

In the first place, setting 

m-l 

S m = - ^ b" (cos [a n TT(Z + h)] cos (a n TTX) } , 

H=U 

R, n ^ b" {cos [a" TT(X + h)] cos (a" nx)} , 
we may write 

(79) ffl*Li!fiL>. .+*.. 

h 



SPECIAL SERIES [IX, 200 

On the other hand, it is easy to show, by applying the law of the mean to the 
function cos(a"7rz), that the difference cos[a"7r(z + h)] cos(a n itx) is less than 
na n | h \ in absolute value. Hence the absolute value of S m is less than 

m-l 

a n b n = n 



ab-l 

n=(l 

and consequently also less than 7r(ab) m /(ab 1), if ab >1. Let us try to find a 
lower limit of the absolute value of B m when h is assigned a particular value. 
We shall always have 

a m x = a m + ,, 

where a m is an integer and m lies between 1/2 and + 1/2. If we set 

p ? 
/j _ 5S , 

a" 1 

where e m is equal to 1, it is evident that the sign of h is the same aa that of 
e m , and that the absolute value of h is less than 3/2a m . Having chosen h in this 
way, we shall have 

a"7f(x + h) a n - n a m 7c(x + h) = a n - m n(a m + e m ). 

Since a is odd and e m = 1, the product a n - (a m -f e m ) is even or odd with 
a m + 1, and hence 

cos[a"7r(z + A)] = ( l) a m + l . 
Moreover we shall have 

coa(a n 7tx) = cos(a"- m a m 7rz) = cos[a"- m *(a m + m )] 
= cos (a" - m a m Tt ) cos (a" - " m n) , 

or, since a"~ m a m is even or odd with a m , 

CQS (fin 7f%\ ( 1 ^""* COS {a n m ^ 7T^ 

It follows that we may write 

( _ 1)g , + i * 

"m 

n 

Since every term of the series is positive, its sum is greater than the first term, and 
consequently it is greater than 6 m since , lies between 1/2 and + 1/2. Hence 



or, since \h\< 3/2a m , 

\ 

If a and b satisfy the inequality 

(80) 
we shall have 



(80) a6>l + , 





3 ab-l 

whence, by (79), 



F(x + h)- F(x) 



> |B|-|S n 



2 



3 x a6-l 



IX, Exs.] EXERCISES 425 

As m becomes infinite the expression on the extreme right increases indefinitely, 
while the absolute value of h approaches zero. Consequently, no matter how 
small e be chosen, an increment h can be found which is less than in abso 
lute value, and for which the absolute value of [F(x + h) F(x)~\/h exceeds any 
preassigned number whatever. It follows that if a and b satisfy the relation (80), 
the function F(x) possesses no derivative for any value of x whatever. 

EXERCISES 

1. Apply Lagrange s formula to derive a development in powers of x of that 
root of the equation y 2 = ay + x which is equal to a when x = 0. 

2. Solve the similar problem for the equation y a + xy m + l = 0. Apply the 
result to the quadratic equation a bx + ex 2 = 0. Develop in powers of c that 
root of the quadratic which approaches a/6 as c approaches zero. 

3. Derive the formula 



l+x \ I 

4. Show that the formula 



Vl + x 1 + x 2 \1 + x/ 2.4 
holds whenever x is greater than 1/2. 

5. Show that the equation 



c 
+ 



2 1 + x 2 2 . 4 \1 + x 2 / 2 . 4 . G \1 + x 2 

holds for values of x less than 1 in absolute value. What is the sum of the series 
when | x | > 1 ? 

6. Derive the formula 



1.2 a + * 1-2.3 



7. Show that the branches of the function sinmx and cosmx which reduce 
to and 1, respectively, when sinx = are developable in series according 10 
powers of sin x : 

f . m 2 - 1 . (m 2 - l)(m 2 - 0) . n 

sm mx = m sm x sm s x + sm 5 x , 

L 1.2.3 1.2.3.4.5 J 

?n 2 . m 2 (m 2 -4) . 

cos mx = 1 sin 2 x H sm 4 x . 

1.2 1.2.3.4 

[Make use of the differential equation 

dy 2 dy 
which is satisfied by u cosmx and by u = sinmx, where y = sinx.J 

8. From the preceding formulae deduce developments for the functions 

cos (n arc cos x) , sin (n arc cos x) . 



CHAPTER X 

PLANE CURVES 

The curves and surfaces treated in Analytic Geometry, properly 
speaking, are analytic curves and surfaces. However, the geomet 
rical concepts which we are about to consider involve only the exist 
ence of a certain number of successive derivatives. Thus the curve 
whose equation is y = f(x) possesses a tangent if the function f(x) 
has a derivative / (#) ; it has a radius of curvature if / (#) has a 
derivative /"(cc); and so forth. 

I. ENVELOPES 

201. Determination of envelopes. Given a plane curve C whose 
equation 

(1) f(x, y, a) = 

involves an arbitrary parameter a, the form and the position of the 
curve will vary with a. If each of the positions of the curve C is 
tangent to a fixed curve E, the curve E is called the envelope of the 
curves C, and the curves C are said to be enveloped by E. The 
problem before us is to establish the existence (or non-existence) of 
an envelope for a given family of curves C, and to determine that 
envelope when it does exist. 

Assuming that an envelope E exists, let (x, y] be the point of tan- 
gency of E with that one of the curves C which corresponds to a cer 
tain value a of the parameter. The quantities x and y are unknown 
functions of the parameter a, which satisfy the equation (1). In 
order to determine these functions, let us express the fact that the 
tangents to the two curves E and C coincide for all values of a. 
Let Sx and 8y be two quantities proportional to the direction cosines 
of the tangent to the curve C, and let dx/da and dy/da be the 
derivatives of the unknown functions x = <(), y = $(&) Then a 
necessary condition for tangency is 

dx dy 
da _ da 

C * 

426 



X, 201] ENVELOPES 427 

On the other hand, since a in equation (1) has a constant value for 
the particular curve C considered, we shall have 

(3) sx + ( ^Sy = 0, 

dx dy " 

which determines the tangent to C. Again, the two unknown func 
tions x = <(), y = ^() satisfy the equation 

f(x, y, a) = 0, 
also, where a is now the independent variable. Hence 

cfdx cfdt/ df 

(4) oIT- + aj+a= 

ex da cy da oa 

or, combining the equations (2), (3), and (4), 



The unknown functions x = <(), y = \f/(d) are solutions of this equa 
tion and the equation (1). Hence the equation of the envelope, in 
case an envelope exists, is to be found by eliminating the parameter a 
between the equations f = 0, df/da = 0. 

Let R(x, y) be the equation obtained by eliminating a between 
(1) and (5), and let us try to determine whether or not this equation 
represents an envelope of the given curves. Let C be the particu 
lar curve which corresponds to a value a of the parameter, and let 
(a- , y ) be the coordinates of the point M ti of intersection of the 
two curves 

(6) /(*, y, a ) = 0, *- = <>. 

t/a 

The equations (1) and (5) have, in general, solutions of the form 
x = <f>(a), y = \j/(a), which reduce to x and y , respectively, for 
a a . Hence for a a Q we shall have 



dx \da/o c // Q \da/o 

This equation taken in connection with the equation (3) shows 
that the tangent to the curve C coincides with the tangent to the 
curve described by the point (x, y*), at least unless df/dx and df/dy 
are both zero, that is, unless the point M Q is a singular point for the 
curve C . It follows that the equation R(x, y) = represents either 
the envelope of the curves C or else the locus of singular points on 
these curves, 



428 PLANE CURVES [X, 202 

This result may be supplemented. If each of the curves C has 
one or more singular points, the locus of such points is surely a part 
of the curve R(x, y) = 0. Suppose, for example, that the point (x, y) 
is such a singular point. Then x and y are functions of a which 
satisfy the three equations 

^\ / Q /> 

f(x,y,a) = Q, ^ = 0, ^ = 0, 

and hence also the equation df/da = 0. It follows that x and y 
satisfy the equation R(x, y) = obtained by eliminating a between 
the two equations / = and df/Sa = 0. In the general case the 
curve R(XJ y) = is composed of two analytically distinct parts, 
one of which is the true envelope, while the other is the locus of 
the singular points. 

Example. Let us consider the family of curves 

/(*,*/,) = /-2/ 2 + (* -) 2 = - 
The elimination of a between this equation and the derived equation 



gives y* y 2 = 0, which represents the three straight lines y = 0, 
y jf. l, y 1. The given family of curves may be generated 
by a translation of the curve y* y 2 + x 2 = along the x axis. 
This curve has a double point at the origin, and it is tangent to 
each of the straight lines y = I at the points where it cuts the 
y axis. Hence the straight line y = is the locus of double points, 
whereas the two straight lines y = 1 constitute the real envelope. 

202. If the curves C have an envelope E, any point of the envelope 
is the limiting position of the point of intersection of two curves of 
the family for which the values of the parameter differ by an infini 
tesimal. For, let 

(7) f(x, y, a) = 0, f(x,y,a + h) = 

be the equations of two neighboring curves of the family. The 
equations (7), which determine the points of intersection of the two 
curves, may evidently be replaced by the equivalent system 

f(x, y, a + h)-f(x,y, a) _ 
f(x, y, a) = 0, - 0, 



X, 202] ENVELOPES 429 

the second of which reduces to Sf/da = as h approaches zero, that 
is, as the second of the two curves approaches the first. This prop 
erty is fairly evident geometrically. In Fig. 37, a, for instance, the 
point of intersection N of the two neighboring curves C and C 
approaches the point of tangency M as C approaches the curve C 





FIG. 37, b 

as its limiting position. Likewise, in Fig. 37, b, where the given 
curves (1) are supposed to have double points, the point of intersec 
tion of two neighboring curves C and C approaches the point where 
C cuts the envelope as C approaches C. 

The remark just made explains why the locus of singular points 
is found along with the envelope. For, suppose that f(x, y, a) is a 
polynomial of degree m in a. For any point M Q (x , ?/ ) chosen at 
random in the plane the equation 

(8) /(.r , 7,0,0 = 

will have, in general, m distinct roots. Through such a point there 
pass, in general, m different curves of the given family. But if the 
point M lies on the curve R (or, y) = 0, the equations 



are satisfied simultaneously, and the equation (8) has a double root. 
The equation R(x, y) = may therefore be said to represent the 
locus of those points in the plane for which two of the curves of 
the given family which pass through it have merged into a single 
one. The figures 37, a, and 37, b, show clearly the manner in which 
two of the curves through a given point merge into a single one as 
that point approaches a point of the curve R(x, y) = 0, whether on 
the true envelope or on a locus of double points. 



430 PLANE CURVES [X, 203 

Note. It often becomes necessary to find the envelope of a family 
of curves 

(9) F(x, y, a, ft) = 

whose equation involves two variable parameters a and b, which 
themselves satisfy a relation of the form <f>(a, ft) = 0. This case 
does not differ essentially from the preceding general case, however, 
for ft may be thought of as a function of a defined by the equation 
< = 0. By the rule obtained above, we should join with the given 
equation the equation obtained by equating to zero the derivative 
of its left-hand member with respect to a : 

dp dFdt __ n 

~0 I 07 7 - " 

ca co da 
But from the relation <f>(a, ft) = we have also 

d(fr d<f> db _ 
da db da 

whence, eliminating db/da, we obtain the equation 



_ 

da db db da 

which, together with the equations F = and <f> = 0, determine the 
required envelope. The parameters a and ft may be eliminated 
between these three equations if desired. 

203. Envelope of a straight line. As an example let us consider the equation 
of a straight line D in normal form 

(11) zcosa + y sin a f(a) 0, 

where the variable parameter is the angle a. Differentiating the left-hand side 
with respect to this parameter, we find as the second equation 

(12) xsinct + y cos a f(a) = 0. 

These two equations (11) and (12) determine the point of intersection of any 
one of the family (11) with the envelope E in the form 

(x=f(a)cosa -f (a) sin a, 
\ y = f(a) sin a + f (a) cos a . 

It is easy to show that the tangent to the envelope E which is described by this 
point (z, y) is precisely the line D. For from the equations (13) we find 

( dx = - [/() + /"()] sin a da , 
I dy = [/(a) + /"(a)] cos a da , 

whence dy/dx = cot a , which is precisely the slope of the line D. 



X, 203] 



ENVELOPES 



431 



Moreover, if s denote the length of the arc of the envelope from any fixed 
point upon it, we have, from (14), 



ds = 



whence 



+ dy* = [/(a) + /"(a)] da- , 
= [//(or) da +/ (<*)]. 



Hence the envelope will be a curve which is easily rectifiable if we merely choose 
for /(a) the derivative of a known function.* 

As an example let us set f(a) = I sin a cos a. Taking y = and x = suc 
cessively in the equation (11), we find (Fig. 38) OA = I sin a, OB = Zcosa, 
respectively ; hence AB = I. The required 
curve is therefore the envelope of a straight 
line of constant length Z, whose extremities 
always lie on the two axes. The formulae 
(13) give in this case 

x = I sin 3 a , y = I cos 3 a- , 
and the equation of the envelope is 
-A* AA? 

if*- 1 

which represents a hypocycloid with four 
cusps, of the form indicated in the figure. 
As a varies from to 7T/2, the point of con 
tact describes the arc DC. Hence the length of the arc, counted from D, is 

s= I 3 1 sin a cos a da = sin 2 a. 
Jo 2 

Let I be the fourth vertex of the rectangle determined by OA and OB, and M 
the foot of the perpendicular let fall from / upon AB. Then, from the tri 
angles AMI and APM, we find, successively, 

AM = AI cos a = I cos 2 a , AP = AM sin a = I cos 2 a sin a . 

Hence OP = OA AP = I sin 3 a, and the point M is the point of tangency of 
the line AB with the envelope. Moreover 

BM = l-AM=lsm*a; 
hence the length of the arc DM = 3BM/2. 




p IG 



* Each of the quantities which occur in the formula for s, s = f (a) + ff(a) daf, 
has a geometrical meaning : a is the angle between the x axis and the perpendicular 
ON let fall upon the variable line from the origin ; f(<x) is the distance ON from the 
origin to the variable line; and / (<*) is, except for sign, the distance MN from 
the point M where the variable line touches its envelope to the foot N of the perpen 
dicular let fall upon the line from the origin. The formula for 5 is often called 
Legendre s formula. 



432 



PL AXE CURVES 



[X, 204 



204. Envelope of a circle. Let us consider the family of circles 

where a, 6, and p are functions of a variable parameter t. The points where a 
circle of this family touches the envelope are the points of intersection of the 
circle and the straight line 

(16) (x - a) a + (y - b) b + pp = . 

This straight line is perpendicular to the tangent M T to the curve C described 
by the center (a, 6) of the variable circle (15), and its distance from the center is 

p dp/ds, where s denotes the length of 
y the arc of the curve C measured from 

some fixed point. Consequently, if the 
line (16) meets the circle in the two 
points N and N , the chord NN is 
/<j[ bisected by the tangent M T at right 
/ angles. It follows that the envelope 
consists of two parts, which are, in 

general, branches of the same analytic 
curve. Let us now consider several 
special cases. 

1) If p is constant, the chord of con 
tact NN reduces to the normal PP to 
the curve C, and the envelope is com 
posed of the two parallel curves C\ and 
C{ which are obtained by laying off the constant distance p along the normal, 
on either side of the curve C. 

2) If p = s + K, we have p dp/ds = p, and the chord NN reduces to the tan 
gent to the circle at the point Q. The two portions of the envelope are merged 
into a single curve T, whose normals are tangents to the curve C. The curve C 
is called the evolute of T, and, conversely, T is called an involute of C (see 206), 
If dp > ds, the straight line (16) no longer cuts the circle, and the envelope is 
imaginary. 

Secondary caustics. Let us suppose that 
the radius of the variable circle is propor 
tional to the distance from the center to a 
fixed point 0. Taking the fixed point O as 
the origin of coordinates, the equation of the 
circle becomes 

where A; is a constant factor, and the equation 
of the chord of contact is 




FIG. 39 




FIG. 40 



E 
(x - a) a + (y - b) b + k*(aa + bb ) = 0. 

If 8 and 8 denote the distances from the 

center of the circle to the chord of contact and to the parallel to it through the 
origin, respectively, the preceding equation shows that 5 = k 2 d . Let P be a 
point on the radius MO (Fig. 40), such that MP = k*MO, and let C be the 



X, - 05] 



CURVATURE 433 



locus of the center. Then the equation just found shows that the chord of con 
tact is the perpendicular let fall from P upon the tangent to C at the center M. 
Let us suppose that k is less than unity, and let E denote that branch of the 
envelope which lies on the same side of the tangent MT as does the point 0. 
Let i and r, respectively, denote the two angles which the two lines MO and 
MN make with the normal MI to the curve C. Then we shall have 

. _ M q _ Mp sin i _ Mq _ MQ _ 1 

= ~ ~~ M~ MP~ k 



Now let us imagine that the point is a source of light, and that the curve C 
separates a certain homogeneous medium in which O lies from another medium 
whose index of refraction with respect to the first is l/k. After refraction the 
incident ray OM will be turned into a refracted ray MR, which, by the law of 
refraction, is the extension of the line NM. Hence all the refracted rays MR 
are normal to the envelope, which is called the secondary caustic of refraction. 
The true caustic, that is, the envelope of the refracted rays, is the evolute of the 
secondary caustic. 

The second branch E of the envelope evidently has no physical meaning ; 
it would correspond to a negative index of refraction. If we set k 1, the 
envelope E reduces to the single point 0, while the portion E becomes the locus 
of the points situated symmetrically with with respect to the tangents to C. 
This portion of the envelope is also the secondary caustic of reflection for inci 
dent rays reflected from C which issue from the fixed point 0. It may be shown 
in a manner similar to the above that if a circle be described about each point of 
C with a radius proportional to the distance from its center to a fixed straight 
line, the envelope of the family will be a secondary caustic with respect to a 
system of parallel rays. 

II. CURVATURE 

205. Radius of curvature. The first idea of curvature is that the 
curvature of one curve is greater than that of another if it recedes 
more rapidly from its tangent. In order to render this somewhat 
vague idea precise, let us first consider the case of a circle. Its 
curvature increases as its radius diminishes ; it is therefore quite 
natural to select as the measure of its curvature the simplest func 
tion of the radius which increases as the radius diminishes, that 
is, the reciprocal l/R of the radius. Let AB be an arc of a circle 
of radius R which subtends an angle o at the center. The angle 
between the tangents at the extremities of the arc AB is also <o, and 
the length of the arc is s = R<a. Hence the measure of the curva 
ture of the circle is w/s. This last definition may be extended to 
an arc of any curve. Let AB be an arc of a plane curve without a 
point of inflection, and w the angle between the tangents at the 
extremities of the arc, the directions of the tangents being taken 
in the same sense according to some rule, the direction from A 



434 



PLANE CURVES 



[X,205 




FIG. 41 



toward B, for instance. Then the quotient w/arc AB is called the 
average curvature of the arc AB. As the point B approaches the 
point A this quotient in general approaches a limit, which is called 

the curvature at the point A. The 
radius of curvature at the point A is 
defined to be the radius of the circle 
which would have the same curvature 
which the given curve has at the point 
A ; it is therefore equal to the recipro 
cal of the curvature. Let s be the 
length of the arc of the given curve 
measured from some fixed point, and 
a the angle between the tangent and 
some fixed direction, the x axis, for example. Then it is clear 
that the average curvature of the arc AB is equal to the absolute 
value of the quotient Aa/A.s ; hence the radius of curvature is given 
by the formula 

, ,. As ds 

R = lim = - 
Aa art 

Let us suppose the equation of the given curve to be solved for y 
in the form y =f(x). Then we shall have 

y"dx 



a = arc tan y , da = y _ " . ds = 
and hence 

( 17 > *= y,, 

Since the radius of curvature is essentially positive, the sign 
indicates that we are to take the absolute value of the expression 
on the right. If a length equal to the radius of curvature be laid 
off from A upon the normal to the given curve on the side toward 
which the curve is concave, the extremity / is called the center of 
curvature. The circle described about 7 as center with R as radius 
is called the circle of curvature. The coordinates (x , y ) of the 
center of curvature satisfy the two equations 



which express the fact that the point lies on the normal at a dis 
tance R from A. From these equations we find, on eliminating x l} 



X, 205] 



CURVATURE 435 



In order to tell which sign should be taken, let us note that if y" is 
positive, as in Fig. 41, y^ y must be positive ; hence the positive 
sign should be taken in this case. If y" is negative, y y is nega 
tive, and the positive sign should be taken in this case also. The 
coordinates of the center of curvature are therefore given by the 
formulae 

1 + y 2 , 1 + y 2 

(18) yi-9*=jf-> **-*=- y fT- 

When the coordinates of a point (x, y) of the variable curve are 
given as functions of a variable parameter t, we have, by 33, 



and the formulae (17) and (18) become 

jz _ (dx* + dy*? 

. j ** y 

} dy(dx* + dif~) 



&x i dxd 2 y-dyd*x 

At a point of inflection y" = 0, and the radius of curvature is 
infinite. At a cusp of the first kind y can be developed according 
to powers of x l/2 in a series which begins with a term in x ; hence 
y has a finite value, but y" is infinite, and therefore the radius of 
curvature is zero. 

Note. When the coordinates are expressed as functions of the arc s of the 

curve, 

x = <P(s) , y = f (a) , 

the functions <f> and f satisfy the relation 

2(S) + V /2 (S)=1, 

since dx 2 + dy 2 = da 2 , and hence they also satisfy the relation 

< (/>" + -fy ^i" = . 
Solving these equations for < and \f/ , we find 



where e = 1, and the formula for the radius of curvature takes on the espe 
cially elegant form 

(20) s[* 



436 PLANE CURVES [X, 206 

206. Evolutes. The center of curvature at any point is the limit 
ing position of the point of intersection of the normal at that point 
with a second normal which approaches the first one as its limiting 
position. For the equation of the normal is 



where X and Y are the running coordinates. In order to find the 
limiting position of the point of intersection of this normal with 
another which approaches it, we must solve this equation simulta 
neously with the equation obtained by equating the derivative of the 
left-hand side with respect to the variable parameter x, i.e. 



The value of Y found from this equation is precisely the ordinate 
of the center of curvature, which proves the proposition. It follows 
that the locus of the center of curvature is the envelope of the 
normals of the given curve, i.e. its evolute. 

Before entering upon a more precise discussion of the relations 
between a given curve and its evolute, we shall explain certain con 
ventions. Counting the length of the arc of the given curve in a 
definite sense from a fixed point as origin, and denoting by a the 
angle between the positive direction of the x axis and the direction 
of the tangent which corresponds to increasing values of the arc, 
we shall have tan a = y , and therefore 



cos a 



On the right the sign -f- should be taken, for if x and s increase 
simultaneously, the angle a is acute, whereas if one of the varia 
bles x and s increases as the other decreases, the angle is obtuse 
( 81). Likewise, if (3 denote the angle between the y axis and the 
tangent, cos (3 = dy/ds. The two formulae may then be written 

dx dy 

COS or = > Sin a = ~ -j 

as as 

where the angle a is counted as in Trigonometry. 

On the other hand, if there be no point of inflection upon the 
given arc, the positive sense on the curve may be chosen in such a 
way that s and a increase simultaneously, in which case R = ds/da 
all along the arc. Then it is easily seen by examining the two 
possible cases in an actual figure that the direction of the segment 



X, 206] 



CURVATURE 



437 



starting at the point of the curve and going to the center of curva 
ture makes an angle a 1 = a -f 7r/2 with the x axis. The coordinates 
( X L > y\) f the center of curvature are therefore given by the formulae 

x l = x + R cos ( a + } = x R sin a, 

\ "/ 

(\ 
a + \ = y + R cos a, 



whence we find 

cfaj = cos ads R cos a da sin a dR = sin a cR, 
efo/! = sin ads R sin a tZa + cos a dR = cos a rf7? . 

In the first place, these formulas show that dy l /dx l = cot a, which 
proves that the tangent to the evolute is the normal to the given 
curve, as we have already seen. Moreover 

ds\ = dx\ + dy\ = dR* , 

or dsi = dR. Let us suppose for definiteness that the radius 
of curvature constantly increases as we proceed along the curve C 
(Fig. 42) from M l to Jl/ 2 , and let us choose the positive sense of 
motion upon the evolute (D) in such a way 
that the arc s 1 of (Z)) increases simultane 
ously with R. Then the preceding formula 
becomes d$i = dR, whence s t = R + C. It 
follows that the arc /!/ 2 = R 2 R 1} and we 
see that the length of any arc of the evolute 
is equal to the difference between the tivo 
radii of curvature of the curve C which cor 
respond to the extremities of that arc. 

This property enables us to construct the 
involute C mechanically if the evolute (Z>) be 
given. If a string be attached to (D) at an 
arbitrary point A and rolled around (D) to / 2 , thence following the 
tangent to M t , the point M 2 will describe the involute C as the 
string, now held taut, is wound further on round (Z>). This con 
struction may be stated as follows : On each of the tangents IM of 
the evolute lay off a distance IM = I, where I + s = const., s being 
the length of the arc AI of the evolute. Assigning various values 
to the constant in question, an infinite number of involutes may be 
drawn, all of which are obtainable from any one of them by laying 
off constant lengths along the normals. 




M 



FIG. 42 



438 



PLANE CURVES 



[X, 207 



All of these properties may be deduced from the general formula 
for the differential of the length of a straight line segment ( 82) 

dl = dar 1 COS (D! (Zcr 2 COS o>. 2 . 

If the segment is tangent to the curve described by one of its 
extremities and normal to that described by the other, we may set 
<0l TT, w. 2 = 7T/2, and the formula becomes dl da^ = 0. If the 
straight line is normal to one of the two curves and Z is constant, 
dl = 0, cos <DI = 0, and therefore cos o> 2 = 0. 

The theorem stated above regarding the arc of the evolute depends 
essentially upon the assumption that the radius of curvature con 
stantly increases (or decreases) along the whole arc considered. If 
this condition is not satisfied, the statement of the theorem must 
be altered. In the first place, if the radius of curvature is a maxi 
mum or a minimum at any point, dR = at that point, and hence 
dxi = dy l = 0. Such a point is a cusp on 
the evolute. If, for example, the radius 
of curvature is a maximum at the point M 
(Fig. 43), we shall have 

arc//! = IM A-!/!, 
arc 77, = IM - 7 2 3/ a , 




whence 



arc /! 7/ 2 = 2 737 



- 7 2 



Hence the difference 7 t M 7 2 M 2 is equal 
to the difference between the two arcs Hi and 77 2 and not their sum. 

207. Cycloid. The cycloid is the path of a point upon the circumference of a 
circle which rolls without slipping on a fixed straight line. Let us take the 




n 



Fio. 44 

fixed line as the x axis and locate the origin at a point where the point chosen on 
the circle lies in the x axis. When the circle has rolled to the point 7 (Fig. 44) 
the point on the circumference which was at has come into the position M, 



X, 207] CURVATURE 439 

where the cirenlar arc IM is equal to the segment OI. Let us take the angle 
between the radii CM and CI as the variable parameter. Then the coordinates 
of the point M are 

x = 01 - IP = R(f> - E sin 0, y - MP = 1C + CQ = R - R cos 0, 

where P and Q are the projections of M on the two lines 01 and IT, respec 
tively. It is easy to show that these formulae hold for any value of the angle 0. 
In one complete revolution the point whose path is sought describes the arc 
OBO\. If the motion be continued indefinitely, we obtain an infinite number 
of arcs congruent to this one. From the preceding formulae we find 

x = R(<}> sin <p) , dx = R(l cos <(>) d<f> , d-x = Rsin0d0 2 , 
y = R(l cos 0) , dy = R sin d0 , d*y = .Rcos0 d0 2 , 

and the slope of the tangent is seen to be 

dy sin ^ 

~ = cot i 

dx 1 cos 2 

which shows that the tangent at M is the straight line M T, since the angle 
MTC = 0/2, the triangle MTC being isosceles. Hence the normal at M is the 
straight line MI through the point of tangency I of the fixed straight line with 
the moving circle. For the length of the arc of the cycloid we find 

(Zs 2 = E 2 d0 2 [sin 2 + (1 - cos 0) 2 ] = 4R 2 sin 2 - d0 2 or ds = 2R sin ~ d0 , 

2 2i 

if the arc be counted in the sense in which it increases with 0. Hence, counting 
the arc from the point as origin, we shall have 



= 4fi( 1-cos-Y 
V 2/ 



Setting <f> = 2it, we find that the length of one whole section OBO t is 8R. The 
length of the arc OMB from the origin to the maximum B is therefore 4.R, and 
the length of the arc BM (Fig. 44) is 4R cos 0/2. From the triangle MTC the 
length of the segment MT is 2R cos 0/2 ; hence arc BM 2MT. 
Again, the area up to the ordinate through M is 

A= f ydx= C E 2 (l-2cos0 + cos 2 0)d0 
Jo Jo 



/3 sin 20\ . 

A= -0-2sm0 + - ^)# 2 . 

V" 4 / 

Hence the area bounded by the whole arc OBOi and the base OOi is 37T.K 2 , that 
is, three times the area of the generating circle. (GALILEO.) 

The formula for the radius of curvature of a plane curve gives for the cycloid 



440 



PLANE CURVES 



[X, 208 



On the other hand, from the triangle MCI, MI = 2R sin 0/2. Hence p = 23fl, 
and the center of curvature may be found by extending the straight line MI 
past I by its own length. This fact enables us to determine the evolute easily. 
For, consider the circle which is symmetrical to the generating circle with 
respect to the point I. Then the point M where the line MI cuts this second 
circle is evidently the center of curvature, since M l = MI. But we have 

arc T M = TtR - arc IM = TtR - arc IM = nE - 01, 
or 

arc T M = OH - OI = IH = T E . 

Hence the point M describes a cycloid which is congruent to the first one, the 
cusp being at B and the maximum at O. As the point M describes the arc 
BOi, the point M describes a second arc B Oi which is symmetrical to the arc 
OB already described, with respect to BB . 

208. Catenary. The catenary is the plane curve whose equation with respect 
to a suitably chosen set of rectangular axes is 



Its appearance is indicated by the arc MAM in the figure (Fig. 45). 




From (21) we find 



FIG. 45 



4 a 2 

If denote the angle which the tangent TM makes with the x axis, the formula 
for y gives 

X X 

_ e a e~ a 2 a 

- , cos0 = = - 
e a + e a e n + e " 

The radius of curvature is given by the formula 

y" a 

But, in the triangle MPN, MP = MNcostf,- hence 



COS 



X, 209] CURVATURE 441 

It follows that the radius of curvature of the catenary is equal to the length of 
the normal MN. The evolute may be found without difficulty from this fact. 
The length of the arc AM of the catenary is given by the formula 



px x _y 

/ e a + e , a I * _ *\ 

dx = - ( e a e 1 . 

Jo 2 2\ / 



or s = y sin <f>. If a perpendicular Pm be dropped from P (Fig. 46) upon the 
tangent MT, we find, from the triangle PMm, 

Mm = MP sin <f> = s . 
Hence the arc AM is equal to the distance Mm. 

209. Tractrix. The curve described by the point m (Fig. 45) is called the 
tractrix. It is an involute of the catenary and has a cusp at the point A. The 
length of the tangent to the tractrix is the distance mP. But, in the triangle 
MPm, mP = y cos</> = a ; hence the length mP measured along the tangent to 
the tractrix from the point of tangency to the x axis is constant and equal to a. 
The tractrix is the only curve which has this property. 

Moreover, in the triangle M TP, Mm x mT = a 2 . Hence the product of the 
radius of curvature and the normal is a constant for the tractrix. This property 
is shared, however, by an infinite number of other plane curves. 

The coordinates (xi, y\) of the point TO are given by the formulae 

e u _ e ~a 

x\ = x s cos (f> = x a 




or, setting e x / a = tan 0/2, the equations of the tractrix are 
(22) xi a cos 6 + a log (tan- 



As the parameter 8 varies from n/2 to n, the point (x l , y v ) describes the arc 
Amn, approaching the x axis as asymptote. As varies from n/2 to 0, the 
point (xi , yi) describes the arc Am n , symmetrical to the first with respect to 
the y axis. The arcs Amn and Am n correspond, respectively, to the arcs AM 
and AM of the catenary. 

210. Intrinsic equation. Let us try to determine the equation of a plane 
curve when the radius of curvature R is given as a function of the arc s, 
E = <f>(s). Let a be the angle between the tangent and the x axis ; then 
R = ds/da, and therefore 



. ds . ds 

da = = 



A first integration gives 



/ 

/ 



ds 

a = tt | - . 
*() 



442 PLANE CURVES [X, 210 

and two further integrations give x and y in the form 

x X Q + I cos a ds , y y + f sin a ds . 
J *o J *o 

The curves defined by these equations depend upon the three arbitrary con 
stants x , 2/o, and a . But if we disregard the position of the curve and think 
only of its form, we have in reality merely a single curve. For, if we first con 
sider the curve C defined by the equations 



the general formulae may be written in the form 
x = x + X cos O-Q 
y = 2/0 + X s i n <*o+ Y cos cr 

if the positive sign be taken. These last formulae define simply a transforma 
tion to a new set of axes. If the negative sign be selected, the curve obtained 
is symmetrical to the curve C with respect to the X axis. A plane curve is 
therefore completely determined, in so far as its form is concerned, if its radius 
of curvature be known as a function of the arc. The equation B #(s) is 
called the intrinsic equation of the curve. More generally, if a relation between 
any two of the quantities B, s, and a be given, the curve is completely deter 
mined in form, and the expressions for the coordinates of any point upon it 
may be obtained by simple quadratures. 

For example, if B be known as a function of a, B =/(), we first find 

ds = /(a) da, and then 

dx = cos a f (a) da, 

dy = sin af(a) da , 

whence x and y may be found by quadratures. If J? is a constant, for instance, 
these formulae give 

x = X Q + B sin a , y = y B cos a , 

and the required curve is a circle of radius B. This result is otherwise evident 
from the consideration of the evolute of the required curve, which must reduce 
to a single point, since the length of its arc is identically zero. 

As another example let us try to find a plane curve whose radius of curva 
ture is proportional to the reciprocal of the arc, B = a?/s. The formulas give 



and then 



r s s ds _ i 

= I * * 



Although these integrals cannot be evaluated in explicit form, it is easy to gain 
an idea of the appearance of the curve. As s increases from to + cc, x and y 
each pass through an infinite number of maxima and minima, and they approach 
the same finite limit. Hence the curve has a spiral form and approaches 
asymptotically a certain point on the line y = x. 




X,2ll] CONTACT OSCULATION 443 

III. CONTACT OF PLANE CURVES 

211. Order of contact. Let C and C be two plane curves which 
are tangent at some point A. To every point m on C let us assign, 
according to any arbitrary law whatever, a point m on C , the only 
requirement being that the point m 
should approach A with m. Taking 
the a.TcAm or, what amounts to 
the same thing, the chord Am as 
the principal infinitesimal, let us first 
investigate what law of correspond 
ence will make the order of the infin 
itesimal mm with respect to Am as 
large as possible. Let the two curves p 

be referred to a system of rectangular 

or oblique cartesian coordinates, the axis of y not being parallel to the 
common tangent AT. Let 

(C) </=/(*), 

(C") Y=F(x) 

be the equations of the two curves, respectively, and let (o , y ) be 
the coordinates of the point A. Then the coordinates of m will 
be [a- -f h, f(x -f A)], and those of m will be [# + k, F(x Q + &)], 
where k is a function of h which defines the correspondence between 
the two curves and which approaches zero with h. 

The principal infinitesimal Am may be replaced by h = ap, for 
the ratio ap/Am approaches a finite limit different from zero as the 
point m approaches the point A. Let us now suppose that mm is 
an infinitesimal of order r -f 1 with respect to h, for a certain 
method of correspondence. Then mm is of order 2r + 2. If 6 
denote the angle between the axes, we shall have 



mm 1 = [F(x + jfe) /(* + *) + (k- h) cos OJ + (k A) 2 sin 2 6 ; 

hence each of the differences k h and F(x + k) f(x + A) must 
be an infinitesimal of order not less than r -f 1, that is, 

k = h -f a h,* + l , F(x + k) -f(x + A) = ph +\ 

where a and /3 are functions of h which approach finite limits as A 
approaches zero. The second of these formulae may be written in 
the form 

F(x + h + a/t^ 1 ) -f(x + A) = fih * 1 . 



PLANE CURVES [X,2ii 

If the expression F(x + h + ah r+l ) be developed in powers of a, 
the terms which contain a form an infinitesimal of order not less 
than r -f 1. Hence the difference 

& = F(x Q + h)-f(x + h) 

is an infinitesimal whose order is not less than r + 1 and may exceed 
r + 1. But this difference A is equal to the distance mn between 
the two points in which the curves C and C" are cut by a parallel 
to the y axis through ra. Since the order of the infinitesimal mm 
is increased or else unaltered by replacing m by n, it follows that 
the distance between two corresponding points on the two curves is an 
infinitesimal of the greatest possible order if the two corresponding 
points always lie on a parallel to the y axis. If this greatest possi 
ble order is r + 1, the two curves are said to have contact of order r 
at the point A. 

Notes. This definition gives rise to several remarks. The y axis 
was any line whatever not parallel to the tangent A T. Hence, in 
order to find the order of contact, corresponding points on the two 
curves may be defined to be those in which the curves are cut by 
lines parallel to any fixed line D which is not parallel to the tan 
gent at their common point. The preceding argument shows that 
the order of the infinitesimal obtained is independent of the direc 
tion of 7), a conclusion which is easily verified. Let mn and mm 
be any two lines through a point m of the curve C which are not 
parallel to the common tangent (Fig. 46). Then, from the triangle 
mm n, 

mm sin 



mn sin mm n 

As the point m approaches the point A, the angles mnm and mm n 
approach limits neither of which is zero or TT, since the chord m n 
approaches the tangent AT. Hence mm /mn approaches a finite 
limit different from zero, and mm is an infinitesimal of the same 
order as mn. The same reasoning shows that mm cannot be of 
higher order than mn, no matter what construction of this kind is 
used to determine m from m, for the numerator sin mnm always 
approaches a finite limit different from zero. 

The principal infinitesimal used above was the arc Am or the 
chord Am. We should obtain the same results by taking the arc 
An of the curve C for the principal infinitesimal, since Am and An 
are infinitesimals of the same order. 



X, 212] CONTACT OSCULATION 445 

If two curves C and C have a contact of order r, the points m 
on C may be assigned to the points m on C in an infinite number 
of ways which will make mm 1 an infinitesimal of order r + 1, for 
that purpose it is sufficient to set k = h + ah a + l , where s^r and 
where a is a function of h which remains finite for h = 0. On the 
other hand, if s < r, the order of mm cannot exceed 5 -f 1. 

212. Analytic method. It follows from the preceding section that 
the order of contact of two curves C and C is given by evaluating 
the order of the infinitesimal 



with respect to h. Since the two curves are tangent at A, 
F(x^) = /() and F (x ) =f (x ). It may happen that others of the 
derivatives are equal at the same point, and we shall suppose for 
the sake of generality that this is true of the first n derivatives : 

(23} \ F(x )=f(x o) , F (* ) =/ (*), 

I F"(* ) =/"(*), -., F^(x ) =/>(x ), 

but that the next derivatives -F (n + 1) (cc ) and / (n + 1) (a- ) are unequal. 
Applying Taylor s series to each of the functions F(x) and f(x), we 
find 



y = 



or, subtracting, 

(24) F ~ y = 1.2.^( 

where c and e are infinitesimals. It follows that the order of contact 
of two curves is equal to the order n of the highest derivatives of F(x) 
and f(x) which are equal for x = x . 

The conditions (23), which are due to Lagrange, are the necessary 
and sufficient conditions that x x should be a multiple root of 
order n + 1 of the equation F(z) =/(<r). But the roots of this 
equation are the abscissae of the points of intersection of the two 



446 PLANE CURVES [X, 212 

curves C and C" ; hence it may be said that two curves which have 
contact of order n have n + 1 coincident points of intersection. 

The equation (24) shows that F y changes sign with h if n is 
even, and that it does not if n is odd. Hence curves which have 
contact of odd order do not cross, but curves which have contact of 
even order do cross at their point of tangency. It is easy to see why 
this should be true. Let us consider for definiteness a curve C 
which cuts another curve C in three points near the point A. If 
the curve C" be deformed continuously in such a way that each of 
the three points of intersection approaches A, the limiting position 
of C has contact of the second order with (7, and a figure shows that 
the two curves cross at the point A. This argument is evidently 
general. 

If the equations of the two curves are not solved with respect to 
Y and y, which is the case in general, the ordinary rules for the 
calculation of the derivatives in question enable us to write down 
the necessary conditions that the curves should have contact of 
order n. The problem is therefore free from any particular diffi 
culties. We shall examine only a few special cases which arise 
frequently. First let us suppose that the equations of each of the 
curves are given in terms of an auxiliary variable 



(X = f(u), 

(c) " A 



and that ^( ) = <K^o) an( ^ A (^o) = < (Xo)> i- e - that the curves are tan 
gent at a point A whose coordinates are f(t ), <f>(t(,)- Iff (to) w not 
zero, as we shall suppose, the common tangent is not parallel to the 
y axis, and we may obtain the points of the two curves which have 
the same abscissae by setting u = t. On the other hand, x x is of 
the first order with respect to t t , and we are led to evaluate the 
order of i/f() <() with respect to t t . In order that the two 
curves have at least contact of order n, it is necessary and sufficient 
that we should have 

(25) ,K* ) = <X*o) , * (<o) = * (<b) , , * (*o) - * Co) , 

and the order of contact will not exceed n if the next derivatives 
<A (n + 1) (*o) and 4> (n + 1) (*o) are unequal. 

Again, consider the case where the curve C is represented by the 
two equations 

(26) *=/(0, y = 



X, 212] 



CONTACT OSCULATION 



447 



and the curve C by the single equation F(x, y} = 0. This case may 
be reduced to the preceding by replacing x in F(x, y) by /() and 
considering the implicit function y ^(t) defined by the equation 

(27) F[/(0, KO] = o. 

Then the curve C 1 is also represented by two equations of the form 

(28) *=/(*)> y = ^CO- 
In order that the curves C and C should have contact of order n at 
a point A which corresponds to a value t n of the parameter, it is 
necessary that the conditions (25) should be satisfied. But the 
successive derivatives of the implicit function i/r() are given by the 
equations 



(29) 



[/ ?+ 2 



Hence necessary conditions for contact of order n will be obtained 
by inserting in these equations the relations 



t = t , x ==y( ), i/^o) == <(^o 

The resulting conditions may be expressed as follows : 
Let 



e ^wo given curves will have at least contact of order n if and 
only if 

(30) F(<o) = 0, F(*o) = 0, ..-, R")(^) = 0. 

The roots of the equation F() = are the values of t which cor 
respond to points of intersection of the two given curves. Hence 
the preceding conditions amount to saying that t = t is a multiple 
root of order n, i.e. that the two curves have n + 1 coincident points 
of intersection. 



PL AXE CURVES [X,213 

213. Osculating curves. Given a fixed curve C and another curve 
C which depends upon n -f- 1 parameters a, b, c, , I, 

(31) F(x, y, a, b, c, , t) = 0, 

it is possible in general to choose these n + 1 parameters in such a 
way that C and C shall have contact of order n at any preassigned 
point of C. For, let C be given by the equations x =/(), y = <KO- 
Then the conditions that the curves C and C should have contact 
of order n at the point where t = t Q are given by the equations (30), 
where 

F(0 = 



If t be given, these n +1 equations determine in general the n +1 
parameters a, b, c, -, /. The curve C obtained in this way is 
called an osculating curve to the curve C. 

Let us apply this theory to the simpler classes of curves. The 
equation of a straight line y = ax -+ b depends upon the two param 
eters a and b ; the corresponding osculating straight lines will have 
contact of the first order. If y =f(x) is the equation of the curve C, 
the parameters a and b must satisfy the two equations 

A x o) = ax + b, f (x ) = a ; 

hence the osculating line is the ordinary tangent, as we should 
expect. 

The equation of a circle 

(32) (x - a) 2 + (y - by - R* = 

depends upon the three parameters a, b, and R ; hence the corre 
sponding osculating circles will have contact of the second order. 
Let y = f(x) be the equation of the given curve C ; we shall obtain 
the correct values of a, b, and R by requiring that the circle should 
meet this curve in three coincident points. This gives, besides the 
equation (32), the two equations 

(33) x -a + (y- b}y = 0, 1+ y 2 + (y - V)y" = 0. 

The values of a and b found from the equations (33) are precisely 
the coordinates of the center of curvature ( 205) ; hence the oscu 
lating circle coincides with the circle of curvature. Since the con 
tact is in general of order two, we may conclude that in general the 
circle of curvature of a plane curve crosses the curve at their point 
of tangency. 



X,2i:<] CONTACT OSCULATION 449 

All the above results might have been foreseen a priori. For, 
since the coordinates of the center of curvature depend only on 
x, y, y\ and y", any two curves which have contact of the second 
order have the same center of curvature. But the center of curva 
ture of the osculating circle is evidently the center of that circle 
itself; hence the circle of curvature must coincide with the oscu 
lating circle. On the other hand, let us consider two circles of 
curvature near each other. The difference between their radii, 
which is equal to the arc of the evolute between the two centers, 
is greater than the distance between the centers ; hence one of 
the two circles must lie wholly inside the other, which could not 
happen if both of them lay wholly on one side of the curve C in 
the neighborhood of the point of contact. It follows that they 
cross the curve C. 

There are, however, on any plane curve, in general, certain points 
at which the osculating circle does not cross the curve ; this excep 
tion to the rule is, in fact, typical. Given a curve C which depends 
upon n + 1 parameters, we may add to the n + 1 equations (30) the 
new equation 



provided that we regard t as one of the unknown quantities and 
determine it at the same time that we determine the parameters 
a, b, c, , I. It follows that there are, in general, on any plane 
curve C, a certain number of points at which the order of con 
tact with the osculating curve C 1 is n -f 1. For example, there are 
usually points at which the tangent has contact of the second order ; 
these are the points of inflection, for which y" = 0. In order to find 
the points at which the osculating circle has contact of the third 
order, the last of equations (33) must be differentiated again, which 
gives 



or finally, eliminating y b, 

(34) (i+yv-*yY"-o. 

The points which satisfy this last condition are those for which 
dR/dx = 0, i.e. those at which the radius of curvature is a maxi 
mum or a minimum. On the ellipse, for example, these points are 
the vertices ; on the cycloid they are the points at which the tan 
gent is parallel to the base. 



450 PLANE CURVES [X,214 

214. Osculating curves as limiting curves. It is evident that an 
osculating curve may be thought of as the limiting position of a 
curve C" which meets the fixed curve C in n + 1 points near a fixed 
point A of C, which is the limiting position of each of the points 
of intersection. Let us consider for definiteness a family of 
curves which depends upon three parameters a, b, and c, and let 
t n + Aj , t + hi, and t + 7t 3 be three values of t near t . The curve 
C which meets the curve C in the three corresponding points is 
given by the three equations 

(35) F(t + AO = , F(t + A 2 ) = , F(t + A,) = . 

Subtracting the first of these equations from each of the others and 
applying the law of the mean to each of the differences obtained, 
we find the equivalent system 

(36) F(^ + A 1 ) = 0, F (f + *i) = 0, F (* +fc 2 ) = 0, 

where & t lies between h l and 7/, 2 , and & 2 between A x and h z . Again, 
subtracting the second of these equations from the third and apply 
ing the law of the mean, we find a third system equivalent to either 
of the preceding, 

(37) Fft + AO-O, F (f + *i) = 0, F"ft, + /,) = 0, 

where ^ lies between 7^ and 2 . As 7? u 7? 2 , and h s all approach 
zero, &!, k 2 , and ^ also all approach zero, and the preceding equa 
tions become, in the limit, 



which are the very equations which determine the osculating curve. 
The same argument applies for any number of parameters whatever. 
Indeed, we might define the osculating curve to be the limiting 
position of a curve C which is tangent to C at p points and cuts C 
at q other points, where 2p + q = n + 1, as all these p + q points 
approach coincidence. 

For instance, the osculating circle is the limiting position of a 
circle which cuts the given curve C in three neighboring points. It 
is also the limiting position of a circle which is tangent to C and 
which cuts C at another point whose distance from the point of 
tangency is infinitesimal. Let us consider for a moment the latter 
property, which is easily verified. 

Let us take the given point on C as the origin, the tangent at 
that point as the x axis, and the direction of the normal toward the 



X, Exs.] 



EXERCISES 



451 



center of curvature as the positive direction of the y axis. At the 
oiigin, y = 0. Hence R = 1/y", and therefore, by Taylor s series, 

1 



where e approaches zero with x. It fol 
lows that R is the limit of the expres 
sion 2 /(2y) = OP 2 /( 2MP ) as the P oint 
M approaches the origin. On the other 
hand, let R l be the radius of the circle 
Ci which is tangent to the x axis at the 
origin and which passes through M. 
Then we shall have 




Fia 



or 



OP = Mm = MP(2R l - MP) , 

;2 



OP 
2MP 



-tti ^ 



M_P. 
2 



hence the limit of the radius 7 a is really equal to the radius of 
curvature R 



EXERCISES 

1. Apply the general formulae to find the evolute of an ellipse ; of an hyper 
bola ; of a parabola. 

2. Show that the radius of curvature of a conic is proportional to the cube 
of the segment of the normal between its points of intersection with the curve 
and with an axis of symmetry. 

3. Show that the radius of curvature of the parabola is equal to twice the 
segment of the normal between the curve and the directrix. 

4. Let F and F be the foci of an ellipse, M a point on the ellipse, MN the 
normal at that point, and N the point of intersection of that normal and the 
major axis of the ellipse. Erect a perpendicular NK to MN at jV, meeting MF 
at K. At K erect a perpendicular KO to MF, meeting MN at 0. Show that 
is the center of curvature of the ellipse at the point M. 

5. For the extremities of the major axis the preceding construction becomes 
illusory. Let .4 CM/ be the major axis and BO B the minor axis of the ellipse. 
On the segments OA and OB construct the rectangle OA EB. From E let fall 
a perpendicular on AB, meeting the major and minor axes at C and D, respec 
tively. Show that C and D are the centers of curvature of the ellipse for the 
points A and B, respectively. 

6. Show that the evolute of the spiral p = ae mu > is a spiral congruent to the 
given spiral. 



452 PLANE CURVES [X, Exs. 

7. The path of any point on the circumference of a circle which rolls with 
out slipping along another (fixed) circle is called an epicycloid or an hypocycloid. 
Show that the evolute of any such curve is another curve of the same kind. 

8. Let AB be an arc of a curve upon which there are no singular points and 
no points of inflection. At each point m of this arc lay off from the point m 
along the normal at m a given constant length I in each direction. Let wtj and 
m 2 be the extremities of these segments. As the point m describes the arc AB, 
the points mi and m 2 will describe two corresponding arcs AiBi and A t B 2 . 
Derive the formulae Si = S 18, S 2 = S + W, where S, <S i , and <S Y 2 are the 
lengths of the arcs AB, A\B\, and A 2 B 2 , respectively, and where is the angle 
between the normals at the points A and B. It is supposed that the arc -Ai-Bi 
lies on the same side of AB as the evolute, and that it does not meet the evolute. 

[Licence, Paris, July, 1879.] 

9. Determine a curve such that the radius of curvatures p at any point M 
and the length of the arc s AM measured from any fixed point A on the curve 
satisfy the equation as = p 2 + a 2 , where a is a given constant length. 

[Licence, Paris, July, 1883.] 

10. Let C be a given curve of the third degree which has a double point 
at 0. A right angle MON revolves about the point O, meeting the curve C in 
two variable points M and N. Determine the envelope of the straight line MN. 
In particular, solve the problem for each of the curves Xy 2 = x 3 and z 3 + y s = f*xy. 

[Licence, Bordeaux, July, 1885.] 

11. Find the points at which the curve represented by the equations 

x a (nw sin u), y = a (n cos w) 

has contact of higher order than the second with the osculating circle. 

[Licence, Grenoble, July, 1885.] 

12. Let m, mi , and m 2 be three neighboring points on a plane curve. Find 
the limit approached by the radius of the circle circumscribed about the triangle 
formed by the tangents at these three points as the points approach coincidence. 

13. If the evolute of a plane curve without points of inflection is a closed 
curve, the total length of the evolute is equal to twice the difference between the 
sum of the maximum radii of curvature and the sum of the minimum radii of 
curvature of the given curve. 

14. At each point of a curve lay off a constant segment at a constant angle 
with the normal. Show that the locus of the extremity of this segment is a 
curve whose normal passes through the center of curvature of the given curve. 

15. Let r be the length of the radius vector from a fixed pole to any point of 
a plane curve, and p the perpendicular distance from the pole to the tangent. 
Derive the formula R = rdr/dp, where R is the radius of curvature. 

16. Show that the locus of the foci of the parabolas which have contact of 
the second order with a given curve at a fixed point is a circle. 

17. Find the locus of the centers of the ellipses whose axes have a fixed direc 
tion, and which have contact of the second order at a fixed point with a given 
curve. 



CHAPTER XI 

SKEW CURVES 

I. OSCULATING PLANE 

215. Definition and equation. Let M T be the tangent at a point M 
of a given skew curve F. A plane through MT and a point M of 
F near M in general approaches a limiting position as the point M 
approaches the point M. If it does, the limiting position of the 
plane is called the osculatiny plane to the curve F at the point M. 
We shall proceed to find its equation. 

Let 

(i) *=A9 y = *(0 *-rtO 

be the equations of the curve F in terms of a parameter t, and let t 
and + h be the values of t which correspond to the points M and 
&f, respectively. Then the equation of the plane MTM 1 is 

A(X -*)+ B(Y-y) + C(Z- 3) = 0, 

where the coefficients ^4, B, and C must satisfy the two relations 
(2) 
(3) 

Expanding f(t + h), <j>(t + A) and \j/(t -f A) by Taylor s series, the 
equation (3) becomes 



After multiplying by 7^, let us subtract from this equation the equa 
tion (2), and then divide both sides of the resulting equation by 
A 2 /2. Doing so, we find a system equivalent to (2) and (3) : 



<) + a] + ^[^"(0 + c s ] = 0, . 

where d, e 2 , and c 3 approach zero with A. In the limit as A 
approaches zero the second of these equations becomes 

(4) ^/(0 + **"(0 + cy (o = o. 

453 



454 SKEW CURVES [XI, 215 

Hence the equation of the osculating plane is 

(5) A(X-x) + B(Y-y-) + C(Z-z) = Q, 

where A, B, and C satisfy the relations 

(Adx + Bdy + C dz = , 
Co) < 

(Ad 2 x + Bd 2 y + Cd*z = 0. 

The coefficients A, B, and C may be eliminated from (5) and (6), 
and the equation of the osculating plane may be written in the form 



X -x Y y Z -z 

dx dy dz 

d?x d* d*z 



= 0. 



Among the planes which pass through the tangent, the osculating 
plane is the one which the curve lies nearest near the point of tan- 
gency. To show this, let us consider any other plane through the 
tangent, and let F(f) be the function obtained by substituting 
f(t + h), $(t -f h), \li(t + h) for X, F, Z, respectively, in the left-hand 
side of the equation (5), which we shall now assume to be the equa 
tion of the new tangent plane. Then we shall have 



where 77 approaches zero with h. The distance from any second 
point M of F near M to this plane is therefore an infinitesimal of 
the second order; and, since F(t) has the same sign for all sufficiently 
small values of h, it is clear that the given curve lies wholly on one 
side of the tangent plane considered, near the point of tangency. 

These results do not hold for the osculating plane, however. For 
that plane, Af" + B<f>" + Cif/" = ; hence the expansions for the 
coordinates of a point of F must be carried to terms of the third 
order. Doing so, we find 

h* 

= 



It follows that the distance from a point of T to the osculating 
plane is an infinitesimal of the third order; and, since F(f) changes 
sign with A, it is clear that a skew curve crosses its osculating plane 
at their common point. These characteristics distinguish the oscu 
lating plane sharply from the other tangent planes. 



XI, 216] OSCULATING PLANE 455 

216. Stationary osculating plane. The results just obtained are not 
valid if the coefficients A, B, C of the osculating plane satisfy the 
relation 

(7) A d s x + Bd*y + Cd*z = 0. 

If this relation is satisfied, the expansions for the coordinates must 
be carried to terms of the fourth order, and we should obtain a 
relation of the form 

A d*x + B d*y + C d* z 



The osculating plane is said to be stationary at any point of F for 
which (7) is satisfied; if A d*x + Bd*y + Cd*z does not vanish 
also, and it does not in general, F(t) changes sign with h and 
the curve does not cross its osculating plane. Moreover the distance 
from a point on the curve to the osculating plane at such a point is 
an infinitesimal of the fourth order. On the other hand, if the 
relation A d*x + Bd*y + Cd*z = is satisfied at the same point, 
the expansions would have to be carried to terms of the fifth order ; 
and so on. 

Eliminating A, B, and C between the equations (6) and (7), we 
obtain the equation 

dx dy dz 



(8) 



d*x d*y 



d s z 



= 0, 



whose roots are the values of t which correspond to the points of F 
where the osculating plane is stationary. There are then, usually, 
on any skew curve, points of this kind. 

This leads us to inquire whether there are curves all of whose 
osculating planes are stationary. To be precise, let us try to find 
all the possible sets of three functions x, y, z of a single variable t, 
which, together with all their derivatives up to and including those 
of the third order, are continuous, and which satisfy the equation 
(8) for all values of t between two limits a and b (a < b). 

Let us suppose first that at least one of the minors of A which 
correspond to the elements of the third row, say dx d^y dy d 2 x, does 
not vanish in the interval (a, b). The two equations 

( dz = C l dx + C 2 dy, 
\ d*z = C l d i x + C 3 <Py, 



456 SKEW CURVES [XI, 216 

which are equivalent to (6), determine C\ and C 2 as continuous 
functions of t in the interval (a, &). Since A = 0, these functions 
also satisfy the relation 

(10) d*z = C\d z x + C z d*y. 

Differentiating each of the equations (9) and making use of (10), 
we find 

d\ dx + dC 2 dy = , dC\ d 2 x + dC z d*i/ = , 

whence dC = dC = 0. It follows that each of the coefficients C l 
and C 2 is a constant ; hence a single integration of the first of 
equations (9) gives 

z C\x + C 2 y + <7 3 , 

where C s is another constant. This shows that the curve F is a 
plane curve. 

If the determinant dxd 2 y dyd 2 x vanishes for some value c of the variable t 
between a and b, the preceding proof fails, for the coefficients Ci and C% might 
be infinite or indeterminate at such a point. Let us suppose for definiteness 
that the preceding determinant vanishes for no other value of t in the interval 
(a, 6), and that the analogous determinant dxd 2 z dzd*x does not vanish for 
t = c. The argument given above shows that all the points of the curve F which 
correspond to values of t between a and c lie in a plane P, and that all the 
points of F which correspond to values of t between c and 6 also lie in some 
plane Q. But dxd^z dzd*x does not vanish for t = c; hence a number h 
can be found such that that minor does not vanish anywhere in the interval 
(c h, c + h). Hence all the points on T which correspond to values of t 
between c h and c + h must lie in some plane R. Since E must have an 
infinite number of points in common with P and also with Q, it follows that 
these three planes must coincide. 

Similar reasoning shows that all the points of F lie in the same plane unless 
all three of the determinants 

dxd^y dyd 2 x, dxd 2 z dzd?x, dyd^z dzd^y 

vanish at the same point in the interval (a, b). If these three determinants do 
vanish simultaneously, it may happen that the curve F is composed of several 
portions which lie in different planes, the points of junction being points at 
which the osculating plane is indeterminate.* 

If all three of the preceding determinants vanish identically in a certain 
interval, the curve F is a straight line, or is composed of several portions of 
straight lines. If dx/dt does not vanish in the interval (a, 6), for example, we 
may write 

d 2 zdx dzd 2 x 



whence 


(dx) 2 
dy = Ci dx , 


(dx)* 

dz C 2 dx, 





"This singular case seems to have been noticed first by Peano. It is evidently of 
Interest only from a purely analytical standpoint. 



XI, 217] OSCULATING PLANE 457 

where C\ and C 2 are constants. Finally, another integration gives 

y = Cix+C"i, 2 = C a z+CJ, 
which shows that T is a straight line. 

217. Stationary tangents. The preceding paragraph suggests the study of 
certain points on a skew curve which we had not previously defined, namely 
the points at which we have 

d2x = d*y = d*z 
dx dy dz 

The tangent at such a point is said to be stationary. It is easy to show by the 
formula for the distance between a point and a straight line that the distance 
from a point of T to the tangent at a neighboring point, which is in general an 
infinitesimal of the second order, is of the third order for a stationary tangent. 
If the given curve T is a plane curve, the stationary tangents are the tangents at 
the points of inflection. The preceding paragraph shows that the only curve 
whose tangents are all stationary is the straight line. 

At a point where the tangent is stationary, A = 0, and the equation of the 
osculating plane becomes indeterminate. But in general this indetermination 
can be removed. For, returning to the calculation at the beginning of 215 
and carrying the expansions of the coordinates of M to terms of the third order, 
it is easy to show, by means of (11), that the equation of the plane through M 
and the tangent at M is of the form 



X-x Y-y Z-z 

f (t) <t> (t) y/() 



= o, 



where ti , e 2 , c 8 approach zero with h. Hence that plane approaches a perfectly 
definite limiting position, and the equation of the osculating plane is given by 
replacing the second of equations (6) by the equation 



Bd s y 
If the coordinates of the point M also satisfy the equation 

d 3 x d 3 y d s z 
dx dy dz 

the second of the equations (6) should be replaced by the equation 
Ad<ix = Bdiy + Cdvz = 0, 

where q is the least integer for which this latter equation is distinct from the 
equation A dx = Bdy + C dz = 0. The proof of this statement and the exami 
nation of the behavior of the curve with respect to its osculating plane are left 
to the reader. 

Usually the preceding equation involving the third differentials is sufficient, 
and the coefficients A, B, C do not satisfy the equation 



Bd*y 
In this case the curve crosses every tangent plane except the osculating plane. 



458 SKEW CURVES [XI, 218 

218. Special curves. Let us consider the skew curves T which satisfy a 
relation of the form 



(12) xdy-ydx = 

where K is a given constant. From (12) we find immediately 



* -yd s x + dxd*y - dyd*x = Kd 3 z. 



Let us try to find the osculating plane of T which passes through a given point 
(a, b, c) of space. The coordinates (x, ?/, z) of the point of tangency must satisfy 
the equation 

a z b y c z 



dx dy dz 



= 0, 



d 2 x d 2 y d*z 
which, by means of (12) and (13), may be written in the form 
(14) ay-bx + K(c-z) = Q. 

Hence the possible points of tangency are the points of intersection of the 
curve F with the plane (14), which passes through (a, 6, c). 

Again, replacing dz, d?z and d 3 z by their values from (12) and (13), the equa 
tion A = 0, which gives the points at which the osculating plane is stationary, 
becomes 

A = \ (dx d 2 y - dy d 2 x) 2 = ; 

hence we shall have at the same points 

d 2 x _ d*y _ 7/d 2 x - xd 2 y _ d 2 z 
dx dy ydx xdy dz 

which shows that the tangent is stationary at any point at which the osculating 
plane is stationary. 

It is easy to write down the equations of skew curves which satisfy (12) ; for 
example, the curves 

x = At m , y = Bt, z = Ct m + n , 

where A, B, C, m, and n are any constants, are of that kind. Of these 
the simplest are the skew cubic x = t, y = i 2 , z t 3 , and the skew quartic 
x t, y = t 3 , z = t*. The circular helix 

x = a cos t , y = a sin t , z = Kt 

is another example of the same kind. 

In order to find all the curves which satisfy (12), let us write that equation in 
the form 

d(xy Kz) = 2ydx. 
If we set 

x =/(), xy-Kz = 4>(t), 

the preceding equation becomes 



XI, 219] ENVELOPES OF SURFACES 459 

Solving these three equations for z, y, and z, we find the general equations of T 
in the form 

(15) =/(<), V = 1%L, Kz 





where f(t) and <f>(t) are arbitrary functions of the parameter t. It is clear, how 
ever, that one of these functions may be assigned at random without loss of 
generality. In fact we may setf(t) = , since this amounts to choosing/() as a 
new parameter. 

II. ENVELOPES OF SURFACES 

Before taking up the study of the curvature of skew curves, we 
shall discuss the theory of envelopes of surfaces. 

219. One-parameter families. Let S be a surface of the family 

(16) f(x,y,z,a)=0, 

where a is the variable parameter. If there exists a surface E which 
is tangent to each of the surfaces S along a curve C, the surface E 
is called the envelope of the family (16), and the curve of tangency 
C of the two surfaces S and E is called the characteristic curve. 

In order to see whether an envelope exists it is evidently neces 
sary to discover whether it is possible to find a curve C on each of 
the surfaces S such that the locus of all these curves is tangent to 
each surface S along the corresponding curve C. Let (x, y, z) be 
the coordinates of a point M on a characteristic. If M is not a 
singular point of S, the equation of the tangent plane to S at M is 

df df df 

dx <- Y -*) + ;( r - y) + (*-)- o. 

i* 1 *^ ^y Cviv 

As we pass from point to point of the surface E, x, y, z, and a are 
evidently functions of the two independent variables which express 
the position of the point upon E, and these functions satisfy the 
equation (16). Hence their differentials satisfy the relation 

(17) -/ dx + ~ du + -/- dz + J- da 0. 
OX cy cz va 

Moreover the necessary and sufficient condition that the tangent 
plane to E should coincide with the tangent plane to S is 

dx dy dz 

or, by (17), 

W t? = 0. 



460 SKEW CURVES [XI, 220 

Conversely, it is easy to show, as we did for plane curves ( 201), 
that the equation R(x, y, z) = 0, found by eliminating the param 
eter a between the two equations (16) and (18), represents one or 
more analytically distinct surfaces, each of which is an envelope 
of the surfaces S or else the locus of singular points of S, or a com 
bination of the two. Finally, as in 201, the characteristic curve 
represented by the equations (16) and (18) for any given value of a 
is the limiting position of the curve of intersection of S with a 
neighboring surface of the same family. 

220. Two-parameter families. Let S be any surface of the two- 
parameter family 

(19) f(x, y, z,a,V) =0, 

where a and b are the variable parameters. There does not exist, 
in general, any one surface which is tangent to each member of this 
family all along a curve. Indeed, let b = <() be any arbitrarily 
assigned relation between a and b which reduces the family (19) to 
a one-parameter family. Then the equation (19), the equation 
b = <f>(a), and the equation 



represent the envelope of this one-parameter family, or, for any 
fixed value of a, they represent the characteristic on the correspond 
ing surface S. This characteristic depends, in general, on <f> (a), 
and there are an infinite number of characteristics on each of the 
surfaces S corresponding to various assignments of <(a). There 
fore the totality of all the characteristics, as a and b both vary arbi 
trarily, does not, in general, form a surface. We shall now try to 
discover whether there is a surface E which touches each of the 
family (19) in one or more points, not along a curve. If such a 
surface exists, the coordinates (x, y, z) of the point of tangency of 
any surface S with this envelope E are functions of the two variable 
parameters a and b which satisfy the equation (19) ; hence their dif 
ferentials dx, dy, dz with respect to the independent variables a 
and b satisfy the relation 



XI, 221] ENVELOPES OF SURFACES 461 

Moreover, in order that the surface which is the locus of the point 
of tangency (x, y, z) should be tangent to S, it is also necessary 
that we should have 



or, by (21), 

Since a and b are independent variables, it follows that the equations 

(22} = , -f- = 

da cb 

must be satisfied simultaneously by the coordinates (x, y, z~) of the 
point of tangency. Hence we shall obtain the equation of the 
envelope, if one exists, by eliminating a and b between the three 
equations (19) and (22). The surface obtained will surely be tan 
gent to S at (x, y, z) unless the equations 

dx dy dz 

are satisfied simultaneously by the values (x, y, z) which satisfy (19) 
and (22) ; hence this surface is either the envelope or else the locus 
of singular points of S. 

We have seen that there are two kinds of envelopes, depending 
on the number of parameters in the given family. For example, 
the tangent planes to a sphere form a two-parameter family, and 
each plane of the family touches the surface at only one point. 
On the other hand, the tangent planes to a cone or to a cylinder 
form a one-parameter family, and each member of the family is 
tangent to the surface along the whole length of a generator. 

221. Developable surfaces. The envelope of any one-parameter family 
of planes is called a developable surface. Let 

(23) z = ax + yf() + <K) 

be the equation of a variable plane P, where a is a parameter and 
where /(a) and <() are any two functions of a. Then the equa 
tion (23) and the equation 

(24) x + yf (a) + * () = 

represent the envelope of the family, or, for a given value of a, they 
represent the characteristic on the corresponding plane. But these 



462 SKEW CURVES [XI, 221 

two equations represent a straight line ; hence each characteristic 
is a straight line G, and the developable surface is a ruled surface. 
We proceed to show that all the straight lines G are tangent to the 
same skew curve. In order to do so let us differentiate (24) again 
with regard to a. The equation obtained 

(25) y/(a) + < = 

determines a particular point M on G. We proceed to show that G 
is tangent at M to the skew curve F which M describes as a varies. 
The equations of F are precisely (23), (24), (25), from which, if we 
desired, we might find x, y, and z as functions of the variable 
parameter a. Differentiating the first two of these and using the 
third of them, we find the relations 

(26) dz = a dx + /(a) d>j , dx + / () dy = , 

which show that the tangent to F is parallel to G. But these two 
straight lines also have a common point ; hence they coincide. 

The osculating plane to the curve F is the plane P itself. To 
prove this it is only necessary to show that the first and second 
differentials of x, y, and z with respect to a satisfy the relations 

dz = a dx 



The first of these is the first of equations (26), which is known to 
hold. Differentiating it again with respect to a, we find 

d*z = ad*x + f(a}d i y + [dx + f(a)dy ]da, 

which, by the second of equations (26), reduces to the second of the 
equations to be proved. 

It follows that any developable surface may be defined as the locus 
of the tangents to a certain skew curve T. In exceptional cases the 
curve F may reduce to a point at a finite or at an infinite distance ; 
then the surface is either a cone or a cylinder. This will happen 
whenever /"(a) = 0. 

Conversely, the locus of the tangents to any skew curve F is a 
developable surface. For, let 



be the equations of any skew curve F. The osculating planes 

y) + C(Z - z) = 



XI, 221] ENVELOPES OF SURFACES 463 

form a one-parameter family, whose envelope is given by the pre 
ceding equation and the equation 

dA(X - x) + dB(Y - y) + dC(Z - 2) = . 

For any fixed value of t the same equations represent the charac 
teristic in the corresponding osculating plane. We shall show that 
this characteristic is precisely the tangent at the corresponding 
point of F. It will be sufficient to establish the equations 

A dx + Bdy + C dz = 0, dA dx + dB dy -f dC dz = . 

The first of these is the first of (6), while the second is easily 
obtained by differentiating the first and then making use of the 
second of (6). It follows that the characteristic is parallel to 
the tangent, and it is evident that each of them passes through 
the point (x, y, z) ; hence they coincide. 

This method of forming the developable gives a clear idea of 
the appearance of the surface. Let AB be an arc of a skew curve. 
At each point M of AB draw the tangent, and consider only that 
half of the tangent which extends in a certain direction, from A 
toward B, for example. These half rays form one nappe Si of the 
developable, bounded on three sides by the arc AB and the tan 
gents A and B and extending to infinity. The other ends of the tan 
gents form another nappe S 2 similar to Si and joined to Si along the 
arc AB. To an observer placed above them these two nappes appear 
to cover each other partially. It is evident that any plane not tan 
gent to F through any point of AB cuts the two nappes Si and S 2 
of the developable in two branches of a curve which has a cusp at O. 
The skew curve F is often called the edge of regression of the 
developable surface.* 

It is easy to verify directly the statement just made. Let us 
take O as origin, the secant plane as the xy plane, the tangent to F as 
the axis of 2, and the osculating plane as the xz plane. Assuming 
that the coordinates x and y of a point of F can be expanded in powers 
of the independent variable 2, the equations of F are of the form 

x = a z z 2 + a 3 z* -\ , y = b 3 z* + , 

for the equations 

dx _ dy _ d z y _ 
dz dz dz* 

* The English term " edge of regression " does not suggest that the curve is a locus 
of cusps. The French terms "arete de rebroussement " and "point de rebroussement " 
are more suggestive. TRANS. 



464 SKEW CURVES [XI 222 

must be satisfied at the origin. Hence the equations of a tangent 
at a point near the origin are 



Setting Z = 0, the coordinates .Y and F of the point where the tan 
gent meets the secant plane are found to have developments which 
begin with terms in z 2 and in 8 , respectively ; hence there is surely 
a cusp at the origin. 

Example. Let us select as the edge of regression the skew cubic x t, y = t 2 , 
z = t 8 . The equation of the osculating plane to the curve is 

(27) *-3 a Jr + 3tr-Z = 0; 

hence we shall obtain the equation of the corresponding developable by writing 
down the condition that (27) should have a double root in t, which amounts to 
eliminating t between the equations 



*- Z = 0. 

The result of this elimination is the equation 

(XY - Z) 2 - 4(X2 - F)(F 2 - JTZ) = 0, 

which shows that the developable is of the fourth order. 

It should be noticed that the equations (28) represent the tangent to the given 
cubic. 

222. Differential equation of developable surfaces. If z = F(x, y) be 
the equation of a developable surface, the function F(x, y) satisfies 
the equation s 2 rt = 0, where r, s, and t represent, as usual, the 
three second partial derivatives of the function F(x, y). 

For the tangent planes to the given surface, 

Z =pX + qY + z px qy, 

must form a one-parameter family ; hence only one of the three 
coefficients p, q, and z px qy can vary arbitrarily. In particular 
there must be a relation between p and q of the form f(p, q) = 0. 
It follows that the Jacobian D( p, q)/D(x, y) = rt s 2 must vanish 
identically. 

Conversely, if F(x, y) satisfies the equation rt s 2 = 0, p and q 
are connected by at least one relation. If there were two distinct 
relations, p and q would be constants, F(x, y) would be of the form 
ax -f by + c, and the surface z = F(x, y) would be a plane. If there 



XI, 223] ENVELOPES OF SURFACES 465 

is a single relation between p and q, it may be written in the form 
a = f(p\ where p does not reduce to a constant. But we also have 

v (rt _ n - D(*-px-qy,p\ 
y(rt 



hence zpx qy is also a function of p, say *l/(p), whenever 
rt s 2 = 0. Then the unknown function F(x, y) and its partial 
derivatives p and q satisfy the two equations 



Differentiating the second of these equations with respect to x and 
with respect to ?/, we find 

ff = L* + ^W + f 

Since p does not reduce to a constant, we must have 



hence the equation of the surface is to be found by eliminating p 
between this equation and the equation 



which is exactly the process for finding the envelope of the family 
of planes represented by the latter equation, p being thought of as 
the variable parameter. 

223. Envelope of a family of skew curves. A one-parameter family 
of skew curves has, in general, no envelope. Let us consider first 
a family of straight lines 

(29) x = az+p, y = bz + q, 

where a, b, p, and q are given functions of a variable parameter a. 
We shall proceed to find the conditions under which every member 
of this family is tangent to the same skew curve T. Let z = <(a) 
be the z coordinate of the point M at which the variable straight 
line D touches its envelope T. Then the required curve T will be 
represented by the equations (29) together with the equation 
z <(<*), an( l the direction cosines of the tangent to T will be pro 
portional to dx/da, dy/da, dz/da, i.e. to the three quantities 

a < () + *() +JP , b<j> (a)+b <l>(a) + q , ^ (a), 



466 SKEW CURVES [XI, 223 

where a , b , p , and q are the derivatives of a, b, p, and q, respec 
tively. The necessary and sufficient condition that this tangent be 
the straight line D itself is that we should have 

dx _ dz_ d_y _ dz 

da da da da 

that is, 



The unknown function <(a) must satisfy these two equations; 
hence the family of straight lines has no envelope unless the two 
are compatible, that is, unless 

a q -b p = 0. 

If this condition is satisfied, we shall obtain the envelope by setting 
<t>(a}=-p /a<=-q</b>. 

It is easy to generalize the preceding argument. Let us consider a 
one-parameter family of skew curves (C) represented by the equations 

(30) F(x, y, a, a) = , *( x , y,z,a) = Q, 

where a is the variable parameter. If each of these curves C is 
tangent to the same curve T, the coordinates (x, y, z) of the point 
M at which the envelope touches the curve C which corresponds to 
the parameter value a are functions of a which satisfy (30) and 
which also satisfy another relation distinct from those two. Let 
dx, dy, dz be the differentials with respect to a displacement of M 
along C ; since a is constant along C, these differentials must satisfy 
the two equations 

dF dF dF J 

-j- dx + -7 dy + -^- dz = 0, 

dx *y ** 

0* , ^ a* A 

-dx + -^-dy + dz = Q. 
dy cz 

On the other hand, let &r, 8y, &s, 8a be the differentials of x, y, z, 
and a with respect to a displacement of M along T. These differen 
tials satisfy the equations 



(32) 

cy 



XI, 223] ENVELOPES OF SURFACES 467 

The necessary and sufficient conditions that the curves C and T 

be tangent are 

dx _ dy _ dz 

~Sx~ 8~y ~~ ~Sz 
or, making use of (31) and (32), 



It follows that the coordinates (x, y, z) of the point of tangency must 
satisfy the equations 

dF d& 

(33) F-0, * = 0, ^ = 0, ^ = 0. 

Hence, if the family (30) is to have an envelope, the four equations 
(33) must be compatible for all values of a. Conversely, if these 
four equations have a common solution in x, y, and z for all values 
of a, the argument shows that the curve T described by the point 
(x, y, z) is tangent at each point (x, y, z) upon it to the correspond 
ing curve C. This is all under the supposition that the ratios between 
dx, dy, and dz are determined by the equations (31), that is, that the 
point (x, y, 2) is not a singular point of the curve C. 

Note. If the curves C are the characteristics of a one-parameter 
family of surfaces F(x, y, z, a) = 0, the equations (33) reduce to 
the three distinct equations 

(34) F.p, |f = 0, 

hence the curve represented by these equations is the envelope 
of the characteristics. This is the generalization of the theorem 
proved above for the generators of a developable surface. 

The equations of a one-parameter family of straight lines are often written 
in the form 

(35) x -x _y -yo_z - z ^ 

a b c 

where XQ , yo , Zo , o, &, c are functions of a variable parameter a. It is easy to 
find directly the condition that this family should have an envelope. Let I 
denote the common value of each of the preceding ratios ; then the coordinates 
of any point of the straight line are given by the equations 

x = x + la , y = y + lb , z = Z Q + Ic , 

and the question is to determine whether it is possible to substitute for I such a 
function of a that the variable straight line should always remain tangent to 



468 



SKEW CURVES 



[XI, 224 



the curve described by the point (z, y, z). The necessary condition for this is 
that we should have 

,q<n Xp + g Z _ 2/6 + b l 26 + c l 
^oo; _ . 

a b c 

Denoting by m the common value of these ratios and eliminating I and m from 
the three linear equations obtained, we find the equation of condition 



(37) 



a b c 



= 0. 



If this condition is satisfied, the equations (36) determine I, and hence also the 
equation of the envelope. 



III. CURVATURE AND TORSION OF SKEW CURVES 

224. Spherical indicatrix. Let us adopt upon a given skew curve F 
a definite sense of motion, and let s be the length of the arc AM 
measured from some fixed point A as origin to any point M, affixing 
the sign -f or the sign according as the direction from A toward 
M is the direction adopted or the opposite direction. Let MT be 
the positive direction of the tangent at M, that is, that which cor 
responds to increasing values of the arc. If through any point O in 
space lines be drawn parallel to these half rays, a cone S is formed 
which is called the directing cone of the developable surface formed 
by the tangents to F. Let us draw a sphere of unit radius about O 
as center, and let 2 be the line of intersection of this sphere with 
the directing cone. The curve 2 is called the spherical indicatrix 





FIG. 48 



of the curve F. The correspondence between the points of these two 
curves is one-to-one : to a point M of F corresponds the point m where 
the parallel to MT pierces the sphere. As the point M describes the 



CURVATURE TORSION 469 

curve F in the positive sense, the point m describes the curve 2 in 
a certain sense, which we shall adopt as positive. Then the corre 
sponding arcs s and o- increase simultaneously (Fig. 48). 

It is evident that if the point O be displaced, the whole curve 2 
undergoes the same translation ; hence we may suppose that lies 
at the origin of coordinates. Likewise, if the positive sense on the 
curve F be reversed, the curve 2 is replaced by a curve symmetrical 
to it with respect to the point ; but it should be noticed that the 
positive sense of the tangent mt to 2 is independent of the sense of 
motion on T. 

The tangent plane to the directing cone along the generator Om is 
parallel to the osculating plane at M. For, let AX + BY + CZ = 
be the equation of the plane Omm , the center of the sphere being 
at the origin. This plane is parallel to the two tangents at M and 
at M ; hence, if t and t + h are the parameter values which corre 
spond to M and M } respectively, we must have 

(38) Af (t) + BV(t) + Cf (0 = 0. 

(39) Af(t + A) + B#(t + k) + Cf (* + A) = 0. 

The second of these equations may be replaced by the equation 

*) - 



B + c = 

h h h 

which becomes, in the limit as h approaches zero, 
(40) Af"() + B<j>"() + Cf 09 = 0. 

The equations (38) and (40), which determine A, B, and C for the 
tangent plane at m, are exactly the same as the equations (6) which 
determine A, B, and C for the osculating plane. 

225. Radius of curvature. Let o> be the angle between the positive 
directions of the tangents MT and M T at two neighboring points 
M and M of F. Then the limit of the ratio w/arc MM , as M 
approaches M , is called the curvature of F at the point M, just as 
for a plane curve. The reciprocal of the curvature is called the 
radius of curvature ; it is the limit of arc MM /to. 

Again, the radius of curvature R may be defined to be the limit 
of the ratio of the two infinitesimal arcs MM and mm , for we have 

arc MM arc MM arc mm chord mm 

VX 



arc mm chord mm 



470 SKEW CURVES [XI, 225 

and each of the fractions (arc raw )/(chord) mm and (chord mm )/<a 
approaches the limit unity as m approaches m . The arcss( = 3/Af ) 
and o-(=mm ) increase or decrease simultaneously; hence 

(> *= 

Let the equations of T be given in the form 
(42) x=f(t), y = *(0, * = lKO 

where is the origin of coordinates. Then the coordinates of the 
point m are nothing else than the direction cosines of MT, namely 

dx dy dz 

a j a > y = 3 

efo ete ds 

Differentiating these equations, we find 



- dx d 2 s ds d*y - dy d*s 

ri - dB= - ,2 - 
s 2 ds 2 



where O indicates as usual the sum of the three similar terms 
obtained by replacing x by x, y, z successively. Finally, expanding 
and making use of the expressions for ds 2 and ds d 2 s, we find 

. Sdx* 



ds 4 

By Lagrange s identity ( 131) this equation may be written in 

the form 

, 2 A* + B* + C* 

* - -*T 

where 

!A = dyd?z dzd^y, B = dzd z x dxd*z, 
C = 

a notation which we shall use consistently in what follows. Then 
the formula (41) for the radius of curvature becomes 



and it is evident that R z is a rational function of x, y, z, x , y , z , 
x", y", z". The expression for the radius of curvature itself is 
irrational, but it is essentially a positive quantity. 



XI, 226] CURVATURE TORSION 471 

Note. If the independent variable selected is the arc s of the 
curve r, the functions /(*), <(), and ^(s) satisfy the equation 

/"(*) + *"() + *"()=! 
Then we shall have 



(45) 



=/ (*) ft = * (), y = 

=/"(*) fe, dp = 4>"(s)ds, d y = 



and the expression for the radius of curvature assumes the partic 
ularly elegant form 

( 44/ ) ^ = [/"()] + [*"(*)? + Cf ()]* 

226. Principal normal. Center of curvature. Let us draw a line 
through M (on T) parallel to w, the tangent to 2 at m. Let MN 
be the direction on this line which corresponds to the positive direc 
tion mt. The new line MN is called the principal normal to T at M : 
it is that normal which lies in the osculating plane, since mt is 
perpendicular to Om and Omt is parallel to the osculating plane 
( 224). The direction MN is called the positive direction of the 
principal normal. This direction is uniquely defined, since the posi 
tive direction of mt does not depend upon the choice of the positive 
direction upon T. We shall see in a moment how the direction in 
question might be defined without using the indicatrix. 

If a length MC equal to the radius of curvature at M be laid off 
on MN from the point A/, the extremity C is called the center of 
curvature of T at M, and the circle drawn around C in the osculat 
ing plane with a radius MC is called the circle of curvature. Let 
a > ft > y be the direction cosines of the principal normal. Then the 
coordinates (aj u y lt z-^ of the center of curvature are 



But we also have 

a = ^_^^i_ p^. dsd z x dxd z s 
da- ds da- ds ds a 

and similar formulae for /? and y . Replacing a by its value in 
the expression for x, we find 

^da&x 

x^x + R 2 - -7^ 
ds 3 



472 SKEW CURVES 

But the coefficient of R 2 may be written in the form 

d*x S dx *- dx 



[XI, 226 



ds* ds* 

or, in terms of the quantities A, B, and C, 

Bdz Cdy 
ds* 

The values of y l and z l may be written down by cyclic permutation 
from this value of x l , and the coordinates of the center of curvature 
may be written in the form 



(46) 



B dz 



C dx A dz 

ds* 
A dii B dx 



These expressions for x lt y : , and z v are rational in x, y, z, x , y , z , 
x" y" z". 

A plane Q through M perpendicular to MN passes through the 
tangent MT and does not cross the curve T at M. We shall proceed 
to show that the center of curvature and the points of T near M lie 
on the same side of Q. To show this, let us take as the independent 
variable the arc s of the curve T counted from M as origin. Then 
the. coordinates X, Y, Z of a, point M of T near M are of the form 

s dx s 2 



the expansions for Y and Z being similar to the expansion for X. 
But since s is the independent variable, we shall have 



dx 
ds 



d z x 
ds 2 



da da d<r _ 1 , 
ds da- ds R 



and the formula for A becomes 



If in the equation of the plane Q, 
a (X - *) + J8 (F - y) 



1.2 



XI, 227] CURVATURE TORSION 473 

A", F, and Z be replaced by these expansions in the left-hand member, 
the value of that member is found to be 

I < + W + rf) + 0(5 + ) = I (! + 

where t] approaches zero with s. This quantity is positive for all 
values of s near zero. Likewise, replacing (X, Y, Z} by the coordi 
nates (x + Ra , y + Rfi , z + .Ry ) of the center of curvature, the 
result of the substitution is R, which is essentially positive. Hence 
the theorem is proved. 

227. Polar line. Polar surface. The perpendicular A to the oscu 
lating plane at the center of curvature is called the polar line. This 
straight line is the characteristic of the normal plane to T. For, in 
the first place, it is evident that the line of intersection D of the 
normal planes at two neighboring points M and M is perpendicular 
to each of the lines MT and M T ; hence it is also perpendicular to 
the plane mOm . As M approaches M, the plane mOm approaches 
parallelism to the osculating plane ; hence the line D approaches a 
line perpendicular to the osculating plane. On the other hand, to 
show that it passes through the center of curvature, let s be the 
independent variable ; then the equation of the normal plane is 

(47) a(X -x) + fi(Y-y) + y(Z -*)*, 

and the characteristic is denned by (47) together with the equation 

(48) | (X - x) + | ( Y - y} + (Z - ) - 1 = . 

This new equation represents a plane perpendicular to the principal 
normal through the center of curvature ; hence the intersection of 
the two planes is the polar line. 

The polar lines form a ruled surface, which is called the polar 
surface. It is evident that this surface is a developable, since we 
have just seen that it is the envelope of the normal plane to F. 
If F is a plane curve, the polar surface is a cylinder whose right 
section is the evolute of F ; in this special case the preceding state 
ments are self-evident. 

228. Torsion. If the words "tangent line" in the definition of 
curvature ( 225) be replaced by the words " osculating plane," a 
new geometrical concept is introduced which measures, in a manner, 
the rate at which the osculating plane turns. Let a> be the angle 
between the osculating planes at two neighboring points M and M ; 



474 SKEW CURVES [XI, 228 

then the limit of the ratio o> /arc MM , as M approaches M , is called 
the torsion of the curve F at the point M. The reciprocal of the 
torsion is called the radius of torsion. 

The perpendicular to the osculating plane at M is called the 
binormal. Let us choose a certain direction on it as positive, we 
shall determine later which we shall take, and let a", ft", y" be 
the corresponding direction cosines. The parallel line through the 
origin pierces the unit sphere at a point n, which we shall now put 
into correspondence with the point M of T. The locus of n is a 
spherical curve , and it is easy to show, as above, that the radius 
of torsion T may be defined as the limit of the ratio of the two corre 
sponding arcs MM and nn of the two curves T and . Hence we 
shall have 

1--^, 

dr 2 

where T denotes the arc of the curve . 

The coordinates of n are a", ft", y", which are given by the formulae 
( 215) 

q"= . A => ft" = ==, y "= 



where the radical is to be taken with the same sign in all three 
formulae. From these formulae it is easy to deduce the values of 
da", dp", dy"; for example, 

da" = (X 2 + - B2 + C*)dAA(AdA + BdB + CdC) 
(A 2 + B 2 + C 2 ) f 

whence, since dr 2 = da" 2 + dft" 2 + dy" 2 , 

- m SA* S**-IS(***)7 

(A 2 + B 2 + C 2 ) 2 
or, by Lagrange s identity, 

S^dC-CdB}* 

(A 2 + B 2 + C 2 ) 2 

where & denotes the sum of the three terms obtained by cyclic per 
mutation of the three letters A, B, C. The numerator of this expres 
sion may be simplified by means of the relations 

Adx+ Bdy + C dz = 0, 
dA dx -f- dB dy + dC dz = 0, 
whence 

dx d d * * 



B dC - C dB CdA - AdC A dB - B dA K 



XI, 228] CURVATURE TORSION 475 

where K is a quantity defined by the equation (49) itself. This gives 

2 _ A W 

~ (A 2 + B* + C 2 ) 2 

where K is defined by (49) ; or, expanding, 



dz dx 



dx dy 

d s x d a y 

d*y), 



dx dy 



dz dx 

d*z d a x 



where o denotes the sum of the three terms obtained by cyclic per 
mutation of the three letters x, y, z. But this value of K is exactly 
the development of the determinant A [(8), 216]; hence 



- A* + B* + C 2 
and therefore the radius of torsion is given by the formula 

^2_|_^2_j_ ^2 



(50) 



T= 



If we agree to consider T essentially positive, as we did the radius 
of curvature, its value will be the absolute value of the second mem 
ber. But it should be noticed that the expression for T is rational 
in x, y, z, x , y , z , x", y", z" ; hence it is natural to represent the 
radius of torsion by a length affected by a sign. The two signs 
which T may have correspond to entirely different aspects of the 
curve F at the point M. 

Since the sign of T depends only on that of A, we shall investigate 
the difference in the appearance of F near M when A has different 
signs. Let us suppose that the trihedron Oxyz is placed so that an 
observer standing on the xy plane with his feet at and his head in 
the positive z axis would see the x axis turn through 90 to his left 
if the x axis turned round into the y axis (see footnote, p. 477). 
Suppose that the positive direction of the binormal MN b has been so 
chosen that the trihedron formed from the lines MT, MN, MN b has 
the same aspect as the trihedron formed from the lines Ox, Oy, Oz ; 
that is, if the curve F be moved into such a position that M coincides 
with O, MT with Ox, and MN with Oy, the direction MN b will coin 
cide with the positive z axis. During this motion the absolute value 
of T remains unchanged ; hence A cannot vanish, and hence it cannot 



176 SKEW CURVES [XI, 228 

even change sign.* In this position of the curve T with respect to 
the axes now in the figure the coordinates of a point near the origin 
will be given by the formulae 



f* = 
(51) \y = 



where e, e , e" approach zero with t, provided that the parameter t is 
so chosen that t = at the origin. For with the system of axes 
employed we must have dy = dz = d 2 z = when t = 0. Moreover 
we may suppose that a t > 0, for a change in the parameter from t to 
t will change a t to a t . The coefficient & 2 is positive since y must 
be positive near the origin, but c z may be either positive or negative. 
On the other hand, f or t = 0, A = 12a 1 J 2 c 3 dt*. Hence the sign of A 
is the sign of e 3 . There are then two cases to be distinguished. If 
c 3 > 0, x and z are both negative for h < t < 0, and both positive 
f or < t < h, where h is a sufficiently small positive number ; i.e. 
an observer standing on the xy plane with his feet at a point P on 



y 

M 



,- M M "X 

/ \ 

I M" ^v 

FIG. 49, a FIG. 49, 6 

the positive half of the principal normal would see the arc MM at 
his left and above the osculating plane, and the arc MM" at his right 
below that plane (Fig. 49, a). In this case the curve is said to be 
sinistrorsal. On the other hand, if c 3 < 0, the aspect of the curve 
would be exactly reversed (Fig. 49, b), and the curve would be said 
to be dextrorsal. These two aspects are essentially distinct. For 
example, if two spirals (helices) of the same pitch be drawn on the 
same right circular cylinder, or on two congruent cylinders, they 
will be superposable if they are both sinistrorsal or both dextrorsal ; 
but if one of them is sinistrorsal and the other dextrorsal, one of 
them will be superposable upon the helix symmetrical to the other 
one with respect to a plane of symmetry. 

* It would be easy to show directly that A does not change sign when we pass from 
one set of rectangular axes to another set which have the same aspect. 



XI, 229] CURVATURE TORSION 477 

In consequence of these results we shall write 

(52) r __4i*<? ; 

i.e. at a point where the curve is dextrorsal T shall be positive, while 
T shall be negative at a point where the curve is sinistrorsal. A dif 
ferent arrangement of the original coordinate trihedron Oxyz would 
lead to exactly opposite results.* 

229. Frenet s formulae. Each point M of T is the vertex of a tri- 
rectangular trihedron whose aspect is the same as that of the trihe 
dron Oxyz, and whose edges are the tangent, the principal normal, 
and the binomial. The positive direction of the principal normal is 
already fixed. That of the tangent may be chosen at pleasure, but 
this choice then fixes the positive direction on the binormal. The dif 
ferentials of the nine direction cosines (a, ft, y), (a 1 , ft , /). (a", ft", y") 
of these edges may be expressed very simply in terms of R, T, and 
the direction cosines themselves, by means of certain formulae due 
to Frenet.f We have already found the formulae for da, dft, and dy : 

/KQN da a dft ft dy y 

( oo ) = > *- = > - = i 

ds R ds R ds R 

The direction cosines of the positive binormal ( 228) are 
~"-- A B " = C 



where e = 1. Since the trihedron (MT, MN, MN b ) has the same 
aspect as the trihedron Oxyz, we must have 



or = 



l 2 + B 2 + C 2 

On the other hand, the formula for da" may be written 
d _ B(B dA -AdB} + C(C dA - A dC) 

(A 2 + B 2 + C 2 )* 
or, by (49) and the relation A = A, 

da " = ^ Cft-By q A 

ds (A 2 + B 2 + C 2 ) 3 ~ ^ + &+ C 12 

* It is usual in America to adopt an arrangement of axes precisely opposite to that 
described above. Hence we should write T = + (A* + B^ + C 2 )/A, etc. See also 
the footnote to formula (54), 229. TRANS. 

t Nouvelles Annales de Mathematiques, 1864, p. 281. 



478 SKEW CURVES [XI, 229 

The coefficient of a is precisely 1/T, by (52). The formulae for 
dp" and dy" may be calculated in like manner, and we should find 







ds T ds T ds T 

which are exactly analogous to (53).* 

In order to find da , d(3 , dy , let us differentiate the well-known 
formulae 

2 + (3 2 + y a = l, 
cm + /3/3 + yy = 0, 



replacing da, d(3, dy, da", d/3", dy" by their values from (53) and 
(54). This gives 

a da + ft dp + y dy = 0, 

ds 
a da + J3 d/3 + y dy + = 0, 



whence, solving for da , dft , dy , 



ds 
- 



ds R T ds R T ds R T 

The formulae (53), (54), and (55) constitute Frenet s formulae. 

Note. The formulae (54) show that the tangent to the spherical 
curve described by the point n whose coordinates are a", /?", y" is 
parallel to the principal normal. This can be verified geometrically. 
Let S be the cone whose vertex is at and whose directrix is the 
curve . The generator On is perpendicular to the plane which is 
tangent to the cone S along Om ( 228). Hence S is the polar cone 
to S. But this property is a reciprocal one, i.e. the generator Om 
of 5 is surely perpendicular to the plane which is tangent to .S" 
along On. Hence the tangent mt to the curve 2, since it is perpen 
dicular to each of the lines On and Om, is perpendicular to the 
plane mOn. For the same reason the tangent nt to the curve is 
perpendicular to the plane mOn. It follows that mt and nt are 
parallel. 

* If we had written the formula for the torsion in the form l/T= A/ (A* + B* + C 2 ), 
Frenet s formulae would have to be written in the form da" /ds = a /T, etc. 
[Hence this would be the form if the axes are taken as usual in America. TRANS.] 



XI, 230] 



CURVATURE TORSION 



479 



230. Expansion of x, y, and z in powers of s. Given two functions 
R = <(s), T = \l/(s) of an independent variable s, the first of which 
is positive, there exists a skew curve T which is completely defined 
except for its position in space, and whose radius of curvature and 
radius of torsion are expressed by the given equations in terms of 
the arc s of the curve counted from some fixed point upon it. A rig 
orous proof of this theorem cannot be given until we have discussed 
the theory of differential equations. Just now we shall merely show 
how to find the expansions for the coordinates of a point on the 
required curve in powers of s, assuming that such expansions exist. 

Let us take as axes the tangent, the principal normal, and the 
binormal at O, the origin of arcs on T. Then we shall have 



(56) 



s 

*=I 



dx 



+ 



= i (*y\ 

1 \C&/I 



+ 



o^z\ 



+ 



/d s x\ 



-ri) + 



) + 



1.2 VdsVo 
s 2 /<?_ 

172 Vo^ 



+ 



123 \ds 3 / o 

1.2.3 W/o + 

g 3 /rf 8 ^\ 
1.2.3\A i /t 



a 
ll 



where #, y, and are the coordinates of a point on F. But 

dx d z x da 

= a, -r-T = -7- 
as as as 

whence, differentiating, 

d*x _ _ a dR 1 la 
~d&~ ~ R 2 ds R \R " T, 

In general, the repeated application of Frenet s formulae gives 



where L n , M n , P n are known functions of R, T, and their successive 
derivatives with respect to s. In a similar manner the successive 
derivatives of y and z are to be found by replacing (a, a , a") by 
(/3, /? , /8") and (y, y , y"), respectively. But we have, at the origin, 
a = 1, ft = 0, y - 0, a n = 0, $ = 1,^ = 0, a = 0, $ = 0, ^ = 1 ; 
hence the formulas (56) become 



(56 ) 






.s 3 dR 



2R BR* ds 



QRT 



480 SKEW CURVES [XI, 231 

where the terms not written down are of degree higher than three. 
It is understood, of course, that R, T, dR/ds, are to be replaced 
by their values for 5 = 0. 

These formulae enable us to calculate the principal parts of cer 
tain infinitesimals. For instance, the distance from a point of the 
curve to the osculating plane is an infinitesimal of the third order, 
and its principal part is s 3 /6RT. The distance from a point on 
the curve to the x axis, i.e. to the tangent, is of the second order, 
and its principal part is s*/2R (compare 214). Again, let us cal 
culate the length of an infinitesimal chord c. We find 



, 

where the terms not written down are of degree higher than four. 
This equation may be written in the form 



which shows that the difference s c is an infinitesimal of the 
third order and that its principal part is s 8 /24# 2 . 

In an exactly similar manner it may be shown that the shortest 
distance between the tangent at the origin and the tangent at a 
neighboring point is an infinitesimal of the third order whose prin 
cipal part is s s /12RT. This theorem is due to Bouquet. 

231. Involutes and evolutes. A curve I\ is called an involute of a 
second curve F if all the tangents to F are among the normals to I\, 
and conversely, the curve F is called an evolute of F^ It is evident 
that all the involutes of a given curve F lie on the developable sur 
face of which F is the edge of regression, and cut the generators of 
the developable orthogonally. 

Let (x, y, z) be the coordinates of a point M of F, (a, ft, y) the 
direction cosines of the tangent MT, and I the segment MMi between 
M and the point M a where a certain involute cuts MT. Then the 
coordinates of M l are 

Xl =X + la, 7/ 1= :7/ + /yS, K l = z + ly, 

whence 

dx l = dx + Ida + a dl, 

d yi = di/ + l</(3 + (3 dl, 
dzi = dz + I dy + y dl. 



XI, 231] 



CURVATURE TORSION 



481 



In order that the curve described by Af x should be normal to 

it is necessary and sufficient that a dx^ + (3 dy l + y dz should vanish, 

i.e. that we should have 



a 



dx 



(ldy + ydz + dl + l(ada + {3dp + y dy) = 0, 



which reduces to ds -\- dl = 0. It follows that the involutes to a 
given skew curve F may be drawn by the same construction which 
was used for plane curves ( 206). 

Let us try to find all the evolutes of a 
given curve F, that is, let us try to pick 
out a one-parameter family of normals to 
the given curve according to some contin 
uous law which will group these normals 
into a developable surface (Fig. 50). Let 
D be an evolute, < the angle between the 
normal MM^ and the principal normal MN, 
and I the segment MP between M and the 
projection P of the point M l on the principal normal. Then the 
coordinates (x i} yi, z^) of M l are 




FIG. 50 



(57) 



" tan <, 
/" tan <, 
ly" tan <, 



as we see by projecting the broken line MPM^ upon the three axes 
successively. The tangent to the curve described by the point M^ 
must be the line MM t itself, that is, we must have 



dx l 



dz l 



Let k denote the common value of these ratios ; then the condition 
dx l = k(x l x) may be transformed, by inserting the values of x^ 
and dx l and applying Frenet s formulae, into the form 



a ds 1 
\ 



- 4) + a ( 
K/ \ 

+ a" d(l 



- kl) 

T I 



tan 



- ^ - kl tan 



> =0. 



The conditions dy^ = ]c(y^ y) and dz l = k(z l z) lead to exactly 
similar forms, which may be deduced from the preceding by repla 
cing (a, a , a") by ((3, /3 , ") and (y, y , y"), respectively. Since the 



482 SKEW CURVES [XI, 231 

determinant of the nine direction cosines is equal to unity, these 
three equations are equivalent to the set 



(58) 



ds 

dl + I tan <f> = kl, 

Ids 

d(l tan <) = kl tan <f> . 



From the first of these I = R, which shows that the point P is the 
center of curvature and that the line PM is the polar line. It fol 
lows that all the evolutes of a given skew curve T lie on the polar sur 
face. In order to determine these evolutes completely it only remains 
to eliminate k between the last two of equations (58). Doing so 
and replacing I by R throughout, we find ds = T d<j>. Hence < may 
be found by a single quadrature : 

(59) + -* 

If we consider two different determinations of the angle < which 
correspond to two different values of the constant < , the difference 
between these two determinations of < remains constant all along T. 
It follows that two normals to the curve T which are tangent to two 
different evolutes intersect at a constant angle. Hence, if we know 
a single family of normals to T which form a developable surface, 
all other families of normals which form developable surfaces may 
be found by turning each member of the given family of normals 
through the same angle, which is otherwise arbitrary, around its 
point of intersection with T. 

Note L If T is a plane curve, T is infinite, and the preceding 
formula gives <f> = <. The evolute which corresponds to < = is 
the plane evolute studied in 206, which is the locus of the centers 
of curvature of F. There are an infinite number of other evolutes, 
which lie on the cylinder whose right section is the ordinary evo 
lute. We shall study these curves, which are called helices, in the 
next section. This is the only case in which the locus of the cen 
ters of curvature is an evolute. In order that (59) should be satis 
fied by taking < = 0, it is necessary that T should be infinite or 
that A should vanish identically ; hence the curve is in any case a 
plane curve ( 216). 



XI, 232] CURVATURE TORSION 483 

Note II. If the curve D is an evolute of T, it follows that T is an 
involute of D. Hence 

ds t = d(MM l ) , 

where s x denotes the length of the arc of the evolute counted from 
some fixed point. This shows that all the evolutes of any given 
curve are rectifiable. 

232. Helices. Let C be any plane curve and let us lay off on the perpendic 
ular to the plane of C erected at any point m on C a length mM proportional to 
the length of the arc a- of C counted from some fixed point A. Then the skew 
curve F described by the point M is called a helix. Let us take the plane of C 
as the xy plane and let 

z=/(o-), y = <t>(v) 

be the coordinates of a point TO of C in terms of the arc o". Then the coordi 
nates of the corresponding point M of the curve F will be 

(60) x=/(<r), y = *(cr), z = K<r, 

where K is the given factor of proportionality. The functions / and <f> satisfy 
the relation / 2 + < 2 = 1 ; hence, from (60), 



where s denotes the length of the arc of T. It follows that s = cr Vl + K 2 + J7, 
or, if s and a- be counted from the same point A on C, s = <r Vl + -K 2 , since H = 0. 
The direction cosines of the tangent to F are 

(61) a 



Since y is independent of (r, it is evident that the tangent to F makes a constant 
angle with the z axis ; this property is characteristic : Any curve whose tangent 
makes a constant angle with a fixed straight line is a helix. In order to prove 
this, let us take the z axis parallel to the given straight line, and let C be the 
projection of the given curve F on the xy plane. The equations of F may always 
be written in the form 

(62) x =/(<r) , y = <t>(v), z = ^(<r) , 

where the functions / and </> satisfy the relation / 2 4- < 2 = 1, for this merely 
amounts to taking the arc o- of C as the independent variable. It follows that 

dS v/V / 2 4- rf/2 _1_ 1/~1 

hence the necessary and sufficient condition that y be constant is that f should 
be constant, that is, that \f/(a-) should be of the form Kcr + z . It follows that 
the equations of the curve F will be of the form (60) if the origin be moved to 
the point x = 0, y = 0, z = z . 

Since y is constant, the formula dy/ds = y /B shows that y = 0. Hence the 
principal normal is perpendicular to the generators of the cylinder. Since it is 
also perpendicular to the tangent to the helix, it is normal to the cylinder, and 
therefore the osculating plane is normal to the cylinder. It follows that the 



SKEW CURVES [XI, 232 

binormal lies in the tangent plane at right angles to the tangent to the helix ; 
hence it also makes a constant angle with the z axis, i.e. y" is constant. 

Since y = 0, the formula dy /ds = - y/B - y"/ T shows that y/R + y"/ T = 0; 
hence the ratio T/R is constant for the helix. 

Each of the properties mentioned above is characteristic for the helix. Let 
us show, for example, that every curve for which the ratio T/R is constant is a 
helix. (J. BERTRAND.) 

From Frenet s formulae we have 

dc^ _dp _dy _ T _ 1 
da" ~ dp" ~ dy 7 ~ R ~ H 
hence, if H is a constant, a single integration gives 

a" = Ha - A , $" = Hp - B, y" = Hy - C , 

where A, B, C are three new constants. Adding these three equations after 
multiplying them by a, /3, y, respectively, we find 

Aa + Bp + Cy = H, 
or 

Aa + Bp + Cy H 



\*A* + B* -r C 2 
But the three quantities 



ABC 



2 + (72 ^/ A 2 + 2 + (72 Vvl 2 + B 2 + C 2 



are the direction cosines of a certain straight line A, and the preceding equa 
tion shows that the tangent makes a constant angle with this line. Hence the 
given curve is a helix. 

Again, let us find the radius of curvature. By (53) and (61) we have 

a _da _ 1 pi 

R ~ ~ds~ ~ - f 

whence, since y = 0, 

<63) i 

This shows that the ratio (1 + K 2 )/R is independent of K. But when K = 
this ratio reduces to the reciprocal 1/r of the radius of curvature of the right 
section C, which is easily verified ( 205). Hence the preceding formula may 
be written in the form R = r(l + K 2 ), which shows that the ratio of the radius 
of curvature of a helix to the radius of curvature of the corresponding curve C 
is a constant. 

It is now easy to find all the curves for which R and T are both constant. 
For, since the ratio T/R is constant, all the curves must be helices, by Bertrand s 
theorem. Moreover, since R is a constant, the radius of curvature r of the 
curve C also is a constant. Hence C is a circle, and the required curve is a 
helix which lies on a circular cylinder. This proposition is due to Puiseux.* 

* It is assumed in this proof that we are dealing only with real curves, for we 
assumed that A 2 + B* + <? 2 does not vanish. (See the thesis by Lyon : Sur les 
courbes a torsion constante, 1890.) 



XI, 233] CURVATURE TORSION 485 

233. Bertrand s curves. The principal normals to a plane curve are also the 
principal normals to an infinite number of other curves, the parallels to the 
given curve. J. Bertrand attempted to find in a similar manner all the skew 
curves whose principal normals are the principal normals to a given skew 
curve F. Let the coordinates x, y, z of a point of F be given as functions of the 
arc . Let us lay off on each principal normal a segment of length I, and let the 
coordinates of the extremity of this segment be X, F, Z ; then we shall have 

(64) X = x + la , Y = y + lp, Z = z + ly . 

The necessary and sufficient condition that the principal normal to the curve I" 
described by the point (X, F, Z) should coincide with the principal normal to F 
is that the two equations 

a dX + p dY+y dZ = 0, 



y (dXd 2 Y - dYd*X) = 

should be satisfied simultaneously. The meaning of each of these equations is 
evident. From the first, dl = ; hence the length of the segment I should be a 
constant. Replacing dX, d 2 JT, dF, in the second equation by their values 
from Frenet s formulae and from the formulas obtained by differentiating 
Frenet s, and then simplifying, we finally find 



whence, integrating, 

(65) I+l- 1 . 

where I is the constant of integration. It follows that the required curves are 
those for which there exists a linear relation between the curvature and the torsion. 
On the other hand, it is easy to show that this condition is sufficient and that 
the length I is given by the relation (05). 

A remarkable particular case had already been solved by Monge, namely 
that in which the radius of curvature is a constant. In that case (65) becomes 
I = R, and the curve T" defined by the equations (64) is the locus of the centers 
of curvature of F. From (64), assuming I R = constant, we find the equations 



which show that the tangent to T is the polar line of F. The radius of curva 
ture R of F is given by the formula 



da" 2 + d/3" 2 + dy"* 

hence R also is constant and equal to R. The relation between the two curves 
F and F is therefore a reciprocal one : each of them is the edge of regression of 
the polar surface of the other. It is easy to verify each of these statements for 
the particular case of the circular helix. 



486 SKEW CURVES [XI, 23* 

Note. It is easy to find the general formulae for all skew curves whose radius of 
curvature is constant. Let R be the given constant radius and let a, /3, 7 be any 
three functions of a variable parameter which satisfy the relation a 2 + /3 2 + y 2 = 1. 
Then the equations 

(66) X = 



where da- = Vda 2 + d/3 2 + (fry 2 , represent a curve which has the required prop 
erty, and it is easy to show that all curves which have that property may be 
obtained in this manner. For a, /3, 7 are exactly the direction cosines of the 
curve defined by (66), and <j is the arc of its spherical indicatrix ( 225). 



IV. CONTACT BETWEEN SKEW CURVES 
CONTACT BETWEEN CURVES AND SURFACES 

234. Contact between two curves. The order of contact of two 
skew curves is defined in the same way as for plane curves. Let F 
and I" be two curves which are tangent at a point A . To each point 
M of F near A let us assign a point M of F according to such a law 
that M and M approach A simultaneously. We proceed to find 
the maximum order of the infinitesimal MM with respect to the 
principal infinitesimal AM, the arc of F. If this maximum order 
is n + 1, we shall say that the two curves have contact of order n. 

Let us assume a system of trirectangular * axes in space, such 
that the yz plane is not parallel to the common tangent at A, and 
let the equations of the two curves be 

( u f(x\. ( Y = F(x), 

/N ./ V / /fN \ 



If x , y , Z Q are the coordinates of A, the coordinates of M and M 1 
are, respectively, 

[x -f h, f(x + A), <f>(x + /*)] , [ar c + k, F(x + k}, 4>(.r + &)] , 

where A; is a function of h which is defined by the law of corre 
spondence assumed between M and M and which approaches zero 
with h. We may select h as the principal infinitesimal instead of 
the arc AM (211); and a necessary condition that MM should 
be an infinitesimal of order n + 1 is that each of the differences 

k-h, F(x () + 7c) - f(x + h} , &(x + fc) - <K*o + A) 

* It is easy to show, by passing to the formula for the distance between two points 
in oblique coordinates, that this assumption is not essential. 



XI,L;a] CONTACT 487 

should be an infinitesimal of order n + 1 or more. It follows that 
we must have 

k-h = ah n + l , F(x + K) -f(x + h) = /3h n + l , 
<D(z + k) - <fr(x + h) = y h n + 1 , 

where a, ft, y remain finite as h approaches zero. Replacing k by 
its value h + ah n+l from the first of these equations, the latter two 
become 

F(x, + h + A + 1 ) - /(*,, + A) = /8A- + 1 , 

$(x + A + <rA" + I ) - <j>( X(> + A) = yA"* 1 . 

Expanding F(x + h + ah n + l ) and <t>(z + A + ah n+l ) by Taylor s 
series, all the terms which contain a will have a factor A" + * ; hence, 
in order that the preceding condition be satisfied, each of the 
differences 

F(x + A) - f(x n + A) , 4>(a; + h) - <f>( X() + A) 

should be of order n -f 1 or more. It follows that if MM is of 
order n + 1, the distance MN between the points M and N of the 
two curves which have the same abscissa x -f h will be at least of 
order n + 1. Hence the maximum order of the infinitesimal in 
question will be obtained by putting into correspondence the points 
of the two curves which have the same abscissa. 

This maximum order is easily evaluated. Since the two curves 
are tangent we shall have 



Let us suppose for generality that we also have 



but that at least one of the differences 



does not vanish. Then the distance MM will be of order n + 1 
and the contact will be of order n. This result may also be stated 
as follows : To find the order of contact of two curves Y and T , con 
sider the two sets of projections (C, C") and (C l , C{) of the given 
curves on the xy plane and the xz plane, respectively, and find the 
order of contact of each set ; then the order of contact of the given 
curves F and I" will be the smaller of these two. 



488 SKEW CURVES [XI, 236 

If the two curves F and F are given in the form 



(F ) X=.f(u), r=*(w), Z=*(M), 

they will be tangent at a point u = t = t if 

*(*o) = *( o) , * ( o) = * (*o) , *(*o) = "KM , * (*o) = <A (V> 

T/* we suppose that f (t ) is not zero, the tangent at the point of 
contact is not parallel to the yz plane, and the points on the two 
curves which have the same abscissa correspond to the same value 
of t. In order that the contact should be of order n it is neces 
sary and sufficient that each of the infinitesimals <$() <(Y) and 
*() \l/(t) should be of order n + 1 with respect to t t , i.e. that 
we should have 

* (*o) = * (*o) , , * ( (*o) = 4> (n) Co) , 
* (*)= * (<), -, * ( " ) (M = A ( " ) (^), 
and that at least one of the differences 



should not vanish. 

It is easy to reduce to the preceding the case in which one of the 
curves F is given by equations of the form 



(67) * 

and the other curve F by two implicit equations 



Resuming the reasoning of 212, we could show that a necessary 
condition that the contact should be of order n at a point of F 
where t = t is that we should have 

(F(*.) = 0, F (*o) = 0, -.., F<">(* ) = 0, 

[Ft^-0, F|(o) = 0, , Fi->(* ) = 0, 
where 

F(0 = nf(t)> *(0 -KO] , f r 1 (0 = F, 



235. Osculating curves. Let F be a curve whose equations are 
given in the form (67), and let F be one of a family of curves in 
2n + 2 parameters a, b, c, , I, which is defined by the equations 

(69) F(x, y,z,a,b,-->,l) = 0, ^ (*, y, z,a,b,c, >, I) = 0. 



XI, 235] CONTACT 489 

In general it is possible to determine the 2n + 2 parameters in such 
a way that the corresponding curve T has contact of order rti with 
the given curve F at a given point. The curve thus determined is 
called the osculating curve of the family (69) to the curve T. The 
equations which determine the values of the parameters a, b, c, , I 
are precisely the 2n + 2 equations (68). It should be noted that 
these equations cannot be solved unless each of the functions F and 
F l contain at least n -f- 1 parameters. For example, if the curves 
F are plane curves, one of the equations (69) contains only three 
parameters ; hence a plane curve cannot have contact of order 
higher than two with a skew curve at a point taken at random on 
the curve. 

Let us apply this theory to the simpler classes of curves, the 
straight line and the circle. A straight line depends on four param 
eters ; hence the osculating straight line will have contact of the 
first order. It is easy to show that it coincides with the tangent, 
for if we write the equations of the straight line in the form 

x = az + p, y = bz + q, 
the equations (68) become 



where (x , y , 2 ) is the supposed point of contact on F. Solving 
these equations, we find 



which are precisely the values which give the tangent. A neces 
sary condition that the tangent should have contact of the second 
order is that x J = az^ , y = bztf, that is, 



x o l/o z o 

The points where this happens are those discussed in 217. 

The family of all circles in space depends on six parameters; 
hence the osculating circle will have contact of the second order. 
Let the equations of the circle be written in the form 

F (x, y, z) = A(x - a) + B(y - b) + C(z - c~) =0, 
F, (x, y, z) = (x- a) 2 + (y - W + ( - c) 2 - /2 2 = 0, 



490 SKEW CURVES [XI, 236 

where the parameters are a, b, c, R, and the two ratios of the three 
coefficients A, B, C. The equations which determine the osculating 

circle are 

A(x - ) -f B(y -b} + C(z e) = 0, 

A *2 +B % +C *1-0, 

at at at 



(x - a) 2 + (y - 

C -H>- 



where x, y, and s are to be replaced by /(), <(), and ^(), respec 
tively. The second and the third of these equations show that the 
plane of the osculating circle is the osculating plane of the curve F. 
If a, b, and c be thought of as the running coordinates, the last 
two equations represent, respectively, the normal plane at the point 
(x, y, z) and the normal plane at a point whose distance from 
(x, y, z) is infinitesimal. Hence the center of the osculating circle 
is the point of intersection of the osculating plane and the polar 
line. It follows that the osculating circle coincides with the circle 
of curvature, as we might have foreseen by noticing that two curves 
which have contact of the second order have the same circle of 
curvature, since the values of y 1 , z , y", z" are the same for the two 
curves. 

236. Contact between a curve and a surface. Let S be a surface 
and T a curve tangent to S at a point A. To any point M of T 
near A let us assign a point M of 5 according to such a law that 
M and M approach A simultaneously. First let us try to find what 
law of correspondence between M and M will render the order 
of the infinitesimal MM with respect to the arc AM a. maximum. 
Let us choose a system of rectangular coordinates in such a way 
that the tangent to T shall not be parallel to the yz plane, and that 
the tangent plane to S shall not be parallel to the z axis. Let 
(*oi y< *o) be the coordinates of A ; Z = F(x, y) the equation of S ; 
y =f(x), z = <fr(x) the equations of T ; and n + 1 the order of the 
infinitesimal MM for the given law of correspondence. The 



CONTACT 491 

coordinates of M are [x + h, f(x + A), <f>(x + &)]. Let X, Y, and 
Z = F(X, Y) be the coordinates of M . In order that MM should 
be of order n + 1 with respect to the arc AM, or, what amounts to 
the same thing, with respect to h, it is necessary that each of the 
differences X - x, Y y, and Z z should be an infinitesimal at 
least of order n + 1, that is, that we should have 

X-x = ah" + 1 , Y-y = /3h n + l , Z - z = F(X, F) - z = yh n + l , 

where a, ft, y remain finite as h approaches zero. Hence we shall 
have 

F(x + ah n + l , y + (3h n + l ) -z = yh n + l , 

and the difference F(x, y) z will be itself at least of order n + 1. 
This shows that the order of the infinitesimal MN, where N is the 
point where a parallel to the z axis pierces the surface, will be at 
least as great as that of MM . The maximum order of contact 
which we shall call the order of contact of the curve and the surface 
is therefore that of the distance MN with respect to the arc AM 
or with respect to h. Or, again, we may say that the order of con 
tact of the curve and the surface is the order of contact between T 
and the curve T in which the surface S is cut by the cylinder which 
projects T upon the xy plane. (It is evident that the z axis may be 
any line not parallel to the tangent plane.) For the equations of 
the curve T are 

y=f(x), Z=F[>, /(*)] = *(*), 
and, by hypothesis, 

*(*) = K*o), * (*) = * (*). 
If we also have 



the curve and the surface have contact of order n. Since the equa 
tion $(x) = <t>(x) gives the abscissae of the points of intersection of 
the curve and the surface, these conditions for contact of order n 
at a point A may be expressed by saying that the curve meets the 
surface in n -f 1 coincident points at A. 

Finally, if the curve T is given by equations of the form x =f(f), 
y = <(<), z = \fr(f), and the surface S is given by a singly equation 
of the form F(x, y, z} = 0, the curve T just defined will have equa 
tions of the form x =/(*), y = <(), * = w(), where ir(t) is a func 
tion defined by the equation 

+(),*(*)] 0. 



492 SKEW CURVES [XI, 237 

In order that F and F should have contact of order n, the infini 
tesimal 7r(Y) \l/(f) must be of order n + 1 with respect to t t ; 
that is, we must have 



Using F() to denote the function considered in 234, these equa 
tions may be written in the form 



These conditions may be expressed by saying that the curve and 
the surface have n + 1 coincident points of intersection at their 
point of contact. 

If 5 be one of a family of surfaces which depends on n + 1 
parameters a, b, c, , I, the parameters may be so chosen that S 
has contact of order n with a given curve at a given point ; this 
surface is called the osculating surface. 

In the case of a plane there are three parameters. The equations 
which determine these parameters for the osculating plane are 

Af (t} + B<j> (t) + C$ (t) + D = 0, 
Af (t) + B# (t) + Cy () = 0, 

Af"(t) + B4"(t) + C^"(f) = 0. 

It is clear that these are the same equations we found before for 
the osculating plane, and that the contact is in general of the second 
order. If the order of contact is higher, we must have 

Af "(t) + Bt" () + Cf"(f) = 0, 
i.e. the osculating plane must be stationary. 

237. Osculating sphere. The equation of a sphere depends on four 
parameters ; hence the osculating sphere will have contact of the 
third order. For simplicity let us suppose that the coordinates 
x, y, K of a point of the given curve F are expressed in terms of the 
arc s of that curve. In order that a sphere whose center is (a, b, c} 
and whose radius is p should have contact of the third order with 
F at a given point (a;, y, z) on F, we must have 

F(*) = 0, F (*) = 0, F"(*) = 0, F"(*) = 0, 
where 

F() = (x - a) 2 -f- (y - i) 2 + (z - c) 2 - p* 



XI, 238] CONTACT 493 

and where x, y, z are expressed as functions of s. Expanding the 
last three of the equations of condition and applying Frenet s 
formulae, we find 

F 0) = (x - a) a + (y - b) {3 + (* - c)y = 0, 

O/ - *) + (* - ) + 1= 0, 



/z \R T R 



These three equations determine a, 6, and c. But the first of them 
represents the normal plane to the curve F at the point (x, y, z) in 
the running coordinates (a, b, c), and the other two may be derived 
from this one by differentiating twice with respect to s. Hence 
the center of the osculating sphere is the point where the polar line 
touches its envelope. In order to solve the three equations we may 
reduce the last one by means of the others to the form 

J D 

(x - a ) a + (y - J)0r, + ( , _ c) yr = T , 



from which it is easy to derive the formulae 

a = x + Ra -T^a", b = y + R? - T ~ 



Hence the radius of the osculating sphere is given by the formula 



If R is constant, the center of the osculating sphere coincides with 
the center of curvature, which agrees with the result obtained in 
233. 

238. Osculating straight lines. If the equations of a family of 
curves depend on n -f 2 parameters, the parameters may be chosen 
in such a way that the resulting curve C has contact of order n with 
a given surface S at a point M. For the equation which expresses 
that C meets S at M and the n -f 1 equations which express that 
there are n -f- 1 coincident points of intersection at M constitute 
n + 2 equations for the determination of the parameters. 



494 SKEW CURVES [XI, EM. 

For example, the equations of a straight line depend on four 
parameters. Hence, through each point M of a given surface S, 
there exist one or more straight lines which have contact of the 
second order with the surface. In order to determine these lines, 
let us take the origin at the point M, and let us suppose that the 
z axis is not parallel to the tangent plane at M. Let z = F(x, y) 
be the equation of the surface with respect to these axes. The 
required line evidently passes through the origin, and its equations 
are of the form 

x _ y _ z _ 

a b c 

Hence the equation cp = F(ap, bp) should have a triple root p = ; 
that is, we should have 

c = ap + bq, 



where p, q, r, s, t denote the values of the first and second deriva 
tives of F(x, y) at the origin. The first of these equations expresses 
that the required line lies in the tangent plane, which is evident 
a priori. The second equation is a quadratic equation in the ratio 
ft/a, and its roots are real if s 2 rt is positive. Hence there are in 
general two and only two straight lines through any point of a given 
surface which have contact of the second order with that surface. 
These lines will be real or imaginary according as s 2 rt is positive 
or negative. We shall meet these lines again in the following 
chapter, in the study of the curvature of surfaces. 

EXERCISES 

1. Find, in finite form, the equations of the evolutes of the curve which 
cuts the straight line generators of a right circular cone at a constant angle. 

Discuss the problem. 

[Licence, Marseilles, July, 1884.] 

2. Do there exist skew curves T for which the three points of intersection 
of a fixed plane P with the tangent, the principal normal, and the binormal are 
the vertices of an equilateral triangle ? 

3. Let T be the edge of regression of a surface which is the envelope of 
a one-parameter family of spheres, i.e. the envelope of the characteristic circles. 
Show that the curve which is the locus of the centers of the spheres lies on 
the polar surface of T. Also state and prove the converse. 

4. Let T be a given skew curve, M a point on T, and a fixed point in 
space. Through draw a line parallel to the polar line to T at M, and lay off 
on this parallel a segment ON equal to the radius of curvature of F at M. Show 



XI, Exs.] EXERCISES 495 

that the curve F described by the point N and the curve T" described by the 
center of curvature of F have their tangents perpendicular, their elements of 
length equal, and their radii of curvature equal, at corresponding points. 

[ROUQUET.] 

5. If the osculating sphere to a given skew curve F has a constant radius a, 
show that F lies on a sphere of radius a, at least unless the radius of curvature 
of F is constant and equal to a. 

6. Show that the necessary and sufficient condition that the locus of the 
center of curvature of a helix drawn on a cylinder should be another helix on a 
cylinder parallel to the first one is that the right section of the second cylinder 
should be a circle or a logarithmic spiral. In the latter case show that all the 
helices lie on circular cones which have the same axis and the same vertex. 

[Tissox, Nouvelles Annales, Vol. XI, 1852.] 

7*. If two skew curves have the same principal normals, the osculating 
planes of the two curves at the points where they meet the same normal make 
a constant angle with each other. The two points just mentioned and the cen 
ters of curvature of the two curves form a system of four points whose anhar- 
monic ratio is constant. The product of the radii of torsion of the two curves 
at corresponding points is a constant. 

[PAUL SERRET ; MANNHEIM ; SCHELL.] 

8*. Let x, y, z be the rectangular coordinates of a point on a skew curve F, 
and s the arc of that curve. Then the curve F defined by the equations 

X = I a"ds, yo I P"ds, z = ty"ds, 

where x , yo, z<> are the running coordinates, is called the conjugate curve to F; 
and the curve defined by the equations 

Z sin0, T= y cos0 + y sin0, Z = z cos0 -f 2 sin0, 



where JT, F, Z are the running coordinates and 6 is a constant angle, is called 
a related curve. Find the orientation of the fundamental trihedron for each of 
these curves, and find their radii of curvature and of torsion. 

If the curvature of F is constant, the torsion of the curve F is constant, and 
the related curves are curves of the Bertrand type ( 233). Hence find the 
general equations of the latter curves. 

9. Let F and I" be two skew curves which are tangent at a point A. From 
A lay off infinitesimal arcs AM and AM from A along the two curves in the 
same direction. Find the limiting position of the line MM . 

[CAUGHT.] 

10. In order that a straight line rigidly connected to the fundamental trihe 
dron of a skew curve and passing through the vertex of the trihedron should 
describe a developable surface, that straight line must coincide with the tangent, 
at least unless the given skew curve is a helix. In the latter case there are an 
infinite number of straight lines which have the required property. 



496 SKEW CURVES [XI, Exs. 

For a curve of the Bertrand type there exist two hyperbolic paraboloids 
rigidly connected to the fundamental trihedron, each of whose generators 
describes a developable surface. 

, Bivista di Mathematical,, Vol. II, 1892, p. 155.] 



11*. In order that the principal normals of a given skew curve should be the 
binormals of another curve, the radii of curvature and the radii of torsion of 
the first curve must satisfy a relation of the form 

A 
where A and B are constants. 



/JL , 1\ = 

\R* TV 



[MANNHEIM, Comptes rendus, 1877.] 

[The case in which a straight line through a point on a skew curve rigidly 
connected with the fundamental trihedron is also the principal normal (or the 
binormal) of another skew curve has been discussed by Pellet (Comptes rendus, 
May, 1887), by Cesaro (Nouvelles Annales, 1888, p. 147), and by Balitrand 
(Mathesis, 1894, p. 159).] 

12. If the osculating plane to a skew curve F is always tangent to a fixed 
sphere whose center is 0, show that the plane through the tangent perpen 
dicular to the principal normal passes through 0, and show that the ratio of 
the radius of curvature to the radius of torsion is a linear function of the arc. 
State and prove the converse theorems. 



CHAPTER XII 

SURFACES 
I. CURVATURE OF CURVES DRAWN ON A SURFACE 

239. Fundamental formula. Meusnier s theorem. In order to study 
the curvature of a surface at a non-singular point M, we shall sup 
pose the surface referred to a system of rectangular coordinates 
such that the axis of z is not parallel to the tangent plane at M. 
If the surface is analytic, its equation may be written in the form 

(1) * = F(x,y), 

where F(x, ?/) is developable in power series according to powers of 
x X Q and y y in the neighborhood of the point M (x , y , ) 
( 194). But the arguments which we shall use do not require the 
assumption that the surface should be analytic : we shall merely 
suppose that the function F(x, ?/), together with its first and second 
derivatives, is continuous near the point (x , y ~) We shall use 
Monge s notation, p, q, r, s, t, for these derivatives. 

It is seen immediately from the equation of the tangent plane 
that the direction, cosines of the normal to the surface are propor 
tional to p, q, and 1. If we adopt as the positive direction of the 
normal that which makes an acute angle with the positive z axis, 
the actual direction cosines themselves A, /*, v are given by the 
formulae 



\ - 
A 



u, = 



Let C be a curve on the surface S through the point M, and let 
the equations of this curve be given in parameter form ; then the 
functions of the parameter which represent the coordinates of a 
point of this curve satisfy the equation (1), and hence their differ 
entials satisfy the two relations 

(3) dz = p dx + q di/ , 

(4) <Pz = p d*x + q d*y + r dx 2 + 2s dxdy + t dy*. 

497 



498 SURFACES [XII, 239 

* 

The first of these equations means that the tangent to the curve C 
lies in the tangent plane to the surface. In order to interpret the 
second geometrically, let us express the differentials which occur in 
it in terms of known geometrical quantities. If the independent 
variable be the arc a of the curve C, we shall have 

dx : _ ^/_o ^f_ d*x_<^ &y__P_ <&* = y 

~fo ~ a do- ~ " do-~ y dv* ~ R da 2 ~ R da 2 ~ R 

where the letters a, ft, y, a , /3 , y , R have the same meanings as in 
229. Substituting these values in (4) and dividing by 
that equation becomes 

y -pa -gp* = 

R^/l + p* + q 2 
or, by (2), 

\a + fji/3 + vy = ra 2 + 2sa@ + 

R 



But the numerator Xa + p.(3 + vy is nothing but the cosine of the 
angle included between the principal normal to C and the positive 
direction of the normal to the surface ; hence the preceding formula 
may be written in the form 

COS 6 ra 2 + 2sa/3 + tfB* 



(5) 



R 



This formula is exactly equivalent to the formula (4); hence it 
contains all the information we can discover concerning the curva 
ture of curves drawn on the surface. Since R and Vl + p* + q 2 
are both essentially positive, cos and ra 2 -f 2saft + tf} 2 have the same 
sign, i.e. the sign of the latter quantity shows whether is acute or 
obtuse. In the first place, let us consider all the curves on the sur 
face S through the point M which have the same osculating plane 
(which shall be other than the tangent plane) at the point M. All 
these curves have the same tangent, namely the intersection of the 
osculating plane with the tangent plane to the surface. The direc 
tion cosines a, /?, y therefore coincide for all these curves. Again, 
the principal normal to any of these curves coincides with one of 
the two directions which can be selected upon the perpendicular to the 
tangent line in the osculating plane. Let o> be the angle which the 
normal to the surface makes with one of these directions ; then we 
shall have 6 = o> or = TT to. But the sign of ra 2 + 2sa/3 + tfP 
shows whether the angle is acute or obtuse ; hence the positive 



XII, 239] CURVES ON A SURFACE 499 

direction of the principal normal is the same for all these curves. 
Since 6 is also the same for all the curves, the radius of curvature 
R is the same for them all ; that is to say, all the curves on the sur 
face through the point M which have the same osculating plane have 
the same center of curvature. 

It follows that we need only study the curvature of the plane 
sections of the surface. First let us study the variation of the 
curvature of the sections of the surface by planes which all pass 
through the same tangent MT. We may suppose, without loss of 
generality, that ra z + 2sa(3 + fy3 2 > 0, for a change in the direction 
of the z axis is sufficient to change the signs of r, s, and t. For all 
these plane sections we shall have, therefore, cos > 0, and the 
angle 6 is acute. If R! be the radius of curvature of the section 
by the normal plane through MT, since the corresponding angle 
is zero, we shall have 

1 __ ra* + 2sa(3 + t/3 2 
R l 



Comparing this formula with equation (5), which gives the radius 
of curvature of any oblique section, we find 

1 cos0 

W^-JT 

or R = R! cos 0, which shows that the center of curvature of any 
oblique section is the projection of the center of curvature of the 
normal section through the same tangent line. This is Meusnier s 
theorem. 

The preceding theorem reduces the study of the curvature of 
oblique sections to the study of the curvature of normal sections. 
We shall discuss directly the results obtained by Euler. First let 
us remark that the formula (5) will appear in two different forms 
for a normal section according as ra 2 + 2saft + t{P is positive or 
negative. In order to avoid the inconvenience of carrying these 
two signs, we shall agree to affix the sign + or the sign to the 
radius of curvature R of a normal section according as the direction 
from M to the center of curvature of the section is the same as or 
opposite to the positive direction of the normal to the surface. 
With this convention, R is given in either case by the formula 

(7) 1 = ra* + 28ap + tF 

R 



500 SURFACES [XII, 239 

which shows without ambiguity the direction in which the center 
of curvature lies. 

From (7) it is easy to determine the position of the surface with 
respect to its tangent plane near the point of tangency. For if 
s 2 rt < 0, the quadratic form ra 2 + 2saft }- tfi 2 keeps the same 
sign the sign of r and of t as the normal plane turns around 
the normal; hence all the normal sections have their centers of 
curvature on the same side of the tangent plane, and therefore all 
lie on the same side of that plane : the surface is said to be convex 
at such a point, and the point is called an elliptic point. On the 
contrary, if s 2 rt > 0, the form m 2 + 2sa(3 + tfi 2 vanishes for two 
particular positions of the normal plane, and the corresponding 
normal sections have, in general, a point of inflection. When the 
normal plane lies in one of the dihedral angles formed by these two 
planes, R is positive, and the corresponding section lies above the tan 
gent plane ; when the normal plane lies in the other dihedral angle, 
R is negative, and the section lies below the tangent plane. Hence 
in this case the surface crosses its tangent plane at the point of 
tangency. Such a point is called a hyperbolic point. Finally, if 
s 2 rt = 0, all the normal sections lie on the same side of the tan 
gent plane near the point of tangency except that one for which 
the radius of curvature is infinite. The latter section usually 
crosses the tangent plane. Such a point is called a parabolic point. 

It is easy to verify these results by a direct study of the differ 
ence n = z z of the values of z for a point on the surface and for 
the point on the tangent plane at M which projects into the same 
point (x, y} on the xy plane. For we have 

z = p(x a* ) + q(ij i/ ) , 
whence, for the point of tangency (x n , ?/ ), 

du _ cz _ ^ M _n 

dx dx dy 
and 

d 2 u (ft u d 2 u 



It follows that if s 2 rt < 0, u is a maximum or a minimum at M 
( 56), and since u vanishes at M. it has the same sign for all other 
points in the neighborhood. On the other hand, if s 2 rt > 0, u 
has neither a maximum nor a minimum at M, and hence it changes 
sign in any neighborhood of M. 



XII, 240] CURVES ON A SURFACE 501 

240. Euler s theorems. The indicatrix. In order to study the varia 
tion of the radius of curvature of a normal section, let us take the 
point M as the origin and the tangent plane at M as the xy plane. 
With such a system of axes we shall have p = q = 0, and the 
formula (7) becomes 

(8) = r cos 2 < + 2s cos <f> sin < + t sin 2 <, 

where < is the angle which the trace of the normal plane makes 
with the positive x axis. Equating the derivative of the second 
member to zero, we find that the points at which R may be a maxi 
mum or a minimum stand at right angles. The following geomet 
rical picture is a convenient means of visualizing the variation of R. 
Let us lay off, on the line of intersection of the normal plane with 
the xy plane, from the origin, a length Om equal numerically to the 
square root of the absolute value of the corresponding radius of cur 
vature. The point ra will describe a curve, which gives an instanta 
neous picture of the variation of the radius of curvature. This ciirve 
is called the indicatrix. Let us examine the three possible cases. 

1) s 2 rt < 0. In this case the radius R has a constant sign, which 
we shall suppose positive. The coordinates of m are = V# cos < 
and r) V-R sin < ; hence the equation of the indicatrix is 



(9) re 

which is the equation of an ellipse whose center is the origin. It is 
clear that R is at a maximum for the section made by the normal 
plane through the major axis of this ellipse, and at a minimum for 
the normal plane through the minor axis. The sections made by two 
planes which are equally inclined to the two axes evidently have the 
same curvature. The two sections whose planes pass through the 
axes of the indicatrix are called the principal normal sections, and 
the corresponding radii of curvature are called the principal radii of 
curvature. If the axes of the indicatrix are taken for the axes of x 
and y, we shall have s 0, and the formula (8) becomes 

= T cos 2 < + t sin 2 <. 
R 

With these axes the principal radii of curvature R l and R 2 correspond 
to <j> = and <f> Tr/2, respectively ; hence 1/Ri = r, 1/R 2 = t, and 

(10) 1 ^cos 2 <ft > sin 2 <fr 
R R R 



502 SURFACES [XII, 240 

2) s 2 rt > 0. The normal sections which correspond to the 
values of <f> which satisfy the equation 

r cos 2 < + 2s cos <j> sin < + t sin 2 < = 

have infinite radii of curvature. Let L(OL^ and L^OL 2 be the inter 
sections of these two planes with the xy plane. When the trace of 
the normal plane lies in the angle L^OL^^ for example, the radius 
of curvature is positive. Hence the corresponding portion of the 
indicatrix is represented by the equation 



where and 77 are, as in the previous case, the coordinates of the 
point m. This is an hyperbola whose asymptotes are the lines 
L[OL l and L Z OL Z . When the trace of the normal plane lies in the 
other angle L ^OL lt R is negative, and the coordinates of m are 



= V R cos <f), t] V R sin <. 
Hence the corresponding portion of the indicatrix is the hyperbola 



which is conjugate to the preceding hyperbola. These two hyper 
bolas together form a picture of the variation of the radius of curva 
ture in this case. If the axes of the hyperbolas be taken as the 
x and y axes, the formula (8) may be written in the form (10), as in 
the previous case, where now, however, the principal radii of curva 
ture R and R 2 have opposite signs. 

3) s 2 rt = 0. In this case the radius of curvature R has a 
fixed sign, which we shall suppose positive. The indicatrix is still 
represented by the equation (9), but, since its center is at the origin 
and it is of the parabolic type, it must be composed of two parallel 
straight lines. If the axis of y be taken parallel to these lines, we 
shall have s = 0, t 0, and the general formula (8) becomes 

- = rcos*$, 
it 

or 



R 



This case may also be considered to be a limiting case of either of 
the preceding, and the formula just found may be thought of as the 
limiting case of (10), when R 2 becomes infinite. 



XII, 241] CURVES ON A SURFACE 503 

Euler a formulae may be established without using the formula (5). Taking 
the point M of the given surface as the origin and the tangent plane as the xy 
plane, the expansion of z by Taylor s series may be written in the form 

rz 2 + 2sxy + ty* 

~o- - + -" 

where the terms not written down are of order greater than two. In order 
to find the radii of curvature of the section made by a plane y = x tan 0, we 
may introduce the transformation 

x = x cos y sin <t> , y = x sin + y cos < , 
and then set y = 0. This gives the expansion of z in powers of x , 
_ r cos 2 <(> + 2s sin <f> cos <f> + t sin 2 

2 2J |- 

1.2 
which, by 214, leads to the formula (8). 

Notes. The section of the surface by its tangent plane is given by the equation 
= rz 2 + 2sxy + ty* + <j> 3 (x, y) + , 

and has a double point at the origin. The two tangents at this point are the 
asymptotic tangents. More generally, if two surfaces S and Si are both tangent 
at the origin to the xy plane, the projection of their curve of intersection on the 
xy plane is given by the equation 

= (r - n)x 2 + 2(8 - Sl )xy + (t - t^y* + -, 

where r\, s\, ti have the same meaning for the surface Si that r, s, t have 
for S. The nature of the double point depends upon the sign of the expression 
(s Si) 2 (r ri)(t ti). If this expression is zero, the curve of intersection 
has, in general, a cusp at the origin. 

To recapitulate, there exist on any surface four remarkable posi 
tions for the tangent at any point : two perpendicular tangents for 
which the corresponding radii of curvature have a maximum or a 
minimum, and two so-called asymptotic, or principal,* tangents, for 
which the corresponding radii of curvature are infinite. The latter are 
to be found by equating the trinomial ra 2 -+-2saft + t{P to zero ( 238). 
We proceed to show how to find the principal normal sections and 
the principal radii of curvature for any system of rectangular axes. 

241. Principal radii of curvature. There are in general two different 
normal sections whose radii of curvature are equal to any given 
value of R. The only exception is the case in which the given 
value of R is one of the principal radii of curvature, in which case 

* The reader should distinguish sharply the directions of the principal tangents 
(the asymptotes of the indicatrix) and the directions of the principal normal sections 
(the axes of the indicatrix) . To avoid confusion we shall not use the term principal 
tangent. TRANS. 



504 SURFACES 



[XII, 241 



only the corresponding principal section has the assigned radius 
of curvature. To determine the normal sections whose radius of 
curvature is a given number R, we may determine the values of 
a, ft, y by the three equations 

Vl + p 2 + a* 

-J = ra* + 2sa/3 + tft\ y =p a + qft, a 2 + /3 2 + y 2 =l. 

It is easy to derive from these the following homogeneous combina 
tion of degree zero in a and ft : 



R <** + ft* + (pa + qpf 

It follows that the ratio ft/ a is given by the equation 

a\l + p* - rD) + 2aft(pq - sD) + (?(!+ <f - tD) = 0, 

where R D Vl +p 2 + q 2 . If this equation has a double root, that 
root satisfies each of the equations formed by setting the two first 
derivatives of the left-hand side with respect to a and ft equal to 
zero : 

S D) =0, 



(12) 

I a(pq- sD} + ft(l + q* - tD) = 0. 

Eliminating a and and replacing D by its value, we obtain an 
equation for the principal radii of curvature : 



On the other hand, eliminating D from the equations (12), we obtain 
an equation of the second degree which determines the lines of inter 
section of the tangent plane with the principal normal sections : 

-pqr] 



From the very nature of the problem the roots of the equations (13) 
and (14) will surely be real. It is easy to verify this fact directly. 
In order that the equation for R should have equal roots, it is 
necessary that the indicatrix should be a circle, in which case all 
the normal sections will have the same radius of curvature. Hence 
the second member of (11) must be independent of the ratio ft fa, 
which necessitates the equations 



pq 



XII, 241] CURVES ON A SURFACE 505 

The points which satisfy these equations are called umbilics. At 
such points the equation (14) reduces to an identity, since every 
diameter of a circle is also an axis of symmetry. 

It is often possible to determine th e principal normal sections 
from certain geometrical considerations. For instance, if a surface 
S has a plane of symmetry through a point M on the surface, it is 
clear that the line of intersection of that plane with the tangent 
plane at M is a line of symmetry of the indicatrix ; hence the sec 
tion by the plane of symmetry is one of the principal sections. For 
example, on a surface of revolution the meridian through any point 
is one of the principal normal sections ; it is evident that the plane 
of the other principal normal section passes through the normal to 
the surface and the tangent to the circular parallel at the point. 
But we know the center of curvature of one of the oblique sections 
through this tangent line, namely that of the circular parallel itself. 
It follows from Meusnier s theorem that the center of curvature of 
the second principal section is the point where the normal to the 
surface meets the axis of revolution. 

At any point of a developable surface, s 2 rt = 0, and the indica 
trix is a pair of parallel straight lines. One of the principal sec 
tions coincides with the generator, and the corresponding radius of 
curvature is infinite. The plane of the second principal section is 
perpendicular to the generator. All the points of a developable 
surface are parabolic, and, conversely, these are the only surfaces 
which have that property ( 222). 

If a non-developable surface is convex at certain points, while other 
points of the surface are hyperbolic, there is usually a line of para 
bolic points which separates the region where s 2 rt is positive from 
the region where the same quantity is negative. For example, on the 
anchor ring, these parabolic lines are the extreme circular parallels. 

In general there are on any convex surface only a finite number of umbilics. 
We proceed to show that the only real surface for which every point is an 
umbilic is the sphere. Let X, p, v be the direction cosines of the normal to the 
surface. Differentiating (2), we find the formulae 

ax _ pqs-(l+q*)r 5X _ pqt-(l + q*)s 

CX ~ (l + p2 +9 2)J Sy 

dfj. _ pqr-(l+p*)s dp 



or, by (15), 

= ^ = = 8 -- 

dy ~ dx ~ dx ~ dy 



500 SURFACES [XII, 242 

The first equation shows that X is independent of y, the second that /x is inde 
pendent of z ; hence the common value of d\/cx, dp/dy is independent of both 
x and y, i.e. it is a constant, say I/a. This fact leads to the equations 



x - X y -Mo Va 2 - (x - x ) 2 - (y - 2/o) 2 

X= - , n 



a 

X X 



Va 2 - (x - x ) 2 - (y - yo) 2 
y -2/0 ) 

Va 2 -(x-x ) 2 -(2/-2/o) 2 
whence, integrating, the value of z is found to be 



z = z + Va 2 - (x - x ) 2 - (y - y )* , 

which is the equation of a sphere. It is evident that if 8\/dx = dp/dy = 0, the 
surface is a plane. But the equations (15) also have an infinite number of 
imaginary solutions which satisfy the relation 1 + p 2 + q 2 = 0, as we can see by 
differentiating this equation with respect to x and with respect to y. 



II. ASYMPTOTIC LINES CONJUGATE LINES 

242. Definition and properties of asymptotic lines. At every hyper 
bolic point of a surface there are two tangents for which the corre 
sponding normal sections have infinite radii of curvature, namely 
the asymptotes of the indicatrix. The curves on the given surface 
which are tangent at each of their points to one of these asymptotic 
directions are called asymptotic lines. If a point moves along any 
curve on a surface, the differentials dx, dy, dz are proportional to 
the direction cosines of the tangent. For an asymptotic tangent 
roP + 2saf3 + tft* = ; hence the differentials dx and dy at any point 
of an asymptotic line must satisfy the relation 

(16) rdx 2 + 2sdxdy + tdif = 0. 

If the equation of the surface be given in the form z = F(x, y), and 
we substitute for r, s, and t their values as functions of x and y, 
this equation may be solved for dy/dx, and we shall obtain the two 
solutions 



We shall see later that each of these equations has an infinite num 
ber of solutions, and that every pair of values (x , y ) determines 
in general one and only one solution. It follows that there pass 
through every point of the surface, in general, two and only two 



XII, 242] ASYMPTOTIC LINES CONJUGATE LINES 507 

asymptotic lines : all these lines together form a double system of 
lines upon the surface. 

Again, the asymptotic lines may be defined without the use of 
any metrical relation : the asymptotic lines on a surface are those 
curves for which the osculating plane always coincides with the tan 
gent plane to the surface. For the necessary and sufficient condition 
that the osculating plane should coincide with the tangent plane to 
the surface is that the equations 

dz p dx q dy = , d 2 z p d^x q d*y = 

should be satisfied simultaneously (see 215). The first of these 
equations is satisfied by any curve which lies on the surface. Dif 
ferentiating it, we obtain the equation 

d*z p d*x q d*y dp dx dq dy = , 

which shows that the second of the preceding equations may be 
replaced by the following relation between the first differentials : 

(18) dp dx + dqdy= 0, 

an equation which coincides with (16). Moreover it is easy to 
explain why the two definitions are equivalent. Since the radius of 
curvature of the normal section which is tangent to an asymptote 
of the indicatrix is infinite, the radius of curvature of the asymp 
totic line will also be infinite, by -Meusnier s theorem, at least unless 
the osculating plane is perpendicular to the normal plane, in which 
case Meusnier s theorem becomes illusory. Hence the osculating 
plane to an asymptotic line must coincide with the tangent plane, 
at least unless the radius of curvature is infinite ; but if this were 
true, the line would be a straight line and its osculating plane 
would be indeterminate. It follows from this property that any 
projective transformation carries the asymptotic lines into asymp 
totic lines. It is evident also that the differential equation is of 
the same form whether the axes are rectangular or oblique, for the 
equation of the osculating plane remains of the same form. 

It is clear that the asymptotic lines exist only in case the points of 
the surface are hyperbolic. But when the surface is analytic the 
differential equation (16) always has an infinite number of solu 
tions, real or imaginary, whether s 1 rt is positive or negative. As a 
generalization we shall say that any convex surface possesses two sys 
tems of imaginary asymptotic lines. Thus the asymptotic lines of an 
unparted hyperboloid are the two systems of rectilinear generators. 



508 SURFACES [XII, 243 

For an ellipsoid or a sphere these generators are imaginary, but 
they satisfy the differential equation for the asymptotic lines. 

Example. Let us try to find the asymptotic lines of the surface 

z = x m y*. 
In this example we have 

r = m(m l)x m - 2 y n , s mnx m - l y n - 1 , t = n(n l)x m y"- 2 , 
and the differential equation (16) may be written in the form 



x dy/ \x dy 

This equation may be solved as a quadratic in (ydx)/(xdy). Let hi and h^ be 
the solutions. Then the two families of asymptotic lines are the curves which 
project, on the xy plane, into the curves 



243. Differential equation in parameter form. Let the equations of 
the surface be given in terms of two parameters u and v : 

(19) x=f(u, v), y = $(u,v}, z = ^(u,v}. 

Using the second definition of asymptotic lines, let us write the 
equation of the tangent plane in the form 

(20) A(X - x) + B(Y - y) + C(Z - z) = 0, 
where A, B, and C satisfy the equations 



- 

dv ov 8v 

which are the equations for A, B, and C found in 39. Since the 
osculating plane of an asymptotic line is the same as this tangent 
plane, these same coefficients must satisfy the equations 

Adx + Bdij + Cdz =0, 
Ad*x + Bd*y + Cd?z = 0. 

The first of these equations, as above, is satisfied identically. Differ 
entiating it, we see that the second may be replaced by the equation 

(22) dA dx + dBdy + dCdz = 0, 

which is the required differential equation. If, for example, we 
set C = 1 in the equations (21), A and B are equal, respectively, 
to the partial derivatives p and q of z with respect to x and y, and 
the equation (22) coincides with (18). 



XII, 244] ASYMPTOTIC LINES CONJUGATE LINES 



509 



Examples. As an example let us consider the conoid z = <f>(y/x). This equa 
tion is equivalent to the system x = u, y = w>, z = </>(), and the equations (21) 

become 

A + Bv = , Bu + C<j> (v) - . 

These equations are satisfied if we set C = u, A = # (), .B = # () ; hence 
the equation (22) takes the form 

utf>"(v)dv 2 24> (v)dudv = 0. 

One solution of this equation is v = const., which gives the rectilinear genera 
tors. Dividing by dv, the remaining equation is 

<t>"(v) dv _ 2 du 
<j> (v) u 

whence the second system of asymptotic lines are the curves on the surface 
defined by the equation w 2 = K<f> (v) , which project on the xy plane into the 
curves 



Again, consider the surfaces discussed by Jamet, whose equation may be 
written in the form 



Taking the independent variables z and u = y/x, the differential equation of 
the asymptotic lines may be written in the form 



JTW \ /(u> 

from which each of the systems of asymptotic lines may be found by a single 
quadrature. 

A helicoid is a surface defined by equations of the form 



The reader may show that the differential equation of the asymptotic lines is 

pf"(p) dp* - 2h dw dp + p*f (p) du* = , 
from which w may be found by a single quadrature. 

244. Asymptotic lines on a ruled surface. Eliminating A , B, and C 
between the equations (21) and the equation 

A /-/2 n* I D sj* .-I* I /"* ^-72 fj _ _ || 

JT. \JU *J ^^ J-f U/ I/ [^ \*> Uv & V/ y 

we find the general differential equation of the asymptotic lines : 

d f d <j> diff 

du du du 

(23) df 8$ d$ =0. 

dv dv dv 



510 



SURFACES 



[XII, 244 



This equation does not contain the second differentials d z u and d 2 v, 
for we have 



CV 



CUCV 



du dv 



ov* 



dv* 



and analogous expressions for d 2 y and d 2 z. Subtracting from the 
third row of the determinant (23) the first row multiplied by d 2 u 
and the second row multiplied by d 2 v, the differential equation 
becomes 



cu 
df_ 



_ _ 

du en 

c_$ cj, 

do cv 



cu cv 



vtr 



= 0. 



Developing this determinant with respect to the elements of the 
first row and arranging with respect to du and dv, the equation 
may be written in the form 

(24) D du 2 + 2D du dv + D" dv* = 0, 

where D, D , and D" denote the three determinants 



(25) 





dx dy dz 

dll du du 




dx dy dz 
cu du du 


- 


dx dy dz 
dv dv cv 


, D = 


dx dy dz 

dv dv dv 




d 2 x d 2 y d 2 z 




c 2 x d 2 y d 2 z 


du 2 du 2 du 2 


du dv du dv du dv 




dx dy dz 






du du cu 




, "= 


dx dy dz 

dv dv dv 







fo 2 dv 2 W 2 





As an application let us consider a ruled surface, that is, a surface 
whose equations are of the form 



where x , y , z , a, /8, y are all functions of a second variable param 
eter v. If we set u = 0, the point (a- , y , z ) describes a certain 
curve F which lies on the surface. On the other hand, if we set 
v const, and let u vary, the point (x, y, z) will describe a straight- 



XII, 245] ASYMPTOTIC LINES CONJUGATE LINES 511 

line generator of the ruled surface, and the value of u at any point 
of the line will be proportional to the distance between the point 
(x, y, z) and the point (x , y , s ) at which the generator meets the 
curve F. It is evident from the formulae (25) that D = 0, that Z> 
is independent of u, and that D" is a polynomial of the second 
degree in u: 



D" 



+ 



Since dv is a factor of (24), one system of asymptotic lines consists 
of the rectilinear generators v const. Dividing by dv, the remain 
ing differential equation for the other system of asymptotic lines is 
of the form 

CL tl> 

(26) - + Lu 2 + Mu + N = 0, 

dv 

where L, M, and N are functions of the single variable v. An equa 
tion of this type possesses certain remarkable properties, which we 
shall study later. For example, we shall see that the anharmonic 
ratio of any four solutions is a constant. It follows that the anhar 
monic ratio of the four points in which a generator meets any four 
asymptotic lines of the other system is the same for all generators, 
which enables us to discover all the asymptotic lines of the second 
system whenever any three of them are known. We shall also 
see that whenever one or two integrals of the equation (26) are 
known, all the rest can be found by two quadratures or by a single 
quadrature. Thus, if all the generators meet a fixed straight line, 
that line will be an asymptotic line of the second system, and all 
the others can be found by two quadratures. If the surface pos 
sesses two such rectilinear directrices, we should know two asymp 
totic lines of the second system, and it would appear that another 
quadrature would be required to find all the others. But we can 
obtain a more complete result. For if a surface possesses two 
rectilinear directrices, a protective transformation can be found 
which will carry one of them to infinity and transform the surface 
into a conoid ; but we saw in 243 that the asymptotic lines on a 
conoid could be found without a single quadrature. 

245. Conjugate lines. Any two conjugate diameters of the indica- 
trix at a point of a given surface S are called conjugate tangents. 
To every tangent to the surface there corresponds a conjugate 
tangent, which coincides with the first when and only when the given 



512 SURFACES [XII, 245 

tangent is an asymptotic tangent. Let z = F(x, y) be the equation of 
the surface S, and let m and m be the slopes of the projections of 
two conjugate tangents on the xy plane. These projections on the 
xy plane must be harmonic conjugates with respect to the projec 
tions of the two asymptotic tangents at the same point of the sur 
face. But the slopes of the projections of the asymptotic tangents 
satisfy the equation 

r + 2s p. + tp. 2 . 

In order that the projections of the conjugate tangents should be 
harmonic conjugates with respect to the projections of the asymp 
totic tangents, it is necessary and sufficient that we should have 

(27) r + s (m + m ) + tmm = . 

If C be a curve on the surface S, the envelope of the tangent 
plane to S at points along this curve is a developable surface which 
is tangent to S all along C. At every point M of C the generator of 
this developable is the conjugate tangent to the tangent to C. Along 
C, x, y, 2, p, and q are functions of a single independent variable a. 
The generator of the developable is defined by the two equations 

Z - z - p(X - x) - q(Y - y) = 0, 
dz + p dx + q dy dp (X x~) dq( Y y} = , 

the last of which reduces to 

Y y _ dp rdx + sdy 

X x dq s dx + tdy 

Let m be the slope of the projection of the tangent to C and m the 
slope of the projection of the generator. Then we shall have 

*y = m y - y = m > 

dx X x 

and the preceding equation reduces to the form (27), which proves 
the theorem stated above. 

Two one-parameter families of curves on a surface are said to 
form a conjugate network if the tangents to the two curves of the 
two families which pass through any point are conjugate tangents 
at that point. It is evident that there are an infinite number of 
conjugate networks on any surface, for the first family may be 
assigned arbitrarily, the second family then being determined by a 
differential equation of the first order. 



XII, 245] ASYMPTOTIC LINES CONJUGATE LINES 513 

Given a surface represented by equations of the form (19), let us find the 
conditions under which the curves u = const, and v = const, form a conjugate 
network. If we move along the curve v = const. , the characteristic of the 
tangent plane is represented by the two equations 

A(X - x) + B(Y -y) + C(Z -z) = 0, 



In order that this straight line should coincide with the tangent to the curve 
u = const., whose direction cosines are proportional to dx/dv, dy/dv, dz/dv, it 
is necessary and sufficient that we should have 



cv dv dv 

dA dx dB dy dC dz _ 
du dv du dv du dv 

Differentiating the first of these equations with regard to u, we see that the 
second may be replaced by the equation 

(28) 



dudv 



dudv 



and finally the elimination of A, JB, and C between the equations (21) and (28) 
leads to the necessary and sufficient condition 

dx dy dz 

du du du 

dx dy dz 

dv dv dv 

du dv cu dv du dv 

This condition is equivalent to saying that x, y, z are three solutions of a 
differential equation of the form 



= 0. 



(29) 



du dv 



du 



where M and N are arbitrary functions of u and v. It follows that the knowl 
edge of three distinct integrals of an equation of this form is sufficient to 
determine the equations of a surface which is referred to a conjugate network. 
For example, if we set M = N = 0, every integral of the equation (29) is 
the sum of a function of u and a function of v ; hence, on any surface whose 
equations are of the form 

(30) x=/(u)+/i(t>), y = *(u)+^(w), = ^(u) + ^i(t), 

the curves (u) and (v) form a conjugate network. 

Surfaces of the type (30) are called surfaces of translation. Any such surface 
may be described in two different ways by giving one rigid curve F a motion of 
translation such that one of its points moves along another rigid curve T . For, 



514 SURFACES [XII, 246 

let MQ, MI , M 2 , M be four points of the surface which correspond, respectively, 
to the four sets of values (u , o), (M, v ), (w , ), (u, v) of the parameters u and u. 
By (30) these four points are the vertices of a plane parallelogram. If is fixed 
and u allowed to vary, the point MI will describe a curve T on the surface ; like 
wise, if M is kept fixed and v is allowed to vary, the point M z will describe 
another curve T on the surface. It follows that we may generate the surface by 
giving F a motion of translation which causes the point M% to describe r", or by 
giving T a motion of translation which causes the point MI to describe T. It is 
evident from this method of generation that the two families of curves (M) and (v) 
are conjugate. For example, the tangents to the different positions of T at the 
various points of T form a cylinder tangent to the surface along F ; hence the 
tangents to the two curves at any point are conjugate tangents. 



III. LINES OF CURVATURE 

246. Definition and properties of lines of curvature. A curve on a 
given surface S is called a line of curvature if the normals to the 
surface along that curve form a developable surface. If z f(x, y) 
is the equation of the surface referred to a system of rectangular 
axes, the equations of the normal to the surface are 



Y=-qZ +(y+qz). 

The necessary and sufficient condition that this line should describe 
a developable surface is that the two equations 



- Z dq + d(y + qz) = 

should have a solution in terms of Z ( 223), that is, that we 
should have 

d(x + pz} _ d(y + qz) 

dp dq 

or, more simply, 

dx + p dz dy -\- q dz 
dp dq 

Again, replacing dz, dp, and dq by their values, this equation may 
be written in the form 

(l + p*)dx + pqdy _ pqdx 



rdx + sdy sdx + tdy 

This equation possesses two solutions in dy/dx which are always 
real and unequal if the surface is real, except at an umbilic. For, 
if we replace dx and dy by a and /3, respectively, the preceding 



XII, 246] LINES OF CURVATURE 515 

equation coincides with the equation found above [(14), 241] for 
the determination of the lines of intersection of the principal normal 
sections with the tangent plane. It follows that the tangents to the 
lines of curvature through any point coincide with the axes of the 
indicatrix. We shall see in the study of differential equations that 
there is one and only one line of curvature through every non- 
singular point of a surface tangent to each one of the axes of the 
indicatrix at that point, except at an umbilic. These lines are 
always real if the surface is real, and the network which they form 
is at once orthogonal and conjugate, a characteristic property. 

Example. Let us determine the lines of curvature of the paraboloid z = 
xy/a. In this example 

a a a 

and the differential equation (33) is 

(a2 + 2/ 2 )dx 2 = (a 2 + z 2 )dy 2 or dx dy =Q 

Vz 2 + a 2 Vy 2 + a 2 

If we take the positive sign for both radicals, the general solution is 

(x + Vz 2 + a 2 )(y 4- Vy 2 + a 2 ) = C, 
which gives one system of lines of curvature. If we set 



(34) X - z Vy 2 + a 2 + y Vx 2 + a 2 , 
the equation of this system may be written in the form 

X + VX 2 + a* = C 
by virtue of the identity 

(z Vy 2 + a 2 + y Vz 2 + a 2 ) 2 + a* = [xy + V(x 2 + a 2 )(j/ 2 + a 2 )] 2 

It follows that the projections of the lines of curvature of this first system are 
represented by the equation (34), where X is an arbitrary constant. It may be 
shown in the same manner that the projections of the lines of curvature of the 
other system are represented by the equation 

(35) z Vy 2 + a 2 - y Vx 2 + a 2 = /*. 

From the equation xy = az of the given paraboloid, the equations (34) and 
(35) may be written in the form 

Vx 2 + z 2 + Vy 2 -f z 2 = C, Vx 2 + z 2 - Vy 2 + z 2 = C . 

But the expressions Vz 2 + z 2 and Vy 2 + z 2 represent, respectively, the dis 
tances of the point (z, y, z) from the axes of z and y. It follows that the lines 
of curvature on the paraboloid are those curves for which the sum or the difference 
of the distances of any point upon them from the axes of x and y is a constant. 



516 SURFACES [XII, 247 

247. Evolute of a surface. Let C be a line of curvature on a sur 
face S. As a point M describes the curve C, the normal MN to the 
surface remains tangent to a curve T. Let (X, Y, Z) be the coor 
dinates of the point A at which MN is tangent to T. The ordinate 
Z is given by either of the equations (32), which reduce to a single 
e< {nation since C is a line of curvature. The equations (32) may 
be written in the form 



z _ z _ (1 + P 2 ) dx + pq dy ^pgdx + (l+ ? 2 ) dy 
r dx + s dy s dx + tdy 

Multiplying each term of the first fraction by dx, each term of the 
second by dy, and then taking the proportion by composition, we 
find 

_ dx 2 + dy* + (p dx + g dy}* 
r dx 2 + 2s dx dy + t dif 

Again, since dx, dtj, and dz are proportional to the direction cosines 
a, ft, y of the tangent, this equation may be written in the form 



_ 

* *5 - 



_ a 2 + /? 2 + (pa + 

- 



m 2 + 2sp + tfi 2 ra 2 + 2safi 



Comparing this formula with (7), which gives the radius of curva 
ture R of the normal section tangent to the line of curvature, with 
the proper sign, we see that it is equivalent to the equation 



(36) Z - z = R = R V , 



where v is the cosine of the acute angle between the z axis and the 
positive direction of the normal. But z + Kv is exactly the value 
of Z for the center of curvature of the normal section under con 
sideration. It follows that the point of tangency A of the normal 
MN to its envelope T coincides with the center of curvature of the 
principal normal section tangent to C at M. Hence the curve F is 
the locus of these centers of curvature. If we consider all the lines 
of curvature of the system to which C belongs, the locus of the cor 
responding curves r is a surface 2 to which every normal to the 
given surface S is tangent. For the normal MN, for example, is 
tangent at A to the curve r which lies on 2. 

The other line of curvature C through M cuts C at right angles. 
The normal to S along C is itself always tangent to a curve T 
which is the locus of the centers of curvature of the normal sections 



XII, 248] 



LINES OF CURVATURE 



517 



tangent to C". The locus of this curve T for all the lines of curva 
ture of the system to which C belongs is a surface 2 to which all 
the normals to S are tangent. The two surfaces 2 and 2 are not 
usually analytically distinct, but form two nappes of the same sur 
face, which is then represented by an irreducible equation. 

The normal MN to S is tangent to each of these nappes 2 and 2 
at the two principal centers of curvature A and A of the surface S 
at the point M. It is easy to find the tangent 
planes to the two nappes at the points A and A 
(Fig. 51). As the point M describes the curve 
C, the normal MN describes the developable 
surface D whose edge of regression is F ; at 
the same time the point A where MN touches 
2 describes a curve y distinct from F f , since 
the straight line MN cannot remain tangent to 
two distinct curves F and F . The developable 
D and the surface 2 are tangent at A ; hence 
the tangent plane to 2 at A is tangent to D 
all along MN. It follows that it is the plane 
NMT, which passes through the tangent to C. 
Similarly, it is evident that the tangent plane 
to 2 at A is the plane NMT through the tan 
gent to the other line of curvature C . 

The two planes NMT and NMT stand at right angles. This fact 
leads to the following important conception. Let a normal OM be 
dropped from any point O in space on the surface S, and let A and 
A be the principal centers of curvature of S on this normal. The 
tangent planes to 2 and 2 at A and A , respectively, are perpendic 
ular. Since each of these planes passes through the given point 0, it 
is clear that the two nappes of the e volute of any surface S, observed 
from any point O in space, appear to cut each other at right angles. 
The converse of this proposition will be proved later. 

248. Rodrigues formulae. If X, /A, v denote the direction cosines 
of the normal, and R one of the principal radii of curvature, the 
corresponding principal center of curvature will be given by the 
formulae 




FIG. 51 



(37) X = 



Z = z + Rv. 



As the point (x, y, z) describes a line of curvature tangent to 
the normal section whose radius of curvature is R, this center of 



518 SURFACES [XII, 249 

curvature, as we have just seen, will describe a curve F tangent to 
the normal MN; hence we must have 

dX _dY _dZ 

X fji v 

or, replacing X, Y, and Z by their values from (37) and omitting the 
common term dR, 

dx -f R d\ _ dy + Rdp. _ dz + Rdy 
X p. v 

The value of any of these ratios is zero, for if we take them by 
composition after multiplying each term of the first ratio by X, of 
the second by p., and of the third by v, we obtain another ratio 
equal to any of the three ; but the denominator of the new ratio is 
unity, while the numerator 

X dx + fj. dy + v dz + R(\ dX + p. dp. + v dv) 

is identically zero. This gives immediately the formulae of Olinde 
Rodrigues : 

(38) dx + R dX = 0, dy + Rdn = 0, dz + Rdv = 0, 



which are very important in the theory of surfaces. It should be 
noticed, however, that these formulae apply only to a displacement 
of the point (a;, y, z) along a line of curvature. 

249. Lines of curvature in parameter form. If the equations of the 
surface are given in terms of two parameters u and v in the form 
(19), the equations of the normal are 

Xx _ Yy_Zz 
A B C 

where A, B, and C are determined by the equations (21). The 
necessary and sufficient condition that this line should describe a 
developable surface is, by 223, 



(39) 



dx dy dz 
ABC 
dA dB dC 



= 0, 



where x, y, z, A, B, and C are to be replaced by their expressions 
in terms of the parameters u and v; hence this is the differential 
equation of the lines of curvature. 



XII, 249] LINES OF CURVATURE 519 

As an example let us find the lines of curvature on the helicoid 



z = a arc tan - , 
x 



whose equation is equivalent to the system 



In this example the equations for A, B, and C are 

A cos + B sin = , Ap sin + Bp cos -f Ca = . 

Taking C = p, we find A = a sin 0, B = a cos 0. After expansion and simpli 
fication the differential equation (39) becomes 

or de- dp 



Choosing the sign + , for example, and integrating, we find 

p + V/o 2 + a 2 = ae-o, or p = - [e-e c - e~(- o)] . 

2 

The projections of these lines of curvature on the xy plane are all spirals which 
are easily constructed. 

The same method enables us to form the equation of the second 
degree for the principal radii of curvature. With the same symbols 
A, B, C, \, n, v we shall have, except for sign, 






f B 2 + C 2 

We shall adopt as the positive direction of the normal that which 
is given by the preceding equations. If .R is a principal radius of 
curvature, taken with its proper sign, the coordinates of the corre 
sponding center of curvature are 

where 



R = p^A* + B 2 + C 2 . 

If the point (x, y, z) describes the line of curvature tangent to the 
principal normal section whose radius of curvature is R, we have 
seen that the point (X, Y, Z) describes a curve F which is tangent 
to the normal to the surface. Hence we must have 

dx + p dA + A dp _ dy + p dB + B dp _ dz + pdC + Cdp 
A B C 



620 



SURFACES 



[XII, 250 



or, denoting the common values of these ratios by dp + K, 

(dx + pdA A K , 
dy + pdB - BK= 0, 
dz + pdC - CK= 0. 

Eliminating p and K from these three equations, we find again the 
differential equation (39) of the lines of curvature. But if we 
replace dx, dy, dz, dA, dB, and dC by the expressions 



ox 

^ d u 
du 



dx 

^ 
dv 



dv, 



dC T dC 
du Hh a~ 

tin cv 



respectively, and then eliminate du, dv, and K, we find an equation 
for the determination of p : 



(41) 



dx 


dA 


dx 


dA 




du 


+ P ~du 


fa 


+ p ^ 


A 


du 


+ p du 


dv 


cv 


B 


o ^ 


dc 


dz 


dC 














C 


du 


du 


du 


cv 





= 0. 



If we replace p by R/ ~vA* + B z -f C 2 , this equation becomes an 
equation for the principal radii of curvature. 

The equations (39) and (41) enable us to answer many questions 
which we have already considered. For example, the necessary 
and sufficient condition that a point of a surface should be a para 
bolic point is that the coefficient of p 2 in (41) should vanish. In 
order that a point be an umbilic, the equation (39) must be satisfied 
for all values of du and dv 

As an example let us find the principal radii of curvature of the rectilinear 
helicoid. With a slight modification of the notation used above, we shall have 
in this example 

= MCOS, y u sin v, z = av, 

A = as mv, B = acosv, C = u, 
and the equation (41) becomes 



whence R (a 2 + u 2 )/a. Hence the principal radii of curvature of the helicoid 
are numerically equal and opposite in sign. 

250. Joachimsthal s theorem. The lines of curvature on certain 
surfaces may be found by geometrical considerations. For example, 
it is quite evident that the lines of curvature on a surface of revolu 
tion are the meridians and the parallels of the surface, for each of 



XII, 251] LINES OF CURVATURE 521 

these curves is tangent at every point to one of the axes of the 
indicatrix at that point. This is again confirmed by the remark 
that the normals along a meridian form a plane, and the normals 
along a parallel form a circular cone, in each case the normals 
form a developable surface. 

On a developable surface the first system of lines of curvature 
consists of the generators. The second system consists of the 
orthogonal trajectories of the generators, that is, of the involutes of 
the edge of regression ( 231). These can be found by a single quad 
rature. If we know one of them, all the rest can be found without 
even one quadrature. All of these results are easily verified directly. 

The study of the theory of evolutes of a skew curve led Joa- 
chimsthal to a very important theorem, which is often used in that 
theory. Let S and S be two surfaces whose line of intersection C 
is a line of curvature on each surface. The normal MN to S along 
C describes a developable surface, and the normal MN to S along 
C describes another developable surface. But each of these normals 
is normal to C. It follows from 231 that if two surfaces have a 
common line of curvature, they intersect at a constant angle along 
that line. 

Conversely, if two surfaces intersect at a constant angle, and if 
their line of intersection is a line of curvature on one of them, it is 
also a line of curvature on the other. For we have seen that if one 
family of normals to a skew curve C form a developable surface, 
the family of normals obtained by turning each of the first family 
through the same angle in its normal plane also form a developable 
surface. 

Any curve whatever on a plane or on a sphere is a lie of curva 
ture on that surface. It follows as a corollary to Joachimsthal s 
theorem that the necessary and sufficient condition that a plane curve 
or a spherical curve on any surface should be a line of curvature is 
that the plane or the sphere on which the curve lies should cut the 
surface at a constant angle. 

251. Dupin s theorem. We have already considered [ 43, 146] 
triply orthogonal systems of surfaces. The origin of the theory of 
such systems lay in a noted theorem due to Dupin, which we shall 
proceed to prove : 

Given any three families of surfaces which form a triply orthogonal 
system : the intersection of any two surfaces of different families is a 
line of curvature on each of them. 



522 SURFACES [XII, 251 

We shall base the proof on the following remark. Let F(x, y, z) = 
be the equation of a surface tangent to the xij plane at the origin. Then 
we shall have, for x y = z 0, dF/dx = 0, dF/cy = 0, but cF/dz does 
not vanish, in general, except when the origin is a singular point. 
It follows that the necessary and sufficient condition that the x and 
y axes should be the axes of the indicatrix is that s = 0. But the 
value of this second derivative s = c 2 z/cx dy is given by the equation 



r\ f\ I r\ o Y I O Ci J"^ I O O V 7 O 

X <7y CX CZ Cy CZ CZ* CZ 

Since p and q both vanish at the origin, the necessary and sufficient 
condition that s should vanish there is that we should have 

dx dy 

Now let the three families of the triply orthogonal system be given 
by the equations 

where F l , F 2 , F 3 satisfy the relation 

f A <3\ 1 ? J _1 2 i 1 2 f\ 

(4:0) -^ o I a 1 < ^ " ^ 

cx ox cy cy cz cz 

and two other similar relations obtained by cyclic permutation of 
the subscripts 1, 2, 3. Through any point M in space there passes, 
in general, one surface of each of the three families. The tangents to 
the three curves of intersection of these three surfaces form a trirec- 
tangular trihedron. In order to prove Dupin s theorem, it will be 
sufficient to show that each of these tangents coincides with one of 
the axes of the indicatrix on each of the surfaces to which it is 
tangent. 

In order to show this, let us take the point M as origin and the 
edges of the trirectangular trihedron as the axes of coordinates ; 
then the three surfaces pass through the origin tangent, respec 
tively, to the three coordinate planes. At the origin we shall have, 
for example, 

l&Y-o. 



Uo, I^Uo, l^) = o. 



XII, 251] LINES OF CURVATURE 523 

The axes of x and y will be the axes of the indicatrix of the surface 
F(x, y, z) = at the origin if (c^F^/dx dy) = 0. To show that this 
is the case, let us differentiate (43) with respect to y, omitting the 
terms which vanish at the origin ; we find 



\ / 1^\ /d_F\\ 



e \dx di/ \ dz / \dy 
or 






From the two relations analogous to (43) we could deduce two 
equations analogous to (44), which may be written down by cyclic 
permutation : 




o _ 

; 



From (44) and (45) it is evident that we shall have also 



which proves the theorem. 

A remarkable example of a triply orthogonal system is furnished 
by the confocal quadrics discussed in 147. It was doubtless the 
investigation of this particular system which led Dupin to the gen 
eral theorem. It follows that the lines of curvature on an ellipsoid 
or an hyperboloid (which had been determined previously by Monge) 
are the lines of intersection of that surface with its confocal quadrics. 

The paraboloids represented by the equation 

.1 o 

y , * - o~ _ 



, 

p X (/ A 

where X is a variable parameter, form another triply orthogonal 
system, which determines the lines of curvature on the paraboloid. 
Finally, the system discussed in 24G, 



= y, 
is triply orthogonal. 



524 SURFACES [XII, 252 

The study of triply orthogonal systems is one of the most interest 
ing and one of the most difficult problems of differential geometry. 
A very large number of memoirs have been published on the subject, 
the results of which have been collected by Darboux in a recent 
work.* Any surface S belongs to an infinite number of triply 
orthogonal systems. One of these consists of the family of surfaces 
parallel to S and the two families of developables formed by the 
normals along the lines of curvature on S. For, let O be any point 
on the normal MN to the surface S at the point M, and let MT 
and MT be the tangents to the two lines of curvature C and C" 
which pass through M; then the tangent plane to the parallel sur 
face through O is parallel to the tangent plane to S at M, and the 
tangent planes to the two developables described by the normals to 
S along C and C are the planes MNT and MNT , respectively. These 
three planes are perpendicular by pairs, which shows that the system 
is triply orthogonal. 

An infinite number of triply orthogonal systems can be derived 
from any one known triply orthogonal system by means of succes 
sive inversions, since any inversion leaves all angles unchanged. 
Since any surface whatever is a member of some triply orthogonal 
system, as we have just seen, it follows that an inversion carries the 
lines of curvature on any surface over into the lines of curvature on 
the transformed surface. It is easy to verify this fact directly. 

252. Applications to certain classes of surfaces. A large number of problems 
have been discussed in which it is required to find all the surfaces whose lines 
of curvature have a preassigned geometrical property. AVe shall proceed to 
indicate some of the simpler results. 

First let us determine all those surfaces for which one system of lines of 
curvature are circles. By Joachimsthal s theorem, the plane of each of the 
circles must cut the surface at a constant angle. Hence all the normals to the 
surface along any circle C of the system must meet the axis of the circle, i.e. 
the perpendicular to its plane at its center, at the same point 0. The sphere 
through C about as center is tangent to the surface all along C ; hence the 
required surface must be the envelope of a one-parameter family of spheres. 
Conversely, any surface which is the envelope of a one-parameter family of 
spheres is a solution of the problem, for the characteristic curves, which are 
circles, evidently form one system of lines of curvature. 

Surfaces of revolution evidently belong to the preceding class. Another 
interesting particular case is the so-called tubular surface, which is the envelope 
of a sphere of constant radius whose center describes an arbitrary curve F. The 
characteristic curves are the circles of radius R whose centers lie on r and 
whose planes are normal to T. The normals to the surface are also normal to T ; 

* Lemons sur les systemes orthogonaux ft les coordonntes curvilignes, 1898. 



XII, 252] LINES OF CURVATURE 525 

hence the second system of lines of curvature are the lines in which the surface 
is cut by the developable surfaces which may be formed from the normals to r. 

If both systems of lines of curvature on a surface are circles, it is clear from 
the preceding argument that the surface may be thought of as the envelope of 
either of two one-parameter families of spheres. Let <Si , S 2 , S 3 be any three 
spheres of the first family, C\ , C 2 , C 3 the corresponding characteristic curves, 
and MI , M 2 , M s the three points in which Ci , C 2 , C 3 are cut by a line of curva 
ture C" of the other system. The sphere <S which is tangent to the surface along 
C" is also tangent to the spheres Si , S 2 , S s at MI , JV/ 2 , M 3 , respectively. Hence 
the required surface is the envelope of a family of spheres each of which touches 
three fixed spheres. This surface is the well-known Dupin cyclide. Mannheim 
gave an elegant proof that any Dupin cyclide is the surface into which a certain 
anchor ring is transformed by a certain inversion. Let 7 be the circle which 
is orthogonal to each of the three fixed spheres Si, S 2 , 83. An inversion whose 
pole is a point on the circumference of 7 carries that circle into a straight line 
00 , and carries the three spheres Si, S 2 , Ss into three spheres 2i, 2 2 , 2 3 
orthogonal to OO 7 , that is, the centers of the transformed spheres lie on OO . 
Let Ci, C 2 , C be the intersections of these spheres with any plane through 
O(y, C a circle tangent to each of the circles C{, C 2 , 3, and 2 the sphere 
on which C is a great circle. It is clear that 2 remains tangent to each of the 
spheres Si, S 2 , 2 3 as the whole figure is revolved about 00 , and that the 
envelope of 2 is an anchor ring whose meridian is the circle C . 

Let us now determine the surface for which all of the lines of curvature of 
one system are plane curves whose planes are all parallel. Let us take the xy 
plane parallel to the planes in which these lines of curvature lie, and let 

x cos a + y sin a = F(a, z) 

be the tangential equation of the section of the surface by a parallel to the xy 
plane, where F(a, z) is a function of a and z which depends upon the surface 
under consideration. The coordinates x and y of a point of the surface are 
given by the preceding equation together with the equation 

dF 

x sin a + y cos a = 

da 
The formulae for x, y, z are 

dF d F 

(46) z = Fcos<r sin a, y = Fs ma-\ cos a, z = z. 

da da 

Any surface may be represented by equations of this form by choosing the 
function F(a, z) properly. The only exceptions are the ruled surfaces whose 
directing plane is the xy plane. It is easy to show that the coefficients A, B, C 
of the tangent plane may be taken to be 

dF 
A = cos a , .B = sin a . C = ; 

dz 

hence the cosine of the angle between the normal and the z axis is 



In order that all the sections by planes parallel to the xy plane be lines of curva 
ture, it is necessary and sufficient, by Joachimsthal s theorem, that each of 



526 SURFACES [XII,253 

these planes cut the surface at a constant angle, i.e. that v be independent of a. 
This is equivalent to saying that F z (a, z) is independent of a, i.e. that F(a, z) 
is of the form 

F(a, z) = t(z) + f (a) , 

where the functions and \f/ are arbitrary. Substituting this value in (46), we 
Bee that the most general solution of the problem is given by the equations 

( x = f(a) cos a ^ () sin a + <f>(z) cos a , 
(47) -| y = \j/(a) sin a + f (a) cos a + <(z) sin a , 



These surfaces may be generated as follows. The first two of equations (47), 
for z constant and a variable, represent a family of parallel curves which are 
the projections on the xy plane of the sections of the surface by planes parallel 
to the xy plane. But these curves are all parallel to the curve obtained by set 
ting <f>(z) = 0. Hence the surfaces may be generated as follows : Taking in the 
xy plane any curve whatever and its parallel curves, lift each of the curves verti 
cally a distance given by some arbitrary law ; the curves in their new positions form 
a surface which is the most general solution of the problem. 

It is easy to see that the preceding construction may be replaced by the 
following : The required surfaces are those described by any plane curve whose 
plane rolls without slipping on a cylinder of any base. By analogy with plane 
curves, these surfaces may be called rolled surfaces or roulettes. This fact may 
be verified by examining the plane curves a = const. The two families of lines 
of curvature are the plane curves z = const, and a = const. 



IV. FAMILIES OF STRAIGHT LINES 

The equations of a straight line in space contain four variable 
parameters. Hence we may consider one-, two-, or three-parameter 
families of straight lines, according to the number of given relations 
between the four parameters. A one-parameter family of straight 
lines form a ruled surface. A two-parameter family of straight 
lines is called a line congruence, and, finally, a three-parameter 
family of straight lines is called a line complex. 

253. Ruled surfaces. Let the equations of a one-parameter family 
of straight lines (G) be given in the form 

(48) x = az+p, y = bz + q, 

where a, b, p, q are functions of a single variable parameter u. Let 
us consider the variation in the position of the tangent plane to the 
surface S formed by these lines as the point of tangency moves along 
any one of the generators G. The equations (48), together with the 
equation z = z, give the coordinates x, y, z of a point M on S in terms 



XII, 253] FAMILIES OF STRAIGHT LINES 527 

of the two parameters z and u ; hence, by 39, the equation of the 
tangent plane at M is 

X x Y y Z z 

a b 1=0, 

a z+p b z + q 

where a , b , p , q denote the derivatives of a, b, p, q with respect 
to u. Eeplacing x and y by az -f p and bz + q, respectively, and 
simplifying, this equation becomes 

(49) (b z + q }(X - aZ -p) - (a z + p )(Y - bZ - q) = 0. 

In the first place, we see that this plane always passes through the 
generator G, which was evident a priori, and moreover, that Jthe plane 
turns around G as the point of tangency M moves along G, at least 
unless the ratio (a z + p )/(b z + <? ) is independent of z, i.e. unless 
a q b p = 0, we shall discard this special case in what follows. 
Since the preceding ratio is linear in z, every plane through a gen 
erator is tangent to the surface at one and only one point. As the 
point of tangency recedes indefinitely along the generator in either 
direction the tangent plane P approaches a limiting position P , 
which we shall call the tangent plane at the point at infinity on that 
generator. The equation of this limiting plane P is 

(50) b (X- aZ -p)-a (Y-bZ -q) = 0. 

Let w be the angle between this plane P and the tangent plane P at 
a point M (x, y, z) of the generator. The direction cosines (a 1 , ft , y ) 
and (a, /?, y) of the normals to P and P are proportional to 

b , - a , a b - ab 
and 

b z + q , -(a z+p 1 ), b(a z+p )-a(b z + q ), 

respectively; hence 

Az + B 
cos u> = aa + (3/3 + yy = j=^ 



Az* + 2Bz + C 
where 

A =a 2 +b 2 +(ab -ba Y, 

B = a p + b q + (ab - ba }(aq - bp ) , 
C=p 2 +q 12 +(aq -bp y. 

After art easy reduction, we find, by Lagrange s identity ( 131), 



- B 2 (a q - b p } Vl -f a* + 

(51) tan CD = - = 

Az + B Az + B 



528 SURFACES [xii, 253 

It follows that the limiting plane P is perpendicular to the tangent 
plane P v at a point O l of the generator whose ordinate z { is given by 
the formula 

_B_ a p + b g + (aV - ba )(ag - bp ~) 
KI ~ A~ a 2 + b 2 + (ab -baj 2 

The point is called the central point of the generator, and the tan 
gent plane P! at O t is called the central plane. The angle 6 between 
the tangent plane P at any point M of the generator and this central 
plane P l is Tr/2 o>, and the formula (51) may be replaced by the 
formula 



tan o 



b p 1 ) Vl 



Let p be the distance between the central point O^ and the point M, 
taken with the sign + or the sign according as the angle which 
Oi M makes with the positive z axis is acute or obtuse. Then we 
shall have p = (z t ) Vl -f a 2 -j- b 2 , and the preceding formula may 
be written in the form 

(53) tan0 = fy, 

where k, which is called the parameter of distribution, is defined by 
the equation 

= a>* >* <- 



The formula (53) expresses in very simple form the manner in which 
the tangent plane turns about the generator. It contains no quantity 
which does not have a geometrical meaning : we shall see presently 
that k may be defined geometrically. However, there remains a cer 
tain ambiguity in the formula (53), for it is not immediately evident 
in which sense the angle should be counted. In other words, it is 
not clear, a priori, in which direction the tangent plane turns around 
the generator as the point moves along the generator. The sense of 
this rotation may be determined by the sign of k. 

In order to see the matter clearly, imagine an observer lying on a 
generator G. As the point of tangency M moves from his feet toward 
his head he will see the tangent plane P turn either from his left 
to his right or vice versa. A little reflection will show that the 
sense of rotation defined in this way remains unchanged if the 
observer turns around so that his head and feet change places. 
Two hyperbolic paraboloids having a generator in common and 



XII, 253] FAMILIES OF STRAIGHT LINES 529 

lying symmetrically with respect to a plane through that generator 
give a clear idea of the two possible situations. Let us now move 
the axes in such a way that the new origin is at the central point O l , 
the new z axis is the generator G itself, and the xz plane is the cen 
tral plane P t . It is evident that the value of the parameter of dis 
tribution (54) remains unchanged during this movement of the axes, 
and that the formula (53) takes the form 

(53 ) tan0 = &*, 

where 6 denotes the angle between the xz plane P l and the tangent 
plane P, counted in a convenient sense. For the value of u which 
corresponds to the z axis we must have a = b = p q = 0, and the 
equation of the tangent plane at any point M of that axis becomes 

(b e + q )X-(a z+p )Y=Q. 

In order that the origin be the central point and the xz plane the 
central plane, we must have also a = 0, q = ; hence the equation 
of the tangent plane reduces to Y = (b z/p ^X, and the formula (54) 
gives k=b /p . It follows that the angle in (53 ) should be 
counted positive in the sense from Oy toward Ox. If the orienta 
tion of the axes is that adopted in 228, an observer lying in the 
z axis will see the tangent plane turn from his left toward his right 
if k is positive, or from his right toward his left if k is negative. 

The locus of the central points of the generators of a ruled surface 
is called the line of strict ion. The equations of this curve in terms 
of the parameter u are precisely the equations (48) and (52). 

Note. If a q = b p for a generator G, the tangent plane is the 
same at any point of that generator. If this relation is satisfied 
for every generator, i.e. for all values of u, the ruled surface is a 
developable surface ( 223), and the results previously obtained can 
be easily verified. For if a and b do not vanish simultaneously, 
the tangent plane is the same at all points of any generator G, 
and becomes indeterminate for the point z = p /a == q /b , i.e. 
for the point where the generator touches its envelope. It is easy 
to show that this value for z is the same as that given by (52) when 
a q = b p . It follows that the line of striction becomes the edge 
of regression on a developable surface. The parameter of distribution 
is infinite for a developable. 

If a = b = for every generator, the surface is a cylinder and 
the central point is indeterminate. 



530 SURFACES [XII, 254 

254. Direct definition of the parameter of distribution. The central 
point and the parameter of distribution may be defined in an entirely 
different manner. Let G and G^ be two neighboring generators cor 
responding to the values u and u -f- h of the parameter, respectively, 
and let GI be given by the equations 

(55) x (a + Aa) z + p -f Aj5, y = (b + Aft) z + q + Ay. 

Let 8 be the shortest distance between the two lines G and G l , a the 
angle between G and G l} and (X, Y, Z) the point where G meets the 
common perpendicular. Then, by well-known formulae of Analytic 
Geometry, we shall have . 

_ _ Aa Ay + Aft Aj? + (a Aft 6 Aa)[(a + Aa) Ay (6 + Aft)Ap] 
(Aa) 2 + (Aft) 2 + (a Aft - 6 Aa) 2 

- _ Aa Ay Aft Ap 

V(Aa) 2 + (Aft) 2 + (a Aft - & Aa) 2 



V(Aa) 2 + (Aft) 2 + ( Aft - 6 Aa) 2 
sin a = . v = , 

Va 2 + b 2 + 1 V(a -f- Aa) 2 + (b + Aft) 2 + 1 

As h approaches zero, Z approaches the quantity x defined by (52), 
and (sin <*)/8 approaches k. Hence the central point is the limiting 
position of the foot of the common perpendicular to G and G l , while 
the parameter of distribution is the limit of the ratio (sin a)/8. 

In the expression for 8 let us replace Aa, A&, A/?, Ay by their 
expansions in powers of h: 

h 2 
Aa = ha + a" -\ ---- 



and the similar expansions for Aft, A/?, Ay. Then the numerator of 
the expression for 8 becomes 



while the denominator is always of the first order with respect 
to h. It is evident that 8 is in general an infinitesimal of the first 
order with respect to h, except for developable surfaces, for which 
a y = b j) - But the coefficient of h s /2 is the derivative of a q b p ; 
hence this coefficient also vanishes for a developable, and the shortest 
distance between two neighboring generators is of the third order 
( 230). This remark is due to Bouquet, who also showed that if 
this distance is constantly of the fourth order, it must be precisely 
zero; that is, that in that case the given straight lines are the 



XII, 255] FAMILIES OF STRAIGHT LINES 531 

tangents to a plane curve or to a conical surface. In order to prove 
this, it is sufficient to carry the development of Aa Ay Aft A/> to 
terms of the fourth order. 

255. Congruences. Focal surface of a congruence. Every two-parameter 
family of straight lines 

(56) x = az+p, y = bz + q, 

where a, b, p, q depend on two parameters a and ft, is called a line 
congruence. Through any point in space there pass, in general, a 
certain number of lines of the congruence, for the two equations (56) 
determine a certain number of definite sets of values of a and ft when 
a-, y, and z are given definite values. If any relation between a and ft 
be assumed, the equations (56) will represent a ruled surface, which 
is not usually developable. In order that the surface be developable, 
we must have 

da dq db dp = 0, 

or, replacing da by (da/da) da -f (da/d/3) dft, etc., 




d n r* 112 ^ n 

(*tr\ \ ^~ ^ ^ ^ 

(57) ** n 



dft ^}\ca dft 

This is a quadratic equation in dft/da. Solving it, we should usu 
ally obtain two distinct solutions, 

dft dft 

(oo) = \j/i (cr, ft) , = \I/2 (<*) p) , 

<x rtnr 

either of which defines a developable surface. Under very gen 
eral limitations, which we shall state precisely a little later and 
which we shall just now suppose fulfilled, each of these equations 
is satisfied by an infinite number of functions of a, and each of them 
has one and only one solution which assumes a given value ft when 
a = a . It follows that every straight line G of the congruence 
belongs to two developable surfaces, all of whose generators are 
members of the congruence. Let F and F be the edges of regression 
of these two developables, and A and A the points where G touches 
F and F , respectively. The two points A and A are called the focal 
points of the generator G. They may be found as follows without 
integrating the equation (57). The ordinate z of one of these points 
must satisfy both of the equations 

z da -f- dp = , .~ db + dq = , 



532 SURFACES [XII, 255 

or, replacing da, db, dp, dy by their developments, 



Eliminating z between these two equations, we find again the equa 
tion (57). But if we eliminate dp /da we obtain an equation of the 
second degree 



whose two solutions are the values of z for the focal points. 

The locus of the focal points A and A consists of two nappes 
2 and 2 of a surface whose equations are given in parameter form 
by the formulae (56) and (59). These two nappes are not in general 
two distinct surfaces, but constitute two portions of the same ana 
lytic surface. The whole surface is called the focal surface. It is 
evident that the focal surface is also the locus of the edges of regres 
sion of the developable surfaces which can be formed from the lines 
of the congruence. For by the very definition of the curve T the 
tangent at any point a is a line of the congruence; hence a is a 
focal point for that line of the congruence. Every straight line 
of the congruence is tangent to each of the nappes 2 and 2 , for it 
is tangent to each of two curves which lie on these two nappes, 
respectively. 

By an argument precisely similar to that of 247 it is easy to 
determine the tangent planes at A and A to 2 and 2 (Fig. 51). 
As the line G moves, remaining tangent to r, for example, it also 
remains tangent to the surface 2 . Its point of tangency A will 
describe a curve y which is necessarily distinct from r . Hence 
the developable described by G during this motion is tangent to 2 
at A , since the tangent planes to the two surfaces both contain the 
line G and the tangent line to y . It follows that the tangent plane 
to 2 at A is precisely the osculating plane of r at A. Likewise, 
the tangent plane to 2 at A is the osculating plane of T at A . 
These two planes are called the focal planes of the generator G. 

It may happen that one of the nappes of the focal surface degen 
erates into a curve C. In that case the straight lines of the con 
gruence are all tangent to 2, and merely meet C. One of the 
families of developables consists of the cones circumscribed about 2 



XH, aw] FAMILIES OF STRAIGHT LINES 533 

whose vertices are on C. If both of the nappes of the focal surface 
degenerate into curves C and C", the two families of developables 
consist of the cones through one of the curves whose vertices lie 
on the other. If both the curves C arid C are straight lines, the 
congruence is called a linear congruence. 

256. Congruence of normals. The normals to any surface evidently 
form a congruence, but the converse is not true : there exists no 
surface, in general, which is normal to every line of a given con 
gruence. For, if we consider the congruence formed by the normals 
to a given surface S, the two nappes of the focal surface are evidently 
the two nappes 2 and 2 of the e volute of S ( 247), and we have seen 
that the two tangent planes at the points A and A where the same 
normal touches 2 and 2 stand at right angles. This is a character 
istic property of a congruence of normals, as we shall see by trying 
to find the condition that the straight line (56) should always remain 
normal to the surface. The necessary and sufficient condition that it 
should is that there exist a function /(a, /3) such that the surface 5 
represented by the equations 

(60) x = az+p, y = bz + q, z=f(a,p-) 

is normal to each of the lines (6 ). It follows that we must have 
8x . d dz 



8x dy dz 
a dp^ b W + W = 

or, replacing x and y by az + p and bz + q, respectively, and divid 
ing by Va 2 + I) 2 + 1, 

dp dq 

a " t~ " ~ 



^( g Va + ft +l)+ -0; 

&* Va 2 



Va 



=0. 



The necessary and sufficient condition that these equations be com 
patible is 



(62) 




534 SURFACES [XII, 256 

If this condition is satisfied, z can be found from (61) by a single 
quadrature. The surfaces obtained in this way depend upon a con 
stant of integration and form a one-parameter family of parallel 
surfaces. 

In order to find the geometrical meaning of the condition (62), it 
should be noticed that that condition, by its very nature, is inde 
pendent of the choice of axes and of the choice of the independent 
variables. We may therefore choose the z axis as a line of the con 
gruence, and the parameters a and ft as the coordinates of the point 
where a line of the congruence pierces the xy plane. Then we shall 
have p = a, q = ft, and a and b given functions of a and ft which van 
ish for a = ft = 0. It follows that the condition of integrability, for 
the set of values a = ft = 0, reduces to the equation da/dft = 8b/da. 
On the other hand, the equation (57) takes the form 



Qj"% **"!* I I "- Oy-l lxl - AW A J ^ !** V/ 

0/3 \tfa <?/?/ da 

which is the equation for determining the lines of intersection of 
the xy plane with the developables of the congruence after a and 
ft have been replaced by and ?/, respectively. The condition 
da/dft = db/da, for a = ft = 0, means that the two curves of this 
kind which pass through the origin intersect at right angles ; that 
is, the tangent planes to the two developable surfaces of the congru 
ence which pass through the z axis stand at right angles. Since the 
line taken as the z axis was any line of the congruence, we may state 
the following important theorem: 

The necessary and sufficient condition that the straight lines of a 
given congruence be the normals of some surface is that the focal planes 
through every line of the congruence should be perpendicidar to each 
other. 

Note. If the parameters a and ft be chosen as the cosines of the angles which 
the line makes with the x and y axes, respectively, we shall have 

a ^ 



VI + a 2 + 6 2 = 



Vl~- a2 - p Vl - a* - p* Vl - a* - 

and the equations (61) become 



(63) 



Vl- *-/3V eft dft 



XII. 2,-i7] 



FAMILIES OF STRAK1IIT LINES 



535 



Then the condition of integrability (62) reduces to the form dq/da = dp/ dp, which 
means that p and q must be the partial derivatives of the same function F(a, p) : 



dF 



dF 

ejs 



where F(a, p) can be found by a single quadrature. It follows that z is the 
solution of the total differential equation 



d(- 

whence 



Badp 



dp 2 



z = Vl - a 2 - p 
where C is an arbitrary constant. 



C + F- a - 
da 



257. Theorem of Malus. If rays of light from a point source are reflected (or 
refracted) by any surface, the reflected (or refracted) rays are the normals to 
each of a family of parallel surfaces. This theorem, which is due to Malus, has 
been extended by Cauchy, Dupin, Gergonne, and Quetelet to the case of any 
number of successive reflections or refractions, and we may state the following 
more general theorem : 

If a family of rays of light are normal to some surface at any time, they retain 
that property after any number of reflections and refractions. 

Since a reflection may be regarded as a refraction of index 1, it is evidently 
sufficient to prove the theorem for a single refraction. Let S be a surface nor 
mal to the unrefracted rays, mM an incident ray which meets the surface of 
separation S at a point M, and MR the refracted ray. By Descartes law, the 
incident ray, the refracted ray, and the normal MN lie in a plane, and the 
angles i and r (Fig. 52) satisfy the relation 
n sin i = sin r. For definiteness we shall sup 
pose, as in the figure, that n is less than 
unity. Let I denote the distance Mm, and 
let us lay off on the refracted ray extended 
a length I = Mm equal to k times I, where 
A; is a constant factor which we shall deter 
mine presently. The point m describes a 
surface S . We shall proceed to show that 
k may be chosen in such a way that Mm is 
normal to S . Let C be any curve on S. 
As the point m describes C the point M 
describes a curve T on the surface 2, and 
the corresponding point m describes another 
curve C" on S . Let s, cr, s be the lengths of the arcs of the three curves C, r, 
C measured from corresponding fixed points on those curves, respectively, 
w the angle which the tangent M TI to r makes with the tangent MT to the 
normal section by the normal plane through the incident ray, and < and <p the 
angles which M T\ makes with Mm and Mm , respectively. In order to find 
cos0, for example, let us lay off on Mm a unit length and project it upon 3/T lt 






FIG. 52 



536 SURFACES [XII, 258 

first directly, then by projecting it upon NT and from M T upon M 7\. This, 
and the similar projection from Mm upon -MTi, give the equations 

cos <f> = sin i cos w , cos <p = sin r cos o . 

Applying the formula (10 ) of 82 for the differential of a segment to the seg 
ments Mm and Mm , we find 

dl = da- cos w sin i , 

dl = da- cos u sin r ds cos , 

where denotes the angle between m M and the tangent to C". Hence, replacing 
dl by k dl, we find 

cos di dcr(k sin i sin r) = ds cos , 
or, assuming k = n, 

ds cos = 0. 

It follows that Mm is normal to C", and, since C is any curve whatever on 
S , 3fm is also normal to the surface S . This surface S is called the anti- 
caustic surface, or the secondary caustic. It is clear that S is the envelope of 
the spheres described about M as center with a radius equal to n times Mm ; 
hence we may state the following theorem : 

Let us consider the surface S which is normal to the incident rays as the envelope 
of a family of spheres whose centers lie on the surface of separation 2. Then the 
anticaustic for the refracted rays is the envelope of a family of spheres with the 
same centers, whose radii are to the radii of the corresponding spheres of the first 
family as unity is to the index of refraction. 

This envelope is composed of two nappes which correspond, respectively, 
to indices of refraction which are numerically equal and opposite in sign. In 
general these two nappes are portions of the same inseparable analytic surface. 

258. Complexes. A line complex consists of all the lines of a three-parameter 
family. Let the equations of a line be given in the form 

(64) x az + p, y = bz + q . 

Any line complex may be defined by means of a relation between a, b, p, q of 
the form 

(65) F(a,b,p,q) = 0, 

and conversely. If F is a polynomial in a, b, p, q, the complex is called an 
algebraic complex. The lines of the complex through any point (x , yo, Zo) form 
a cone whose vertex is at that point ; its equation may be found by eliminating 
a, 6, p, q between the equations (64), (65), and 

(66) x = az + p , yo = bz + q. 
Hence the equation of this cone of the complex is 

/7\ vi x ~ x y ~ y x z ~ xz Z/oZ - yz 

(yi) r I ) > ) 

\z ZD z - ZQ z z z Z 

Similarly, there are in any plane in space an infinite number of lines of the 
complex ; these lines envelop a curve which is called a curve of the complex. 
If the complex is algebraic, the order of the cone of the complex is the same as the 



XII, -5,s] FAMILIES OF STRAIGHT LINES 537 

class of the curve of the complex. For, if we wish to find the number of lines of 
the complex which pass through any given point A and which lie in a plane P 
through that point, we may either count the number of generators in which P 
cuts the cone of the complex whose vertex is at A, or we may count the number 
of tangents which can be drawn from A to the curve of the complex which lies 
in the plane P. As the number must be the same in either case, the theorem is 
proved. 

If the cone of the complex is always a plane, the complex is said to be linear, 
and the equation (65) is of the form 

(68) Aa + Bb + Cp + Dq + E(aq - bp) + F = 0. 

Then the locus of all the lines of the complex through any given point (XQ, 3/0, ZQ) 
is the plane whose equation is 

( A(x - XQ) + B(y - y ) + C(x z - z x) 

I + D(y z - z 2/) + E(y Q x - x y) + F(z - z ) = 0. 



The curve of the complex, since it must be of class unity, degenerates into a 
point, that is, all the lines of the complex which lie in a plane pass through a 
single point of that plane, which is called the pole or the focus. A linear com 
plex therefore establishes a correspondence between the points and the planes 
of space, such that any point in space corresponds to a plane through that point, 
and any plane to a point in that plane. A correspondence is also established 
among the straight lines in space. Let D be a straight line which does not 
belong to the complex, F and F the foci of any two planes through D, and A 
the line FF . Every plane through A has its focus at its point of intersection <p 
with the line D, since each of the lines </>F and <f>F evidently belongs to the 
complex. It follows that every line which meets both D and A belongs to the 
complex, and, finally, that the focus of any plane through I) is the point where 
that plane meets A. The lines D and A are called conjugate lines; each of them 
is the locus of the foci of all planes through the other. 

If the line D recedes to infinity, the planes through it become parallel, and 
it is clear that the foci of a set of parallel planes lie on a straight line. There 
always exists a plane such that the locus of the foci of the planes parallel to it 
is perpendicular to that plane. If this particular line be taken as the z axis, 
the plane whose focus is any point on the z axis is parallel to the xy plane. By 
(69) the necessary and sufficient condition that this should be the case is that 
A = JB = C = D = 0, and the equation of the complex takes the simple form 

(70) aq-bp + K=0. 

The plane whose focus is at the point (x, y, z) is given by the equation 

(71) Xy -Yx + K(Z-z) = Q, 

where JT, Y, Z are the running coordinates. 

As an example let us determine the curves whose tangents belong to the 
preceding complex. Given such a curve, whose coordinates x, y, z are known 
functions of a variable parameter, the equations of the tangent at any point are 

X - x _ Y-y _ Z-z 
dx dy dz 



538 SURFACES [XII, Exs. 

The necessary and sufficient condition that this line should belong to the given 
complex is that it should lie in the plane (71) whose focus is the point (x, y, z), 
that is, that we should have 

(72) xdy ydx = Kdz. 

We saw in 218 how to find all possible sets of functions x, y, z of a, single 
parameter which satisfy such a relation ; hence we are in a position to find 
the required curves. 

The results of 218 may be stated in the language of line complexes. For 
example, differentiating the equation (72) we find 

(73) xd*y -yd*x = Kd*z, 

and the equations (72) and (73) show that the osculating plane at the point 
(x, y, z) is precisely the tangent plane (71); hence we may state the following 
theorem : 

// all the tangents to a skew curve belong to a linear line complex, the osculating 
plane at any point of that curve is the plane whose focus is at that point. 

(APPELL.) 

Suppose that we wished to draw the osculating planes from any point in 
space to a skew curve F whose tangents all belong to a linear line complex. Let 
M be the point of contact of one of these planes. By Appell s theorem, the 
straight line M belongs to the co