co
CD
:3-
iS
:<d-
o
o
CD
co
<r
TREATISE
ON
INFINITESIMAL CALCULUS;
CONTAINING
DIFFERENTIAL AND INTEGRAL CALCULUS, CALCULUS OF VARIA-
TIONS, APPLICATIONS TO ALGEBRA AND GEOMETRY,
AND ANALYTICAL MECHANICS.
BY
BARTHOLOMEW PRICE, M.A,F.R.S.,
FELLOW AND TUTOR OF PEMBROKE COLLEGE, AND
8EDLEIAN PROFESSOR OF NATURAL PHILOSOPHY, OXFORD.
VOL. II.
INTEGRAL CALCULUS,
AND
CALCULUS OF VARIATIONS.
: Les progres de la science ne sont vraiment fructueux, que quand ils amenent
aussi le progres des Traites etementaires." CH. DUPIN.
OXFORD:
AT THE UNIVERSITY PRESS.
M.DCCC.LIV.
PREFACE.
I HE present volume contains Integral Calculus" the
Calculus of Variations, and Differential Equations ; as
a scientific inquiry into these subjects, it differs so
much from the English treatises in ordinary use, that
it is incumbent on me to say a few words of explana-
tion.
The Integral Calculus has been for the most part
established on an inversion of the rules of the Differ-
ential Calculus ; it has had scarcely any principles of
its own, and of these none independent of those of the
Differential Calculus ; the student has been obliged
to burden his memory with certain rules which he
mechanically applies; he has not been taught to
deduce them from first principles, because he has
had no principles pregnant with such rules ; and of
them, at least in the early stages of his knowledge,
he can give neither intelligible account nor interpre-
tation ; and it is only when he arrives at the first
geometrical application that he gets an insight into
the meaning of the processes ; and his view is even
then obscured by an expansion into a series, which
a 2
iv PREFACE.
he no sooner obtains than he omits all terms, save
one, of it.
Now in a science replete with applications so large
and so important as those of the Integral Calculus,
such a method is unsatisfactory, not to say unphilo-
sophical ; and it is neither desirable nor necessary to
leave it in this state. Most foreign mathematicians
have been alive to the defects, and have succeeded in
remedying them : why then should Englishmen be
behind ? Professor De Morgan is, as far as I know,
the first English author who constructed the science
on a more philosophical basis ; and in his large Trea-
tise on the Differential and Integral Calculus, the
Integral Calculus is established on sound principles,
and placed early in the course. For purely scientific
reasons such an arrangement may be the best, but it
may fairly be questioned whether it is convenient for
didactic purposes : I have chosen to place it after the
Differential Calculus.
In the following treatise the Integral Calculus is
considered a part of Infinitesimal Calculus, and as
such, is founded on an intelligible conception of Infi-
nitesimals ; it is thus a branch of the science of con-
tinuous number ; its principles are involved in, and
effluent from, that fundamental idea ; it assumes the
existence of an infinitesimal element-function, formed
according to an assigned law, the law being involved
in the symbolical form of the infinitesimal ; and the
primary problem is, to determine the finite number
or function of number of which the given infinitesi-
mal is the constituent elemental part ; that is, Given
the infinitesimal element, to find the finite quantity of
which it is the infinitesimal element. The required
PREFACE. v
result can evidently only be definite, when the sum
of the infinitesimal elements is to be taken between
certain fixed limits, which are at a finite distance
apart. Thus the primary problem is one of summa-
tion of a series, of which the law is given, (for the
symbolical form of the element-function, or type-term,
determines that,) and the first and the last terms are
given, and the sum of these infinitesimal element-
functions or differentials is called the Definite Inte-
gral. The notion of a Definite Integral is therefore
the fundamental one of the Integral Calculus; and
the work of the Calculus is, to discover rules for the
formation of these, to construct the code of laws
which they are subject to, and to investigate the con-
ditions necessary for their application to other subject-
matter. Hence it is that the Definite Integrals of
simple element-functions are investigated in the early
part of the Treatise from first principles, and it is
only when I have rigorously proved in the most
general case that the Definite Integral may be found
by an inversion of the process of Differentiation that
I have considered myself free to make use of the
knowledge of the Differential Calculus, which has
been (usually) previously acquired. By these means
our labour is diminished, and nothing of principle is
lost, because the rules thus found might have been
discovered directly from the peculiar principles of the
Integral Calculus.
In support of the view of the subject here taken, I
allege that on this conception of Infinitesimal ele-
ments, and on this conception only, is the Integral
Calculus applied to the problems of Rectification,
Quadrature and Cubature, and in proof of this allega-
vi PREFACE.
tion 1 appeal to the processes of Chaps. VI, VJ1 and
VIII; in them the infinitesimal element-function
exists previous to the finite function, and the latter is
found by the summation of an infinite number of the
former. And this is undoubtedly the process, and
the only intelligible process, of determining the finite
results of an ever- varying law : it is, I assert, on the
notion of Infinitesimals only, that the problems of
varying velocity can be intelligibly treated by the
Integral Calculus.
The course of the treatise is therefore this. The
principles of the subject having been formally stated
and exemplified, rules directly applicable to known
functions are constructed, and the Geometrical Pro-
blems, which are the most simple applications of the
Calculus, are solved. These problems suggest a wide
extension, viz. integrals, multiple, and of many varia-
bles ; an adequate discussion of the properties of
which requires a transformation of them into their
equivalents in terms of other variables. Hereby I
am led to a concise account of two new systems of
reference, those of Gauss and Lame ; and these have
up to the present time scarcely been introduced into
any English Treatise. An inquiry into properties of
unknown functions, dependent on the fulfilment by
a definite integral of certain given conditions, ori-
ginates the Calculus of Variations, and in the course
of it I have taken the opportunity of discussing at
some length the properties of geodesic lines on an
ellipsoid ; for we have herein an instance of the ad-
vantage of the new systems of reference.
The second part of the volume contains Differential
Equations ; element-functions, that is, involving two
PEEFACE. vii
or more dependent variables. I wish that this part
of the work were more perfect ; but the subject is
surrounded with difficulties, and I can do little else
than exhibit such detached portions of it as have
yielded to the powers of Analysis.
As the science of number is progressive, so the
state of this branch of it is very incomplete ; its
boundaries are being advanced in all directions ; and
although we have to lament the great loss of two
such men as Abel and Jacobi, yet the reader of the
Journals of Liouville and Crelle knows that the la-
bours of others are not fruitless. Much therefore is
omitted, either because it is not suited to an elementary
treatise, or because it is so isolated as to be beyond
the range of our idea. And two subjects, Definite
Integrals (peculiarly so called), including the Beta-
and Gamma-functions, and Elliptic Transcendents,
are very imperfectly, and only incidentally discussed.
The latter subject, especially with the new Theorems
of Abel, is not adapted to such a work as the present ;
and the former, if fully treated of, would have en-
larged the Volume to an undue size. I must also
observe that many of the processes might have been
shortened by an use of the method of Determinants ;
and Mr. Spottiswoode's Treatise* on the subject would
supply all the necessary information ; but I have
reason to think that English students are at present
little acquainted with the subject ; so that to them at
least results deduced by it would have been unintelli-
* " Elementary Theorems relating to Determinants," by William
Spottiswoode, M. A., of Balliol College, Oxford, Longman and Co.,
London, 1851. A new edition of the work, much enlarged, is likely
to appear shortly in Crelle's Journal.
viii PREFACE.
gible : I trust however that the method will ere long
be introduced into the course of mathematical study ;
in which case the processes may be abridged by each
reader for himself.
I am, as in the first Volume, under obligation to
many friends for assistance and advice ; to Professor
Stokes of Pembroke College, Cambridge, to Mr. W.
Spottiswoode, M. A., of Oxford, to Mr. H. J. S. Smith,
Fellow of Balliol College, Oxford ; to Professor De
Morgan, to M. Moigno, to M. Duhamel ; and to many
others whose contributions are acknowledged in va-
rious parts of the Treatise. And I am also bound to
express my sense of obligation to M. Liouville and
M. Crelle, on account of their valuable Journals.
The Chapters mark the salient divisions of the
matter; the Articles are numbered continuously
throughout the Volume, and their numerals are placed
in the inner corners on the top of the pages. Brack-
eted numerals are also attached to the more impor-
tant equations and are separate for each Chapter;
and reference is for the most part made to the num-
bers of the Article and of the equation.
PEMBROKE COLLEGE, OXFORD,
Aug. 4, 1854.
ANALYTICAL TABLE OF CONTENTS.
PART I.
EXPLICIT FUNCTIONS OF ONE VARIABLE.
CHAPTER I.
EXPLANATION OF DEFINITE AND INDEFINITE INTEGRATION.
Art. Page
1. Differentiation is a process of disintegration .............. 1
2. Integration is a process of summation .................. 2
3. Definite and indefinite integrals. Description of integral cal-
culus .......................................... 3
4. Correct value of a definite integral .................... 4
5. Relation of the definite and indefinite integrals ........... 5'
6. The definite integral is independent of the mode of division . . 5
7. Relation of symbols of differentiation and integration ...... 6
8. Fundamental theorems of definite integrals .............. 8.
9. Examples of definite integrals determined from first principles 1 1
CHAP. II.
RULES FOR THE INTEGRATION OF ALGEBRAICAL FUNCTIONS.
10. Fundamental theorems of indefinite integrals ............ 15
SECTION 1. Integration of Algebraical Functions.
1 1 . Integration of x n dx ................................ 16
12. Integration of x~ l dx ............................... 17
13. Examples in illustration ................. I .......... 18
dx dx * f*
14. 15. Integration of - 5 , + -5 - 5 .................. *^
22 - 2
x n dx dx 01
16. Integration of - and of - , . ............ **
(a + bx) m x n (a + bx) m
17. Examples in illustration ............................. 23
PRICE, VOL. II. b
X ANALYTICAL TABLE
SECTION 2. Integration of Rational Fractions.
18. Definition of rational fractions, and simplification 24
1 9, 20. Decomposition into partial fractions when the roots of the
denominator are all unequal 25
21, 22. Decomposition into partial fractions when there are in the
denominator sets of equal roots 30
dx
23, 24. Integration of n , , 36
X ~T~ I
x m dx
25. Integration of n , -, 43
*F ~T~ 1
26. Integration of rational fractions by various artifices 43
27-30. Integration by reduction of
dx dx x m dx
45
SECTION 3. Integration of irrational Algebraical Functions.
31-35. Integration of
+ dx dx dx dx
48
36. Proof of identity of results apparently incongruous ........ 50
37, 38. Integration of - - - r
(a + bxcx2)*
39,40. Integration of (a 2 xrfdx,
37, 38. Integration of - - - r , - , ....... 51
52
41 . Integration of - ............................ 54
(a 3 + * 2 )*
42. Examples in illustration ..................... 55
SECTION 4. Integration of Irrational Functions by Rationalization.
P
43-45. Rationalization of x m (a + bx n )i dx, and examples ...... 56
46, 47- Other forms admitting of rationalization .............. 59
SECTION 5. Integration of Irrational Functions by Reduction.
48. General remarks on the process ...................... 60
x n dx
49. Integration of - r ............................ 60
(a 2 -or 2 )*
OF CONTENTS. x i
x n dx
50. Integration of 7 61
(a 2 + x 2 ) 2
51 , 52. Integration of - r 61
(2axx2)*
53, 54. Integration of (a 2 + x 2 )%dx 63
dx dx
55, 56. Integration of , 64
x n dx
57. Integration of r 65
(a + &r)*
x n dx
58. Integration of 66
CHAP. III.
INTEGRATION OF LOGARITHMIC AND CIRCULAR FUNCTIONS.
SECTION 1 . Exponential and Logarithmic Functions.
59. Integration of a* dx and e*dx 68
60, 61. Integration of x n e ax dx, and x^e^dx 68
62. Various examples of integration of exponential functions .... 69
63, 64. Integration of logarithmic functions 70
SECTION 2. Circular Functions.
65-67- Integration of fundamental circular functions 71
68, 69. Integration of (sin x) n dx and (cos x) n dx 76
70. Integration of (smx)~ n dx and of (cos x)~ n dx 78
71, 72. Integration of (sin x) m (cos x) n dx 80
73. Integration of (tan x) n dx and of (cot x) n dx 82
74. Integration of x n sin x dx, and of x n cos x dx 82
75. 76. Integration of e a * (cos x) n dx, and of e"* (sin x) n dx 83
77' Integration of f(x) sin-^cfcr, f(x) tan -1 #(fo, &c 85
dx
78. Integration of = 85
(a + b cos x) n
79. Integration by substitution 86
CHAP. IV.
VARIOUS PROPERTIES OF DEFINITE INTEGRATION.
80. Great importance of definite integrals 89
SECTION 1. Further Researches into the Theory of Definite Integrals.
81. Elementary transformations of definite integrals 90
bz
xii ANALYTICAL TABLE
82. Further theorems of definite integrals 92
83. The case of a definite integral becoming infinite for a certain
value, included within the limits 95
84. Cauchy's principal value of a definite integral 96
SECTION 2. Examples of Definite Integrals.
85. Values of definite integrals deduced from indefinite integrals 98
86. Expansion of a function by means of definite integration . . 101
87. Proof of Taylor's Series founded on definite integration .... 103
88. A similar proof of Maclaurin's Series 105
SECTION 3. Methods of approximating to the Value of a Definite
Integral.
89. 90. Integration by series 106
91. Bernoulli's series for approximation 108
92-94. Other methods of approximation 109
CHAP. V.
SUCCESSIVE INTEGRATION.
95. The problem proposed Ill
96. The nth integral requires the introduction of n constants . . Ill
97. A series equivalent to Taylor's Series is deduced 112
98-100. The calculus of operations applied to successive inte-
gration 114
CHAP. VI.
INTEGRAL CALCULUS APPLIED TO THE RECTIFICATION OF
CURVED LINES.
101 . Elementary geometrical problems solved 118
SECTION 1 . Rectijication of Plane Curves referred to Rectangular
Coordinates.
102. Investigation of the general expression of the length-element 121
103. Examples of rectification 122
104. Discussion of properties of the arc of an ellipse 127
105. Fagnani's Theorem 128
106,107- Geometrical interpretation of the analytical equations 129
SECTION 2. Rectification of Plane Curves : Polar Coordinates.
108. Investigation of the general expression of the length-element 131
109. Examples in illustration 132
1 10. Value of length-element in terms of r and p 133
OF CONTENTS. xiii
SECTION 3. Rectification of Non-Plane Curves.
111. Investigation of the general length-element, and examples 134
SECTION 4. Determination of the Equation of a Curve by means of a Re-
lation between the length and the Coordinates to any Point on it.
112. Investigation of the general equation, and examples 135
SECTION 5. Involutes of Plane Curves.
113. 114. Investigation of general properties of involutes 137
1 15. Examples of involutes 139
116. Involutes of curves referred to polar coordinates, and ex-
amples 141
CHAP. VII.
QUADRATURE OF SURFACES, PLANE AND CURVED.
SECTION 1. Quadrature of Plane Surfaces; Rectangular Coordinates.
117- Investigation of the surface-element, and explanation of the
process of double integration 143
118. Examples of quadrature of plane surfaces 145
119. The order of integrations changed, and examples 149
1 20. Quadrature of a plane surface contained between two given
curves 150
121. Examples in illustration 151
122. Quadrature determined by means of substitution 153
SECTION 2. Quadrature of Plane Surfaces; Polar Coordinates.
] 23. Investigation of the general differential expression of a sur-
face-element 154
124. Examples illustrative of it 156
125. The order of integrations inverted 157
126. Cases of various and curved limits 158
127. Investigation of the surface-element in terms of r and. p . . 159
128. Quadrature of a surface between a curve, its evolute, and
two bounding radii of curvature 161
SECTION 3. Quadrature of Surfaces of Revolution.
129. Investigation of the surface-element 162
130. Examples illustrative of the process 163
xiv ANALYTICAL TABLE
131, 132. Changes in the expression for the surface-element ac-
cording as the axis of revolution is that of y or is
parallel to a coordinate axis 164
SECTION 4. Quadrature of Curved Surfaces.
133. Investigation of the surface-element 166
134. The equivalent form, if the equation to the surface be
F (x, y, z) == c 167
135. The form deduced from the last article, when the equation
to the surface is z =/(#, y) ; and the order of integration
explained 168
136. Examples illustrative of the processes 169
CHAP. VIII.
CUBATURE OF SOLIDS.
SECTION 1. Cubature of Solids of Revolution.
137- Investigation of the volume- element, when the axis of x is
the axis of revolution 171
138. Examples in illustration 171
139. Investigation of volume- element, when axis of y is that of
revolution 173
140. 141. Extension of the method to volumes bounded by other
surfaces 174
SECTION 2. Cubature of Solids bounded by any curved Surface.
142. Investigation of volume-element, and explanation of the
processes of triple integration 176
143. Examples of cubature 178
144. Necessity of caution as to the order of integrations 181
145. Modification of general form of volume-element, and examples 181
146. Volume-element in terms of polar coordinates 183
147- Examples in illustration 186
CHAP. IX.
GENERAL PROPERTIES OF MULTIPLE INTEGRALS, AND THEIR
TRANSFORMATIONS.
148. Explanation of symbolization 188
SECTION 1. Transformation of Multiple Integrals.
149. Particular examples of transformation 189
OP CONTENTS. xv
150. Remarks on elimination by means of a system of linear
equations, and on the final derivative and its svmbols 195
151. General result derived from explicit functions 196
152. General result derived from implicit functions 199
153. Illustrative examples 200
154. Investigation of a method for determining the new limits in
a transformed multiple integral 202
SECTION 2. New Systems of Curvilinear Coordinates.
155. Explanation of Gauss' system of reference 204
156. Length-element of a curve in terms of the new coordinates 205
157. Particular values of length-element, and direction-cosines in
reference to the new axes 206
158. Surface-element in terms of the new coordinates 207
159-161. Geometrical explanation of the preceding formulae .. 208
162. M. Lame's system of elliptical coordinates 210
163. The three confocal surfaces of the second order intersect
orthogonally 212
164. Length-element and volume-element in terms of the new
coordinates 212
165. The three confocal surfaces intersect each other along their
lines of curvature : the surface-element of the ellipsoid . . 214
166. Jacobi's modification of Lame's coordinates 215
SECTION 3. Miscellaneous Illustrations of preceding Principles.
167- Means of transforming multiple integrals, so that variable
limits may become constant 216
168. Simplification of other definite integrals 217
169. Quadrature of the surface of the ellipsoid 217
170-172. Definite integrals expressing the whole surface of the
ellipsoid ; M. Catalan's Theorem 220
173. An integral involving an irrational function transformed by
a process due to Jacobi into a rational function 223
CHAP. X.
ON THE VARIATION OF DEFINITE INTEGRALS DUE TO THE VARIATION
OF PARAMETERS INVOLVED IN THE ELEMENT-FUNCTION, AND IN
THE LIMITS.
174. Variation of a definite integral due to the variation of a
constant contained in the element-function 225
175. Evaluation of certain definite integrals by the process .... 226
xvi ANALYTICAL TABLE
176. Variation of a definite integral due to the integration of it
with reference to a constant contained in the element-
function 228
177- Examples in illustration 228
178. Variation of a definite integral due to that of a constant
which is contained in the element-function and in the
limits 230
179. Geometrical interpretation of the same 231
180. Statement of the problems which originate the calculus of
variations . 233
CHAP. XL
EXPOSITION OF THE PRINCIPLES OF THE CALCULUS OF VARIATIONS.
181. The calculus of variations is a calculus of continuous func-
tions 234
182. The view of it by the light of a problem 235
183. Difference as to operations and symbols between the dif-
ferential calculus and the calculus of variations 236
184. Further differences and coincidences 237
185. Our ignorance limits the calculus of variations to certain
forms of definite integrals 238
186. Formal enuntiation of the principle of the calculus in its
limited state 239
187- Symbolization of the calculus of variations 241
188. Geometrical interpretation of fundamental operations .... 242
242
yi
189. Variation of / F(X, dx, d 2 x, . . y, dy, d 2 y, . . )
Jo
190. Modification of the preceding when x is equicrescent .... 245
191. Further modification when x undergoes no variation .... 245
192. Variation of / F (x, y, y, y", . . ) 246
Jo
193. Geometrical interpretation of the result of the last Article . 248
194. Modification of the result when the variations become dif-
ferentials 250
ri
195. Variation of / F(X, dx, d*x, . . y, dy, d 2 y, . . z, dz, d 2 z, . . ) . . 251
Jo
196. Modification of the result, when an equation of condition is
given 262
197- Variation of / F(X, y, y, y", . . z, z, z", . . ) 254
Jo
OF CONTENTS. xvii
198. Modification of the result when an equation of condition is
given 256
1 99. Calculation of a variation of a variation 257
200. Variation of a product of differentials 258
201. 202. Variation of a definite double integral due to the varia-
tions of the limits and of the element-function 259
203. Examination of the several parts of the result 263
204. Geometrical interpretation of the result 263
205. The calculus of variations considers a function of an infinite
number of variables 265
CHAP. XII.
APPLICATIONS OF THE CALCULUS OF VARIATIONS TO PROBLEMS OF
MAXIMA AND MINIMA.
SECTION 1. Critical values of definite integrals, whose element-functions
involve variables and their differentials.
206. The problem stated, and the methods of the differential
calculus shewn to be insufficient 267
207- Determination of conditions for maxima and minima ... 267
208. Modification of the result when derived-functions, and not
differentials, enter into the element-function 269
209. The number of arbitrary constants contained in the final
result 270
210. Particular cases 271
211. Problems of relative maxima and minima 271
212. Determination of maxima and minima, when the element-
function of the definite integral contains three variables 272
213-219. Problems of absolute maxima and minima 273
220-223. Problems of relative maxima and minima 283
224, 225. A problem solved on the principle of Art. 205 290
SECTION 2. On Geodesic Lines.
226. The equations of a geodesic line in terms of rectangular
coordinates 293
227. The osculating plane of a geodesic is a normal plane of the
surface 294
228. Another equation of a geodesic 294
PRICE, VOL. II. C
Xviii ANALYTICAL TABLE
229. The radius of absolute curvature of a geodesic is equal to
the radius of curvature of the normal section which
touches the geodesic 295
230. The radius of torsion of a geodesic 296
231. 232. Joachimsthal's theorems of geodesies on an ellipsoid 296
233. Geodesic parallels and geodesic circles 299
234. The equations of geodesies in terms of the Gaussian coor-
dinates 300
235. Modifications of the preceding results 301
236. 237- The equations of a geodesic on an ellipsoid in terms of
Lamp's elliptical coordinates 303
238. Length of a geodesic on an ellipsoid 307
239. Various theorems of geodesies on an ellipsoid 307
240. Identity of the two first integrals found in the previous
Articles 309
SECTION 3. Investigation of critical values of a definite Integral,
whose element-function involves derived-functions.
241. Determination of the necessary conditions 311
242. Investigation of particular forms 312
243. Solution of various problems 313
SECTION 4. Discriminating conditions of Maxima and Minima.
244. General considerations as to the required conditions .... 316
245. Statement of the requisites, and Jacobi's mode of satisfying
them 317
246. Proof that 8H u dx is an exact differential 322
247. The form of its integral 323
248. 249. The integral can always be found 325
250, 251 . Two particular cases wherein the criteria are applied 326
252. Application of the criterion to the general case 330
253. The criterion applied to a case of relative critical value . . 331
SECTION 5. Investigation of the critical values of a double definite
Integral.
254. Determination of the necessary criteria 332
255. 256. Application to examples 332
OF CONTENTS. x i x
PART II.
INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO AND
MORE VARIABLES.
CHAPTER XIII.
DIFFERENTIAL EdUATIONS OF THE FIRST ORDER.
SECTION 1. General considerations of Differential Equations.
257- Meaning of the term " differential equation" ; definitions of
order, degree 335
258. Geometrical interpretation of an integral of a differential
equation 336
259. Similar interpretation of a partial differential equation .... 339
260. The analytical origin of a differential equation 340
261. The complete integral of a differential equation of the wth
order and first degree requires n arbitrary constants . . 341
262. Definition of general integral, particular integral, singular
solution 342
263. 264. Integration by separation of the variables 343
SECTION 2. Exact Total Differential Equations.
265. Criterion of exactness or condition of integrability 345
266. Examples in illustration 346
267- The criterion is satisfied when the variables are separated :
definite integrals of total differential equations 347
268. Total differential equations of three variables ; criterion of
exactness and process of integration 348
269. Examples of integrals 350
270. The definite integral of a total differential equation of three
variables 351
271. A differential equation of n independent variables : number
of conditions necessary for exactness 352
SECTION 3. Homogeneous equations of two variables.
272. Integration of homogeneous equations by separation of the
variables 353
273. Examples of integration 354
274. Geometrical interpretation of homogeneous equations .... 355
c 2
xx ANALYTICAL TABLE
275. The substitution required by separation of the variables
shewn to be equivalent to multiplication by an inte-
grating factor 355
276. An a posteriori proof that an homogeneous equation when
thus modified is an exact differential 356
277- Another form reducible to an homogeneous equation ... 357
SECTION 4. The first Linear Differential Equation.
278. The variables are separated by means of a substitution. . . . 358
279. Examples of integration 359
280. Bernoulli's equation 360
SECTION 5. Partial Differential Equations of the first order and degree.
281. Method of integrating partial differential equations, and of
introducing an arbitrary functional symbol 361
282. Examples of such integration 363
283. Geometrical illustration of the process 367
284. Partial differential equations of any number of variables . . 368
SECTION 6. Integrating Factors of differential Equations.
285. Every differential equation of the first degree has an inte-
grating factor 371
286. And the number of such integrating factors is infinite .... 372
287- Mode of determining integrating factors 373
288. Integrating factor of a homogeneous equation of n dimen-
sions and two variables 374
289. Examples in illustration 376
290,291. Integrating factor of the linear equation of the first
order 377
292. Examples of other forms wherein the integrating factor
can be found 380
293. Integrating factors of equations of three variables 381
294. Examples in illustration 383
295. Application of the method to homogeneous equations .... 386
296. Another method of integrating differential equations of
three variables 388
297- Geometrical interpretation of the criterion of integrability 390
298. Firstly by Monge's theorem 391
299-301. Secondly by Bertrand's theorem 393
302. A method of integration, when the condition of integrability
is not satisfied 397
OF CONTENTS. xxi
SECTION 7- Singular Solutions of Differential Equations.
303, 304. The general value of the integral of a differential equa-
tion 399
305. Only one general form of function satisfies a differential
equation 401
306. Criteria of singular solutions and examples 402
307- Another form of criterion 404
308. A solution being given, to determine whether it is singular
or not 405
309. Relation between the general integral and the singular so-
lution, and the means of deducing the latter from the
former 407
310. Examples of the process 408
311. Geometrical interpretation of the relation 409
312. Examination of another criterion 410
SECTION 8. Differential Equations of first order and higher degree.
313. General method of integration . , 412
314. Particular forms. Clairault's form 414
315. Geometrical interpretation of Clairault's form 416
316. An extension of Clairault's form 419
317- The case where the coefficients of the powers of y are ho-
mogeneous 422
318. Difficulties of solving partial differential equations of higher
degrees 424
319. Particular cases wherein the solution is possible 424
SECTION 9. Particular processes.
320. 321. By means of substitution 427
322. Euler's differential equation 429
323. Determination of unknown functions by differentiation and
integration 431
324. On the general equation of rectifiable curves 432
325-327. Riccati's equation, and its equivalents ' . . 433
CHAP. XIV.
INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS HIGHER
THAN THE FIRST.
SECT i ON 1 . General properties.
328. General remarks . 439
xxii ANALYTICAL TABLE
329, 330. Conditions that a differential equation of two variables
should be integrable once, independently of the func-
tional relation between its variables 440
331. Similar conditions that it should be integrable TO times . . . 442
332. Similar conditions applicable to functions of more variables 444
333. Application of the conditions to linear equations 445
SECTION 2. Linear differential equations.
334. Proof that the integration of a linear equation with a second
member is dependent on the integration of the same
equation without the second member 446
335. Application of the process to an equation of the third order 448
336. Expression of the result in terms of the particular integrals
of the equation without the second member 449
337- Examples of application of the process 451
338. If TO particular integrals of an equation without the second
member are known, the integration of the equation with
the second member depends on that of an equation of
the (n m)th order 452
339. The general integral of an equation with the second member
may be expressed in terms of n particular integrals of
the equation without the second member and of a par-
ticular integral of itself 453
340. 341. Analogies between linear differential equations and al-
gebraical equations 454
342. Construction of a linear differential equation, when particu-
lar integrals are given 455
SECTION 3. Linear differential equations with constant coefficients.
343. First method of integration 458
344. The determination of the constants 460
345. Modification if the roots are impossible 462
346. Modification if two or more roots of the characteristic are
equal 464
347- Second method of integration 466
348. Examples in illustration of the process '. 469
349. The form of the result when the right-hand member of the
equation is a constant 470
350. Third method of integration by the calculus of operations . 471
351. Modification of the result, if two or more roots are equal. . 474
352. Examples , 475
OF CONTENTS. xxiii
353. Other modes of employing the operative symbols 477
354. Lagrange's method of variation of parameters 479
SECTION 4. Particular forms of Linear Differential Equations
with Variable Coefficients.
355. Integration of a linear differential equation whose coefficients
are the successive powers of a binomial 482
356. Integration of a linear differential equation, whose coeffi-
cients are simple binomials 484
357. Examples of the process : Riccati's equation 486
SECTION 5. Differential Equations of higher orders and degrees.
358. Integration oif n (x) = v{f n - l (x),f n - 2 (x)} 489
359. Integration of f(x,y',y") = 0, and of f(y,y',y") .... 492
360. Homogeneous equations of the second order 493
361. Two other particular forms 495
SECTION 6. Integration of Partial Differential Equations of
higher orders.
362. Monge's method 496
363. Examples in illustration 498
364. Integration of a linear partial differential equation of the
wth order 500
365. Integration by the calculus of operations 500
CHAP. XV.
SOLUTION OF GEOMETRICAL PROBLEMS INVOLVING
DIFFERENTIAL EQUATIONS.
366. Problems involving differential equations of the first order 503
367- Trajectories of plane curves referred to rectangular coor-
dinates 504
368. Trajectories of plane curves referred to polar coordinates. . 507
369. Other cases of trajectories 508
370. Trajectories of surfaces 509
371 . Geometrical problems involving partial differential equations
of the first order 511
372. Integration of the differential equation of the lines of cur-
vature on an ellipsoid 512
373. Geometrical problems involving differential equations of the
second order . 516
xxiv ANALYTICAL TABLE OF CONTENTS.
374. To find the surface every point of which is an umbilic .... ">18
375. The surface of revolution, the principal radii of curvature at
every point of which are equal and have opposite signs 519
CHAP. XVI.
INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS.
376. Simultaneous differential equations of the first order .... 521
377- The number of arbitrary constants is the same as that of
the simultaneous equations 523
378. Integration of linear simultaneous equations 523
379. Integration of linear simultaneous equations with constant
coefficients and of the first order 525
380. Linear simultaneous equations of higher orders and of con-
stant coefficients solved by the calculus of operating
symbols 528
381. The same method applies to simultaneous linear partial
differential equations 529
382. Some particular examples 529
CHAP. XVII.
INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES.
383. Application of Taylor's theorem 533
384. Application of Maclaurin's theorem 534
385. The method of undetermined coefficients 536
386. The solution of Riccati's equation effected by the method 537
387- A particular case of the last equation 538
388. The general solution of particular forms of Riccati's equa-
tion expressed in terms of a definite integral 539
INTEGRAL CALCULUS.
PART I.
INTEGRATION OF EXPLICIT FUNCTIONS
OF ONE VARIABLE.
CHAPTER I.
INTRODUCTORY; EXPLANATION OP DEFINITE AND INDEFINITE
INTEGRATION.
ARTICLE 1.] Consider the following problem: let x be the
length of a line OP (see Fig. 1) which varies continuously from
OP O = #o, up to OP H = x n ; on OP O , OP, op n let squares be de-
scribed, viz. OR O , OR, OR n , so that OR = x? ; let OP be increased
by an infinitesimal PQ = dx, and on OQ let a square be de-
scribed ; then the increase of # 2 due to the infinitesimal increase
of x is 2x dx : and suppose that a similar process of augmenta-
tion is performed on all values of x from <r up to x n ; the effect
of this will be that the square x<? will grow into the square # n 2
by infinitesimal augments, each of which is of the form 2# doe,
wherein x receives the successively-increased values. From
another point of view however the effect of such a process is,
that the finite gnomonic area OR W OR O will be resolved into
infinitesimal elements, or infinitesimal gnomons, each of which
is of the form PR'S ; and a? 2 will be resolved into elements
2# dx, corresponding to values of x from x = XQ up to x = x n ;
and if # = 0, the whole square OR will by the process become
resolved into its gnomonic infinitesimal elements.
But let us forego geometrical illustration, and consider the
following more general problem of pure number : Let f(x) be
PRICE, VOL. II. B
2 DIFFERENTIATION A PROCESS OF DISINTEGRATION. [2.
a function of x finite and continuous for all values of x between
x n and a? ; and let the difference x n a? be divided into n equal
and finite parts each of which is equal to A a?, so that x n a?
= wAa?; then by Art. 101, equation (21), of Vol. I,
/(a? + Aa?) -/(a? ) = Aa? /'(a? + 0Aa?)
/(a? + 2 Aa?) /(a? + Aa?) = Aa? /'(a? + Aa? +
(1)
Aa? /'{a? + (n l)Aa?
Let Aa? become infinitesimal, that is, become dx; then adding
the several members of (1), and remembering that
OCii '^ XQ -\~ 71 CLOC y
j ($11) """/ x^O/ ^~ J v^O/ $& \J v^O ~T" (^0[/)CLOO -p J \JX?Q ~\~ A (JLOCj CvX ~\- *
. . . + /'(a? n dx) dx. (2)
that is, the process of growth by infinitesimal increase, on
which principle equations (1) are constructed, is equivalent
to the resolution of /(a? w ) /(a? ) into infinitesimal elements as
exhibited in equation (2).
2.] The Differential Calculus therefore is a method by which
any given function may be resolved into its infinitesimal com-
ponent elements, and these are of such a nature, that the ag-
gregate of an infinity of them is required to constitute a finite
quantity ; the general form of all the elements is the same, as
appears from the above examples, and therefore any one is a
type of all : but the form of the typical element varies, as the
finite function varies, the law of connexion depending on the
process of Differentiation. The subject then on which Diffe-
rentiation is performed is the finite function, and which becomes
resolved into its elements. The process of Differentiation is
therefore one of Disintegration.
Suppose however that the data are changed : that an infini-
tesimal element is given, which is the type of others, and that
the sum of them between certain terms which are given is to
be determined ; if the terms, or limits as they are called, are at
a finite interval apart, there will be an infinity of infinitesimal
elements between them of which we have to find the sum ; and
which will in general be a finite quantity. The process there-
fore required for such a Problem is necessarily a summation
3.] THE PROBLEM OF INTEGRAL CALCULUS. 3
of series, and is inverse to that of the Differential Calculus : in
this case the infinitesimal element is given, and the finite func-
tion, of which it is the element, is sought ; and the particular
form of the problem varies as the form of the function changes ;
that is, the particular mode of summation will be different for
different functions, although the nature of the process will be
the same in all cases.
Suppose for example that the element is %xdx, and that the
sum of all such is required, as x gradually increases from x
to x n : for the sake of simplicity, let x n X Q be divided into
n equal infinitesimal parts each of which = i, so that
X n #o
tl/yj "~~ Ct/O filly . il - ,
n
and in the result n = oo ; then the sum
_ (
= %l
n(n 1) .
( ) X n XQ X n XQ
~ ~
= x n 2 XQ Z , when n = oo ;
that is, according to the interpretation of fig. 1, the sum of all
such gnomonic pieces as are indicated by 2# dx is the dif-
ference between two squares ; and if the lower limit = 0, then
the sum = # n 2 .
3.] The process by which such sums are found is termed
Integration, being, as it is, the putting together the parts of
which a whole is composed ; and the sum of the series of infini-
tesimal elements between given terms is called a Definite Inte-
gral, the values of the variable which assign the first and last
terms being called the limits of Integration. It will also be
observed that, as the form of each term of the series is the
same, if a general term be given, it may be considered the
type of all, and therefore by it the law of the series is assigned ;
and hereby can be determined the general form of the sum
of a series of terms, although the first and last terms may not
be given ; this general sum is called the Indefinite Integral.
B 2
4 DEFINITION OF INTEGHAL CALCULUS. [4.
The Integral Calculus is the aggregate of the rules by which
Integrals are determined, and the code of laws subject to which
Differentials and Integrals in their mutual relations may be
applied to questions of Geometry and Physics.
The following problem therefore is that which primarily the
Integral Calculus has to solve :
4.] Let f(x) be a function of x finite and continuous for all
values of x between XQ and x n , and not changing its sign be-
tween these limits ; and let x n be greater than XQ. Suppose the
general type of an infinitesimal element, or the infinitesimal
element-function, as I shall call it, to be f(x) dx ; it is required
to find the sum of the series of such infinitesimal elements, as
x increases from XQ to x n .
Let x n x Q be divided into n infinitesimal parts, and let
Xi, x<z, . . x n _i be the values of x corresponding to the points
of division. Let S be the definite integral ; then observing
that f(x) is multiplied by the increment immediately following
x y we have
... + /(a?_i) (x n -#_!). (3)
But suppose F(#) to be that function of x whose derived-func-
tion is /(x), then if x and XQ are two values of x differing by
an infinitesimal , , -, . ,
= f(X Q ) (a?,. XQ), (4)
which equation is in fact the mathematical definition of a de-
rived-function. Similarly
F(a? 2 ) F(a?i) = /(a?i) (a? a #1), "\
' ' ......... [ (5)
Tf(X n ) - P(j? n _!) = /(^ n _i) (X n - X n -l) . }
Substituting these in equation (3)
S = F(a?i) -*(#<>) + *(ff)-F (*i) +. . . +F(# n )-F (#_!) .
= F(a? n ) F(a? ) ........... (6)
The symbolization best suited to the problem is the follow-
ing ; d, as in the Differential Calculus, expresses the differential
or infinitesimal element of a function or of a quantity; thus
df(x) is the element of /(a?). / (the long S) symbolizes the
general sum of an infinite number of terms, of which each is
6.] PRINCIPLES AND SYMBOLS OF INTEGRATION. 5
an infinitesimal; so that if f(x] dx, (which be it observed is the
product of two quantities, one finite and the other infinitesi-
mal,) be the type of the elements, the sum of an infinite
number of which is to be determined, that sum is represented
by lf(x) dx ; and as thus far the limits are not introduced, this
symbol represents the indefinite Integral. And if x n and XQ
are the limits of the Integral, x n being the last or superior, and
XQ the first or inferior limit, the definite integral between these
limits of f(x) dx is conveniently represented by
J**f(9)d*. (7)
Since v(x) is that function whose derived is f(x), let us repre-
sent, as in the Differential Calculus, f(x) by v'(x), so that in
equation (6), S is equal to the sum of infinitesimal elements
of which F'(#) dx is the type ; and therefore (6) becomes
I
Jxn
// /y __ in / /yi \ __ -to t /y \ /Q \
\JvdL T \W7iy "~^ * \tJbfyJ . \^/
5.] If the superior limit be x, x being a general value of the
variable, subject to the conditions that F'(#) is finite and con-
tinuous and does not change sign, for all values of x between
XQ and #, then
x
) dx = F(#) P(J? O ). (9)
p
I
'
and omitting F(^ O ), which is constant, the indefinite Integral
of F'(#) dx is F(#), and we have
fo(x)dg = F(a?). (10)
Hence it follows that the definite Integral of F'(#) dx between
the limits x n and XQ is the value of the indefinite Integral when
x = x n , less its value when x XQ', on this account it is fre-
quently and conveniently expressed as follows
F'(O?) dx = "*<*)*" = F(a? B ) - F(* O ). (11)
6.] Perhaps it may be supposed that the definite Integral
depends on the number and magnitude of the elements Xi X Q)
x 2 #!, ... x n #_!, and therefore on the number of the
parts ; but if the elements be infinitesimal, and their number
therefore infinite, the mode of division makes no difference in
the definite integral, as may thus be shewn :
6 DEFINITE INTEGRATION. [7.
Whatever another mode of division is, we may consider it to
be a subdivision of the first, and thus its elements to be parts
of the former elements, Suppose then x\ <r to be divided
into n parts, and 1} 2 , 3, -i to be the values of x corre-
sponding to the points of division, and let F'(#) dx be the in-
finitesimal element -function: then the sum of all the infini-
tesimal elements corresponding to the successive values of x
between XQ and x\ is
the value of which, by the process identical with that of Art. 4, is
And as analogous results are true for each of the other ele-
ments # 2 x, ... x n x n _i, so will the sum be true ; and there-
fore equation (11) is true, independently of the particular mode
of division by which the elements are formed. Hence we have,
subject to the condition that F'(#) is finite and continuous and
does not change sign within the limits x n and <r
(#2 #1) + "
(*- *n-l). (12)
... + F(a? n ) - F(d? B _j),
= F(a? B ) - F(O?O). (13)
It will be observed that in the series (12), the terms do not
go as far as F' (x n ) ; in the definite integral therefore expressed
by (13) the value of the element-function at the inferior limit
is included, and that at the superior limit is excluded.
If the limits for which the integral has to be calculated are
such that F'(#) changes sign at, say, x k , then the values of the
integral must be calculated separately from <r to x k) and from
x k to x n .
7.] To return to the consideration of the Indefinite Integral :
by equation (10) ,,
/ F'(#) dx = F(#) ;
that is, the operation symbolized by I dx when performed on
i
jf(x) changes it into v(x); but by the Differential Calculus -
dx
7-] INDEFINITE INTEGRATION. 7
is the symbol of an operation which being performed on F(#)
/j
dx and -y- are so related
dx
that one represents a process the reverse of that indicated by
the other ; that is, according to the index law which the symbol
-r- is subject to,
r / d \ - 1
.'. f = d~\ (15)
and / represents an operation which is the reverse of differen-
tiation *.
Hence according to the notation of derived functions
F' (x) dx = v(x),
f"(x) dx = if'(x),
dx = F"- 1
/ F n
and in this mode of viewing the subject, the symbol / dx must
be considered as a complex character, and indicative of a
certain analytical process to be performed on a certain func-
tion, and which is the reverse of Derivation.
Hence the problem of Integration resolves itself into this;
viz. to determine the function which, when differentiated, pro-
duces the infinitesimal element-function of the terms of the
series ; and therefore as this is a process the reverse of Differen-
tiation, we may make use of our knowledge of the Differential
Calculus, and as far as possible invert its rules, and these will
thereby become those of the Integral Calculus ; such processes
we shall enter on in the next Chapter, and thereby obtain in-
definite integrals, from which definite integrals may be deduced
by means of equation (11). In this aspect of the Calculus an-
other point requires explanation. Since an arbitrary constant
when connected with a function of x by addition or subtraction
* See Vol. I. Art. 365.
8 FUNDAMENTAL THEOREMS. [8.
disappears in Differentiation, so in the reverse process such a
constant must be introduced ; and thereby we have
v(x)-dx = r(x) + c;
the same result also follows from equation (9), wherein F(# O )
being independent of x is constant with respect to it. But as
the Integral Calculus might exist previously to the Differential,
for the infinitesimal element may exist previously to and inde-
pendently of the finite function, so its principles ought to have
an independent basis. We shall therefore in the first place in-
vestigate certain properties of definite integrals, which will be
required in the sequel, and also integrate some infinitesimal
element-functions from first principles.
8.] THEOREM I. If an infinitesimal element -function has
a constant quantity as a factor, the definite integral will also
have the same constant factor.
Let the infinitesimal element-function be a F'(#) dx, wherein
is a constant quantity ; then
x) dx a F (x n ) a
= a {v(x n ) -
- n I v'(r\ r?r
(I I * V^/ Wtt'. {l.\JJ
A constant factor therefore may be taken outside the sign of
integration; and similarly may, if required, be removed from
without to within the sign of Integration.
The following are particular cases :
(JC /*ii*
" v'(x) dx = I V(#) dx. (17)
/*' v'(x)dx 1 /* ,
/ - = - / F (a?) dx. (18)
Jx c c J Xo
The same theorem is of course true of an indefinite integral.
THEOREM II. The integral of the algebraic sum of any
number of infinitesimal element-functions is equal to the alge-
braic sum of the integrals of the same element-functions.
Let F'(#) flkr, f'(x)dx, <'(#)</#, ... be any infinitesimal ele-
ment-functions finite and continuous, and not changing sign,
between the limits x n and # > then
8.] INTEGRATION BY PARTS. 9
''(#) /(#) 0'(#) } dx
~\ x n
J V
<*0
r x n r#
x} dx I f(x) dx + / <b'(x) dx . . . (19)
/ /
The same theorem is also true of indefinite integrals.
Hence, and by means of the former theorem,
Jx
) dx + A/^I ( Xn f(x} dx. (20)
THEOREM III. If the infinitesimal element-function be of
the form /(#) x F'(,T) dx, then
C x n f~ ~\ x n F x n
I /(%) x f'(x}dx \f(x) x F(^) / F(a?) xf(x)dx.
x Jar Jf
For convenience of notation, let f(x} u, F(^) = v, v'(x~) dx
= dv, and let UQ, u^, u%...u n) VQ, Vi, v z ...v n be the several values
of u and v corresponding to X Q) x\, x z ...x n ; then
... V n (U n M n _i
and since the differences between VQ, v\, v%, ... v n are infinitesi-
mal, if we take i to be the general symbol of an infinitesimal,
we have
Vl = VQ + l,
PRICE, VOL. II.
10 INTEGRATION BY SUBSTITUTION. [8.
so that
f*"
I u dv = u n v n U Q V O {v (HI w ) +
^T
the last term of which equality must be neglected, because it
contains infinitesimals of a higher order than those of the pre-
/**
ceding term, and the preceding term is / vdu; therefore
Jx
C x n f x *
I udv = u n v n u Vo \ vdu, (21)
Jx Jx
[-I*,. fn
uv\ - vdu. (22)
J ^ "*
And therefore resubstituting
" 7 " (^) xf(x)dx. (23)
This theorem is known by the name of integration by parts,
and is of very frequent use ; the form which it assumes in the
case of an indefinite integral is
) x /(a?) dx = /(a?) x *(o?) - J F (a?) x /(a?) dx ; (24)
or if u and v are two functions of x, then
udv = uv I vdu, (25)
THEOREM IV. If, in order to determine the Integral of
v'(x) dx, it be convenient to introduce another variable z, related
to x by the equation z = <f> (a?), so that
=/(*), dx=f(z)dz, *(x) = *(f(z)),
and if ^ n and z are the values of z corresponding to x n and XQ,
then
z
that is, for the definite integral determined by means of x, we
9-] DEFINITE INTEGRALS FROM FIRST PRINCIPLES. 11
may take as its equivalent the other definite integral determined
by means of z.
Let Zij Zz,...z n -i be the values of z corresponding to x\ t
$2, ... x n -\, then, the elements of x being infinitesimal, we have
dz', (26)
that is, the two definite integrals are equal; and the latter
therefore may be used for the former ; and vice versd.
This method is called Integration by substitution, and is
of course true for the indefinite integrals as far as variable
quantities enter into the functions ; the arbitrary constants will
however frequently assume different although equivalent forms.
Other Theorems on definite integrals we shall reserve to
Chapter IV.
9.] The determination of definite integrals from first principles.
f x *
Ex. 1. To determine / dx.
Let, as in Art. 4, x n X Q be divided into n infinitesimal parts,
and let x\, x 2> ... x n _\ be the values of x corresponding to the
points of division ; then
r x n
Ex. 2. To determine / x a dx.
J
Let, as heretofore, a?i, x z , x$, ...^ n -i be the values of x corre-
c z
12 EXAMPLES OF DEFINITE INTEGRALS FEOM [9.
spending to the (n 1) points of division of x n XQ', and as the
mode of division does not affect the result provided that the
elements are infinitesimal, let us suppose that <r , x\, ...x n form
a geometrical progression whose common ratio differs infinitesi-
mally from unity : that is, let
Xi= Xo(\+l) .'. XI XQ = XQ t,
x z = xi (1 + i) x z Xi = Xi i,
X n #_! =
wherefore i is infinitesimal ; and we have
/*
/
Now,
XQ)
= xf l i + xf l i + ... + x a n + _\i,
+
r /*\ a+l
Iw
The denominator being evaluated according to the method
applied in Art. 30 of Vol. I ;
/* r a+1 r2 +l
f\a dx== ^ b_.
L a + l
^ a + l
Ex. 3. Determine
Let a? n a7o be divided into infinitesimal elements, as in the
last example; then
9-] FIRST PRINCIPLES. 13
'x ^ XQ Xi X n _i
= i + i + ... + t,
= ni;
M
and since = (1 + i) n , n log (1 + i) = log x n log x 0)
rX n ^y,
' L, 7 =
= log x n log x Q :
since log (1 + i) = i, when e is infinitesimal, by Cor. I. Art. 21
of Vol. I.
Ex. 4. Determin
/**
ine / a x dx.
Jx
Let a? n XQ be divided into ra equal parts, each of which is
equal to i ; so that
x n Xo = m; then
a x dx = a*o
i
log e a
since by Art. 32 of Vol. I, a* 1 = log e a.
Hence we have . Xn
I " e x dx = e x e x o.
+ ' .V*.
Ex.5. Determin
/**
ine /
Jx
Let a? n X Q be divided into w equal elements, each of which
is equal to i ; so that
wfi ~ <* fit?
14 DEFINITE INTEGRALS FROM FIRST PRINCIPLES. [9.
f*m
.' . I CQSXdx = i COS XQ + i COS (#0 + i) + ...
Jx
+ i cos {#0 + (n 1) i},
= i | cos XQ -f cos (XQ + I) + ...
+ cos{# + (n 1) i} \
n 1 .\ ni
and therefore, if n = oo , and i be infinitesimal,
o ^n + ^0 ^n XQ
cos a?6?a? = 2 cos sin >
The preceding examples then give the following indefinite
Integrals :
I dx = x + c,
/dx ,
= log* + c,
r a*
I a x dx = I -- [- c,
J logo
/ e x dx = e x + c,
I cos x dx = sin x + c.
10.] KULES FOR INTEGRATION. 15
CHAPTER II.
CONSTRUCTION OF RULES FOR INTEGRATION OF
ALGEBRAICAL FUNCTIONS.
10.] WE proceed then in the present and next Chapters to
deduce the rules of the Integral by inverting those of the
Differential Calculus : and first we shall from this point of view
exhibit the forms of indefinite integrals which correspond to
the theorems on definite integrals which have been proved in
Article 8.
THEOREM I. Since
d.aT?(x) = a F'(#) dx,
.' . \ay[(x)dx = a~s(x),
= a \v'(x)diK; (1)
that is, in the integration of an infinitesimal element-function,
one of whose factors is a constant, the constant may be placed
outside the sign of integration.
THEOREM II. Since
d.{i(x} f(x} ...} = d.F(a?) d.f(x) ...
= J>'(x~)dx f(x)dx ...
f'(x}dxf(x}dx...} = F(a?) f(x) .... (2)
that is, the Integral of an algebraic sum of infinitesimal ele-
ment-functions is equal to the sum of the Integrals of the several
infinitesimal element-functions.
I {a F'(#) dx cf(x} dx} = a F(#) + cf(x),
16 INTEGRATION OF ALGEBRAICAL FUNCTIONS. [ll.
THEOREM III. Since
d.r(x) x f(x) = f(x) x v'(x)dx + F(#) x f(x)dx,
.'. f(x) x ?'(x)dx d.(v(x) x/Oz-)) F(O?) xf(x}dx,
Jf(x) x *'(x)dx = F(#) x /(a?) r j*(*) x f'(x)dx; (3)
and therefore if
/(a?) = U, TS(X) = V,
ludv =. uv lvdu', (4)
THEOREM IV. Since
/
<{/(*)} /(*)^ = *{/(*)}; (5)
which is the theorem in the Integral Calculus corresponding to
that of the differentiation of compound functions as explained
in Art. 31 of Vol. 1, and which may also thus be proved :
Let f(x) = z .-. f(x)dx = dz,
= *(*),
= *{/(*)} (6)
Integration by this last process, as is manifest, is equivalent
to the Integration of a Compound Function, and is of great
importance; for hence it follows that those formulae of Inte-
gration which are true for x and simple functions of x, are
true also for compound functions.
SECTION I. Integration of Fundamental Algebraical Functions.
11.] Integration of x*dx.
Since d.x m = mx m -*dx,
r
mx m ~ l dx = x m ,
x m ~ l dx .
m
12.] INTEGRATION OF ALGEBRAIC FUNCTIONS. 17
If therefore n be substituted for m 1, that is, if m = n + 1,
",,*, = ! + .* (7)
_/",
Therefore to integrate x n dx, add unity to the index, divide by
the index so increased, and by dx.
Of this result the following are particular cases :
(1.) Let n be negative; that is, for n substitute n,
f Cdx #-<-i) 1
/ x~ n dx = = -- = = -- - -- , .
J J x n n l (nl)x n - 1
(2.) Let n be fractional, n = -,
p+q
therefore
Cdx _ J_
A**-!*
12.] The formula (7) is true for all integral and fractional,
positive and negative values of n, with the exception of, n = 1 ;
in which case the right-hand member becomes op , and the
formula ceases to give an intelligible result : we must therefore
return to the principles of definite integration, and by means
of them obtain the true integral.
r-n>n + l-,x n+l _ n + 1
x-dx = [ ] = x - -- ^L_ = when n= _L
'
Evaluating therefore the indeterminate fraction by the rules
of Chapter V, Vol. I, and observing that n is the variable,
* Henceforth we shall not add the arbitrary constant c; it is of the same
*rm in all cases, and therefore a repetition of it is superfluous.
PRICE, VOL. II. D
18 INTEGRATION OF ALGEBRAIC FUNCTIONS. [13.
'* d_X _ log,. X . X n+ 1 log, XQ .Xp n+l
= log e x log e x Q) when n = 1,
= log e (J-); (8)
a result identical with that of Ex. 3, Art. 9 ; and therefore
[dx ,
I = log e #. (9)
J x
13.] Extending therefore the results of Art. 11 and 12 to
Compound Functions, as we are authorized to do by Theorem
IV of Art. 10, we have
// f( x \ "J.w+1
f f(x)\ n f'(x}dx = (10)
IOP- SffrM C\'\\
1U 5 \J v* / / \ ii )
Hence the Integral of a fraction, whose numerator is the dif-
ferential of the denominator, is the Napierian Logarithm of the
denominator.
Ex. 1. (a + bx) n dx T I (a + bx} n d(a + bx)
(a + bx) n+1
Ex. 2.
(a + bx z ) n+l
Ex.3. j(a m x m ) n x m - l dx=- - 1 (a m x m } n d(a m x m )
(a m x m ) n+l
Ex. 4. ( a + bx + cx z ) n (b + 2c#) dx
= (a
(a + bx+ cx 2 ) n+l
n+ 1
1 4-] INTEGRATION OF ALGEBEAIC FUNCTIONS. 19
Ex. 5. f X * dX ^ = - \ /"(fl3-*)-*rf ( 3 -^),
* (a 3 x*y* 6J
= _|(fl-*8)*.
f dx 1 Cd(a + bx) 1 .
Ex. 6. / - r- = T / v . ' - - log (a + bx).
J a + bx bj a + bx b
C(b + 2 ex) dx I'd (a + bx + ex 2 )
Ex.7. / - z r - *= , - ir L
J a + bx+cx 2 J a + bx + cx z
Ex.9, --dx = {I+x + x 2 + .,.+x n -i}dx,
~~ SO v
x 2 x n
= X+- + ......... -I- _.
2 n
// */*
14.1 Integration of 5 5 . and of similar forms.
a 2 + x 2
x adx
Since a. tan" 1 - =
a I ^,2
T *
adx
-\ x f '
a J a'*
*X L . X - _.
- = - tan- 1 - . (12)
And therefore by Art. 13,
l t . ._! n ov
= tan x . (lo)
The following is a general form, which admits of being reduced
to (] 2) when the roots of the denominator are impossible :
f dx _ 1 f_
J a + bx + ex 2 ~~ cJ a
dx
b
c c
= ir _
cJ 4<acb e '
( 2
D 2
20 INTEGRATION OF ALGEBRAIC FUNCTIONS. [15.
dx 1 2c
I
tan
a + x + ca: c c-brf (4,ac-b 2 )*
2
tan- 1 - , (14)
since 4ae b' 1 is positive, when the roots of the denominator
are impossible.
Ex.1. -_ = tan- 1 *.
+x 2
2 f
' 4 - J
dx
2.3*
/
J2-
Cm-4-nx , C mdx C xdx
Ex. 4. / ? dx = I -= = + w / -= 5,
/ nt | '** ^/ flf* -U ;>;* / fl* -(- SC
m A . x n
/jfyy ft V
15.1 Integration of -= -, -= 5, and of similar forms.
J a? 2 a 2 a 2 a? 2
i i r i i >
Since -5 5 = s -s ?
ar a^ 2a CiP a # + a J
/* rf* If/* cfa? C dx ~\
'"' 7*2 a 2 = JJaUa? a ~7* + aJ
1 f fd(xa) rd(x + a)")
~ 2a(.J x a J x + a )
= o-1 log( a) log( +
i4 ff V.
1 , a /1K .
s loe . (15)
2a *
1 6.] INTEGRATION OF ALGEBRAIC FUNCTIONS. 21
Hence, by virtue of Theorem IV, Art. 10,
[ fWd* ! loa . //(a)- \
J {/(*)}*- a 2 ~2 5 V(*) + ''
1 1 f 1 1 )
Again, since -5 - 5 = < -- 1 -- <-
a 2 ^ 2 20 (.a + x ax)
/dx I C f dx f dx ~\
a 2 x 2 ~2a\.Ja + x J a x)
1_ C Cd(a + x) _ Cd(a x)-\
2a \J a + x J a x S
) log (a a?)
i /1CN
= log - . (16)
2a ax
Also when the roots of the denominators of the following form
are real and unequal it admits of being reduced to one or other
of the above forms :
r dx i r
J a + bx + ex 2 " cJ
log ^ v ' , (17)
-m- -m i UOC -1 I
EX. 1. / 7-2 = j-
log
2(ai)*
=/^
16.] Integration of JT-. and of ; r-r-, m and
w> n m '
being integers.
22 INTEGRATION OP ALGEBRAIC FUNCTIONS. [l6.
x n dx
To integrate
(a + bx} m '
za
Let a + bx = z .-. x =
o
.'. bdx dz dx = -r
o
/x n dx r/z a\ n dz
a + bx m ~ J V b i ~bz
r fj\n
l^Ldz. (18)
z m
to integrate which, (z a) n must be expanded by the Binomial
Theorem, and each term of the expansion, having been divided
by z 7 ", must be integrated separately ; and the substitution of
a + bx for z made subsequently.
To integrate -. r~r~-
/v>n ( n I f^y>\m
dj \Ui "|" Utb J
Let x - . . dx = 2" ' whereby we have
C dx _ Cz n+ *
and which is of the form (18), and may be integrated accord-
ingly-
Ex. 1. r X ' dx
(a + bx) 3 '
T . Z ~ a
Let a + bx = z . . x = r
b
bdx = dz fa
dx = -;-
o
C x 2 dx f(za) 2 dz
*'* i (a + bx) 3 ~ J b 2 bz*
= b*J~~z r ~
'''dz
I CCl 2a a 2 \
= n 1 1 1 -H 35 r
b 3 J (.z z 2 z 3 J
1 7.] INTEGRATION OF ALGEBRAIC FUNCTIONS. 23
f x*dx \_ r 2a _ a 2 )
Ja + bx}* ~P I g2> "T"~2^J
Ex.2.
x\a +
1
Let x = -
z 2 dz
y b
Let o + 02 = y .. z = 2 -
/*(y-
" J a 3
I (-W,
~ dy
2b a + bx
CL X
17.] Examples of the preceding methods of Integration.
fib \ , 6 2c 5
Ex.1. / (a 5 + cx*\ dx = aa? + = + -=-<r 5 .
j \ a? 3 2^ 2 5
/f^i2 ->-)O /v>4 ^5
(1+^)(1-^)^ = |- + |--|-|-.
,., fx^dx a? 3
Ex. 3. / - = a? + tan- 1 ^.
j - 2 + 1 3
Ex 4 /" ^-'^ Zl \
J(a + ba? n ) n n(n l)b (a + bx n ) n ~ l
f (a x)dx ,1
Ex.5. 1-2 L__ - (2^ ayS)-.
J - 2 *
24
Ex.6.
Ex.7.
Ex.8.
INTEGRATION OF RATIONAL FRACTIONS. [l8.
J 3 x* + 7 12
r dx 2 _ 1 2a? + l
i r dx
JI+x + x 2 3* 3*
f(m + nx}dx n
Ja + bx+ca? 2c &(
2mc nl
2c
J a + bx + c# 2 '
SECTION 2. Integration of Rational Fractions by decomposition
into partial Fractions, and by formula of reduction.
18.] A rational fraction is of the form
go x m qi x m ~ l + #2 x m ~* + q m _ l x
(19)
the numerator and denominator being algebraical expressions
involving only positive and integral powers of x, and q 0) qi, ...
q m , Pi, Pz, ---Pn being constants.
Now when m is greater than, or equal to n, (19) may by
common division be reduced to the sum of an integral alge-
braical expression, and of a fraction whose denominator will be
the same, and whose numerator will be of dimensions lower by
at least unity than the denominator : the integral part may be
integrated by the methods of the last Section ; and the frac-
tional part by the method which we now proceed to explain.
The most general form therefore of such a fraction is
^^ " + p \x +p' (20)
which, for the sake of convenience of reference, we shall sym-
bolize by
suppose the n roots of f(x) to be i, a^, ... a n , which may be
either real or impossible ; and all may be unequal or there may
be one or more systems of equal roots. With a view to subse-
quent integration it is necessary to explain a method of resolv-
ing a fraction such as (20) into other and more simple fractions;
1 9.] INTEGRATION OF RATIONAL FRACTIONS. 25
and as different processes must be applied, according as all
the roots are unequal, or as some (not all) are equal to each
other, so shall we divide our inquiry into two parts. It will
appear that the processes are equally applicable, whether the
roots be real or impossible ; and if there be systems of equal
roots, processes similar to that which is applied to one set must
be applied to each of the others.
19.] Let all the roots of /(<#) be unequal, so that
/(#) = (x flj) (x a z } ...... (xa n }. (22)
Tjl / /Vt\
Then -{ may be resolved into a series of fractions of the form
x
, ,
where N b N 2 , ...N W are constants and to be determined.
The possibility and legitimacy of such an identity as (23)
may thus be shown :
Let the right-hand member of (23) be reduced to a common
denominator, which will be f(x} ; then as the identity (not
equality only) of the two members of the equation is to be per-
fect, the two numerators must be identical ; and by the hypo-
thesis they may be so : for as F (x) is a function of not more
than (n 1) dimensions, it may have n terms, but cannot have
more : and therefore involves n coefficients, some of which how-
ever may in certain cases have zero values : and the numerator
of the right-hand member will be also of (n I) dimensions
and will have n coefficients, involving n undetermined constants
NX, N 2 , ... N W ; by equating, therefore, the coefficients of the
same powers of x on both sides of the equation, there will be
n different equations, whereby N!, N 2 , ...N n may be determined,
and which of course it is possible to do.
Multiplying both sides of (23) by/(#), we have
...
(x i) (x 2 ) (% a n y
And as the two sides are identical, they are the same for all
values of x; let therefore x = a l , and since oc a l is a factor
of f(x), all the terms of the right-hand side vanish except the
first; and that becomes ^, and must therefore be evaluated
PRICE, VOL. II. E
26 INTEGRATION OF RATIONAL FRACTIONS. [19.
by the method of Chapter V. Vol. I, whence we have, when
x = i,
(25)
C- -1 1 -f F ( ft 2)
similarly it x = a^, N 2 =
x = a n , N W =
i
Whence we have
/( 2 )
(26)
i ^"^ ^ , F ( g ") _ /07\
T /// \ ~ r~" > * t '.*"/\~ , > \*' j
/(a?) /(i) a? - i /( 2 ) a? - 2 /() a? -
and therefore
c?# F( 2 ) /* c?^
F(a n ) dx
/F(#) _ v(ai) C
f&) X = f(a^J
/dx
- . we have by equa-
2? i
tion (11) , dx
which form is, when a\ is real, as convenient as the result
admits of: but if a k be impossible, then, to avoid the Loga-
rithms of impossible quantities, we reduce as follows :
Let a + /3 \/ 1, a ftV 1 be a pair of conjugate roots,
and let the coefficients of the partial fractions corresponding
to these roots, and found as above, be P + Q\/ 1, P Q v 1,
so that the two partial fractions are
p-f Q-S/ 1 P Q\/ 1
and
x (a + /3\/l) # (a ft* I)
let these be compounded into a single fraction with a quadratic
denominator, whence we have
and therefore
L -iZ^. ( 30)
2O.] INTEGRATION OF RATIONAL FRACTIONS. 27
20.] Examples of the preceding method :
p i f xdx
J x* 5x
6'
To decompose into its partial fractions 5
.a? 2
the roots of /(#) are 2 and 3 ; and
: .-. the coefficient of - = is 2
2^5
----- of 3 is 3,
x o
f
*'* J ^ 2 -
xdx
= 2 log (a? 2) + 3 log (a? 3)
Ex. 2.
In this example the roots of the denominator are 1, 2, 3.
coefficient of =- is 1,
is 5,
5 is 5 ;
(x*+I)dx f dx K f dx t f dx
* _6/"-^L+5/"-
/ *y>_i_ I / w i O f 'r
/ / ~|~ i / W ~T~ <W / cC<
= log
r R *._o
Ex. 3.
The roots of the denominator are 0^ 2, 4; and
F<T 5r 2
E 2
INTEGRATION OF RATIONAL FRACTIONS. [ao.
. . the coefficient of - is -
x 4
is 3
x + 2
1 . 11
~ 18 T>
4
(5x 2)dx 1 Cdx ndx 11 f
'' # 3 +6# 2 +8<z> = ~4,Jlv + Jx + 2~~4J
dx
- logr + 3 log^ + 2) - log(,r + 4)
= log
Ex.4. /^ 2 %*-2-
The roots of the denominator are 1, + V 2, V 2.
f(x)
the coefficient of
1 1
1 . 1
== is
2-V-2
12
v 3-
V2
I f dx
12
I f(2-V-2
~ 12J l^-yT^
= log
20.] INTEGRATION OF RATIONAL FRACTIONS.
dx
Ex. 5. f-
J a
The roots of the denominator are 0, +\/ 1,
F(a?) 1
\/ 1,
the coefficient of - is -.
x 4
~6-
is
IS
J_
24'
J^
24 ;
dx
1 [dx 1 ff I _1 _ -
-'- 4>J T 6J \x_ yzi + ^/Zf J
Ex.6. f-^.
J a? 3 1
The roots of the denominator are 1,
and
^
24
l+V3 1 A/^
.-. the coefficient of r is =,
xl 3
is
x
30 INTEGRATION OF RATIONAL FRACTIONS. [21-
.*. the coefficient of . is
C dx 1 C dx
i* Ju """"* A O*7 tt/ X
6
* IT
. 4- . f Cfa?
x-- x
= -log(#-l)--/ __-<**
lojr O M / _
1\ 2 3
4
1, 1 1
= slog(# 1) ^ log (# 2 4- a? 4-1) ^=tan -1
It will be observed that the coefficient of the second fraction
corresponding to a pair of conjugate roots is deduced from that
of the first by changing the sign of the impossible part.
21.] Let two or more of the roots of /(#) be equal; then
the preceding process of resolution does not admit of being
applied directly, because in the right-hand member of (23)
there will not be n undetermined constants, and therefore the
number of unknown constants is not sufficient to render the
two numerators identical ; in this case we proceed as follows :
Suppose that m roots of f(x) are equal to i, and let the
other roots be a m +i, a m+2 , ... o n ; so that
f(x) = (xa^ (x-a m+l ) (xa m+z ) ... (xa n ). (31)
F /2>\
Then if - 7 be resolved into a series of fractions of the form
(23), the numerator of that which has (x ai) m in the denomi-
nator, must be of (m \) dimensions or involve m undetermined
constants, otherwise the equation cannot be an identity; we
must therefore suppose
~ l -f B 2 a?"*- 2 + . . . + B m _! X + B OT N m+1
... +-_. (32)
x a n
But, for the purposes of integration, it is more convenient, and
21.] INTEGRATION OF RATIONAL FRACTIONS. 31
it is allowable, to assume the numerator of the first partial
fraction in the form
Mi + Ma(a? ai)+M 8 (# i) 2 + ... +M m (r-a 1 ) w - 1 ; (33)
so that
g(a?) MI M 2 M OT N OT+I
f(x) ~ (x- ai ) m (x- ai ) m -* #-! a?-a m+1 "
... + -^-. (34)
a? a n
As to the numerators of these partial fractions, those cor-
responding to simple factors in the denominators must be de-
termined by the method of the preceding articles ; and for the
determination of MI, M 2 , ... M W , let
(x a m+l ) (xa m+2 ) ... (xa n ) = <t>(x);
and let (33), which is the numerator of the fraction which has
(x a\) m in the denominator, be symbolized by -fy(x), so that
Q
^ '
Q being a function of a? of n m 1 dimensions, which however
it will not be requisite to determine, and <$>(x) being the pro-
duct of all the factors in the denominator short of the set of
equal factors, so that
/(a?) = (a?-ai)* <*>(#). (36)
By the Theorem in Vol. I, Art. 119, Equation (14)
observing that, as ^(#) is rational and involves only positive
powers of the variable and is of (m 1) dimensions, all derived-
functions of it after the (m l)th vanish, and that therefore
the series (37) has only m terms. Hence, dividing both sides
of (37) by (x-a{) m
1 (x ai) m ~ l 1.2 (x ai) w ~ 2
~p ...... (- ~ ;r ~ / -i \ ^ \^' u /
32 INTEGRATION OP RATIONAL FRACTIONS. [21.
As to v/f(<), be it observed, that equating the numerators in
equation (35) we have
F(a?) = V(#) x </>(#) + QOe-ai) 1 ", (39)
But as (38) involves ^(#) and its derived functions up to the
(m l)th order, and these when x= a, the latter term in the
right-hand member of (40) will vanish for all values for which
^r(x) is used; we may therefore, for all purposes for which we
shall have to use \^(#), employ the following equation,
that is, ty(x) is equal to the numerator of the original fraction,
divided by its denominator short of the set of equal factors.
Substituting therefore in (34),
_ _
/(a?) (a? i) m 1 (a? fli)"'- 1 1.2 (off tti)" 1 - 2
V*- 1 (fli) 1 F(a m+1 ) 1
F( n )
- + ^3-
and therefore
* * * I 1 O O / I
but
-m& (r-l) (a?
. f Y (^ dx= V'foi) x ___ _ _
J f(x) ' (w 1) (a? !>-! (m 2) (a? "- 2 "*
If the denominator contains other sets of equal factors, the
21.] INTEGRATION OP RATIONAL FRACTIONS. 33
series of partial fractions corresponding to them must be de-
termined in a manner precisely analogous to that applied
above.
The method also is applicable to sets of eqiial factors involv-
ing impossible roots, in which case we may combine terms of
the series (44) corresponding to conjugate factors : for suppose
a-f/3A/ 1, a fiV 1 to be a pair of conjugate roots; and let
m equal factors corresponding to each enter into the denomi-
nator of the original fraction ; then the first terms of the series
(44) will be
1 \|f (a + /3 v 1) 1 ty( a
_ and
(45)
the sum of which two fractions, short of the common factor
s
+ /3 \/^l) (x a + [B J^l} m ~ l + ^(aft \/^l) (x a /3 x/^l
{(x a) 2 + /3 2 } m - 1
(46)
and similarly may the other results be combined : but in order
to avoid the logarithms of impossible quantities, the last terms
of the series corresponding to a pair of conjugate roots are
1.2.3... (m-1) ^.a^yHT 1.2.3... (m-1) ^_ a
which may be combined into a single fraction ; and of which the
numerator is {\lr m ~ l (a + p -v/^T) + ^r m ~ l (a ft \/^l)} (x a)
+ W m - l (a + p V^l) - V n ~ l (a-p \/^l)} /3 V^l and whose
denominator is (x a) 2 + /3 2 , and of which the coefficient is
and thus the integral of the corresponding element -function
will be of the form
PRICE, VOL, II.
< 47 >
34 INTEGRATION OF RATIONAL FRACTIONS. [22.
22.] Examples illustrative of the preceding.
f
EX ' L /7
J
To determine the coefficient of -:
x
+ x 2 + 2 r 1
r; ' the coefficient of - is 2.
l J ar
To determine the terms of the series (42) corresponding to
-l) 2 ,
6# 2
To determine the terms of the series (42) corresponding to
(*+!)*
by series (43)
1 5 1 -I
^~ T , i f X ,
1 3
-T) + log
////y)
CNp
(a?-l) 2
Ex
j (a? i) 4 (#'+!)
To determine the coefficients of ; and
x V
A/ 1 '
22.] INTEGRATION OF RATIONAL FRACTIONS. 35
.-. the coefficient of is -r.
1 1
To determine the terms of the series (42) corresponding to
(*-!),
X
Ex ' 3 ' /(^TT^
To determine the coefficient of - - - and of
OS V 1 #
.-. the coefficient of - - - is -
8
To determine the terms of the series (42) corresponding to
F 3
36 INTEGRATION OF RATIONAL FRACTIONS. [23.
#3 + 1 a?*+3o? 8 -2o?
2
(a?) =
I)
A &
L ) J \X J. ) A J \X L ) -L . A J (X JL
1 v 1 ""I
-1 ' 8
1
>-l 4
1 _j
" 4
As the process of decomposition and subsequent integration
is the same in all cases, it is unnecessary for us to encumber
our pages with other examples : the student however must ex-
ercise himself in them, and the ordinary collections will yield a
copious supply.
There are however two particular cases of similar decom-
position which exhibit remarkable peculiarities ; viz. those in
which the denominators are of the forms x n 1, and x n + \ :
and first we will take the simplest forms wherein the numerators
are unity.
doc
23.1 To determine the integral of .
__ i
First let n be odd ; then, in Art. 60, Vol. I, it is proved that
the roots of x n 1 = are,
1. cos 1- V 1 sin , cos \-V 1 sin ,
n ' n n ' n
nI /= . n\
. . . cos TT + v 1 sin TT :
23.] INTEGRATION OF RATIONAL FRACTIONS. 37
Now -^ = - 7 = - = -) because x n =l for all the roots
f(ac) nx n ~ l nx n n
of x n 1, and these are the only values of x for which we have
to consider the function.
.. the coefficient of - =- is -.
x 1 n
of
1 1 f 2** , . 27O
is - < cos -- 1- V 1 sm f.
, = . 2?r n (. n n )'
, = .
x ~ cos -- v 1 sin
n n
of
2?r / ^ .
a? cos -- 1- v Isua
1 . 1 f 2* , . 27T~)
. is - 4 cos -- V 1 sm J- .
/ . 2ir w (. n n j
Combining the pairs of conjugate partial fractions according
to equation (29) the first pair becomes
f\
27T
cos 2
n
f - 2* i") '
n < # 2 2# cos 1- 1 \
(. n )
and similarly will the other pairs of conjugate partial fractions
be compounded ; so that the following series will be formed
2 IT 4<n-
2#cos 2 2#cos 2
n n
27T . 4?r
7T 7T _
2# cos -- i-l a? 2 2# cos -- 1- 1
n n
2#cos
-
^ 2 2x COS
2#cos - TT 2
+ ...... + - - 5- <>
ZTT /. _ XTT\ ,
r , i fj cos , I2ir 2 cos I da
I ax 1 I ax * r w
' J x n 1 ~ nj x 1 w J
a? 2 2# cos [- ]
2-n-\
LII i e* i * COS )
n ' i \ n '
f . *> i
'' /
a? cos ) + ( sin J
n ' V ^
+ (49)
38 INTEGRATION OF RATIONAL FRACTIONS. [23.
277
, 277 27T
2 sin x cos
n , _ n
tan- 1 - - -- h
n . 2-Tr
sin
n
1 n I , . n I
n , ( . n ,\
... +-COS - 77 log \X 2 2#COS - 77+11
n n \ n
n-l
_ X COS - 7T
2 . n 1 . n /Km
-- sin - TT tan- 1 . (50)
n n , nl
SUl - 7T
If n be even, then equation (48) by means of Art. 60, Vol. I,
becomes
2x cos 2
27T
a? 2 2xcos
n
2#COS - 7T 2
-
a? 2 2,r cos - TT + 1
n
A , f dx 1 C dx I f dx
And / - r = / :r T / ^
J x n 1 nj x 1 wJ^ + 1
(51)
1 w 2 / w 2
cos TT log I a? 2 2 x cos TT -|-
n n \ w
w-2
_ <r COS 77
2 . w 2 w
sui 77 tan" 1 (52)
n n . n2
sm 77
n
24.] INTEGRATION OF RATIONAL FRACTIONS.
39
_ ?r
The roots of a? 3 1 are, 1, cos -^- + v 1 sin ;
o 3
.. the coefficient of - =- is ^r,
X 1 O
,,
f
I
x
/ . ^ . rx
I cos -5- + v -1 sm 1
\ o o I
/ . 27T-
f
of
2ir i =- . %TT
x cos-s v 1 sin--
1 C 2v /-^ .
is = ] cos V Ism
o (, 6
5- -;
6 J
C dx 1 C dx 1 /*
2-n
-= -- 2
2
27T
2 .
27T-
-- = tan
,
- 1
/ ^p
24."] To determine / - ^.
J a? w + l
Let w be even, then, in Art. 60, Vol. I, it is proved that the
, -. Tf . / v . TT 3w / - . STT
roots oi a? w + l. are cos H v Ism-, cos -- 1- v Isui ,
~
n n
cos
w 1 / ^ . w 1
ir + V 1
sin - TT.
n
40 INTEGRATION OF RATIONAL FRACTIONS. [24.
p / nfl\ nn QQ
Now -sM = , = = , since x n = 1 for all the
/ (x) nx n ~ l nx n n
roots of x n + 1 = 0; therefore the coefficient of
1 1 C 77 / r . 77 )
-^ is < cos - + v 1 sin - 5- ,
77 / - .77 w t n n)
t T
x cos v 1 sin -
n n
of
1 1 f 77 / - . 77-i
is s cos v Isin-V,
77 / r- .77 n (. n n)
cos |- v 1 sin
n n
and combining the pairs of conjugate partial fractions, accord-
ing to equation (29) the first pair is
77 / r . 77 77 / = . 77
cos- + v 1 sin- cos v Isin-
n n n n
n I I 77 / r . 77\ / 77 / =- . 77\
f x I cos h v 1 sm - 1 x I cos v 1 sin -
\- \ n n' \ n n'
77
which becomes -^ -^ .
n
'' 2 x cos [- 1
n
and the other pairs give similar results : so that
377
- (2,rcos -- 2) dx A%x cos -- 2) dx
C dx If \ _ n I ^ [ n
J x n + 1 ~ nj " TT ~nj'
^x cos - TT 2 dx
_
2,2? COS - 7T + 1
(53)
each of which must be integrated according to the process
indicated in the last article.
Again, let n be odd : then the roots of x n + I = are
7T . / T . TT 3?r / - . 3-rr
cos h v Isin-, cos -- Kv Inn , ......
n ~ n n - n
n2 / - . n2
cos 77 + v Ism 77, 1.
n n
24.] INTEGRATION OF RATIONAL FRACTIONS. 41
so that if the conjugate partial fractions are compounded by a
process similar to that employed when n is even, the last pair
becomes
n 2
, 2a?cos 77 2
n
n n 2
X Z %X COS 77 + 1
and therefore when n is odd
. (2^ cos -- 2) dx
C dx 1 C \ n '
J x n + l ~ ~ n J " 77
# 2 2#cos- 1
n
,. (2,2? cos 77 2) dx
1 C \ n I f dx
/ H / - . (54)
nJ n2 nJ x + \
X z 2tf COS 77 + 1
Ex.1.
The roots of x* + 1 = are
TT / - . TT STT / = . STT
cos - + V 1 sm -r, cos -r- + v 1 sin -r- ;
4 - 4 4 - 4
x x
therefore the coemcient of
1 1 / TT / r . TT\
is T (cos T + V 1 sm T ),
7T / - r .77 4^4 4/
cos - V 1 sin -
4 4
C 1 1/77 / . 77 \
oi - is - ( cos - v 1 sm T ) .
w . / - .77 4 \ 4 4''
cos -r + \ 1 sin ~
4 4
and the pairs of conjugate partial fractions compound into
77
--
, and - T
/I ^ c*iiu "T
tf 2 -2*
PRICE, VOL. II.
42 INTEGRATION OF RATIONAL FRACTIONS. [24.
377
- 2} dx ~[2xcos. 2) dx
4 ' 1 TV 4
~ I 4 J .
877
2#cos- + 1 ,r 2 2#cos
= - cos ^ log \x z 2x cos 7 + 1 )
sin
4
1 37r ! / 2 3?T ,
- cos log (x 2 2% cos +
STT
i o . < COS
1 . STT \ 4
sin -r- tan -1
sm-r-
4
24 J .877
= ^-= log ( - ' "" v ._ ) -| 7= tan
A / f) * "* . T / r\ /r*
Ex.2. /^4-
/T - / I ^ & V/WO .^ <W I M/O/ -. *
dx 1 Tj 5 / 1 T
B + 1 OJ _ 77... OJ
2) c?a? /. (2a? cos
5
877
2^? cos -
5
1 f dx
f 5Jx +
-= COS - log (x 2 ^X COS - + 1 )
n r \ n /
77
2. X COS p-
. 77 \ 5
+ rr sm -tan- 1 -
55 j . 77
sm-r
5
1 3?7, / 3?7 \
^ cos log ( <r 2 2^" cos -^- + 1 j
877
, cos _
2 . 377, , \ 5
+ F sin -IT- tan- 1
5 5 j . 877
sin -
5
26.] INTEGRATION OF RATIONAL FRACTIONS. 43
25.1 To determine the integrals of and of -,
-I r n ~,n _i_ 1 '
<*> A !li -f- J.
where m is less than n.
rr, . Cx m dx
To determine / - .
The roots of the denominator are those given in Art. 23 ; and
since
X
f'(x) ~~ nx n ~^- nx n n
we may determine without difficulty the coefficients of the
several partial fractions, and the form which a pair of con-
jugate partial fractions assumes when they are compounded.
. ., . x m dx , v(x)dx
By a similar process may we integrate -, and - ,
when F(O?) is integral and rational and of not more than (n I)
dimensions.
To the general forms of Arts. 23 and 24 also may be reduced
C x m dx 1 r x m dx
J a + bx n " ^tJ tT~ n "'
a
by substituting as follows;
i
b I a\ n
let -x n = z n .*. x =. (-} z
a \b'
i
(o, \ n
A/ ^ z>
whereby the integral becomes
a r z m dz
^1 J z n + 1 '
b n
26.] The process however for obtaining the integrals of many
infinitesimal-elements of the preceding forms may oftentimes
be much simplified by a judicious substitution; the selection
of which must be left to the ingenuity of the student, because
no general rules can be given. A careful study of the fol-
G 2
44 INTEGRATION OF RATIONAL FRACTIONS. [26.
lowing cases will probably indicate the course whereby we may
be led to such a simplification.
f da: f x*dx 1 f d.x*
Jx(a* + x*) ~7tf 3 (a 3 + # 3 ) == 3Jx*(a 3 + x 3 )'
which, if x 3 = z, becomes - / ^ - . ; and the integral may
3J z (a 3 + z)
be determined by the method of Art. 19.
[x*dx 1 f d.x* ^_ x*
' 7^?^ 3V(a 3 ) 2 +(# 3 ) 2 3a 3t< a 3 '
by means of equation (13), Art. 14.
F q /* dx i. /* ^'^ -f " _ z.
1 Jx(a+ke*? ~ = b 2 J x*(k + **)*' b~
_\_ r d.x*
~~ iwJ x*(k+x 3 ) 2 '
and which last integral may be found by the method of Art. 21.
/* xdx \ C d.x 2 . . . , . -
Ex - 4 - = whlch 1S of the form
equation (15), Art. 15,
1 . x z -a z
: 4^ 10 S-
Ex. r ^ d "
r^ 2 (ir
.5. =-
J x* 1
,
2
1 1
+
4 ^-1
x^dx 1 1'' "jr 1
i =r = xtan- 1 ^ + 7 log --
* \ 2 4 &
Ex 6 ( X * dx - {(x X 1
^ X>b - 7i + ^ 2 -J i* i + ^J
27.] INTEGRATION OF RATIONAL FUNCTIONS. 45
27.] The following integrals might be determined by one or
other of the preceding methods : but the process of integration
by parts leads to a result more convenient, and better suited
for finding the definite integral.
dx
Integration of 2 2w .
In the formula, / u dv = uv I v du
let u = O 2 + 2 )-" dv = dx
2n x dx
r di
I ( npa I
/ I vU f~
dx
+
dx _ 2 C dx
~ a
/*
V (
I_|_ a 2)n + l (0 + *) ' V(^ 2 + <Z 2 ) n
for w write n 1,
?#
C dx x 1 2n3 C dx
i _ __ _ _ _ ___ _i ___ I __ (55)
J(a> 2 + 2 )' 1 2(w-l) 2 (a? 2 + 2 )"- 1 " 1 " 2 2w-2J(^ 2 + a 2 )"- 1
Now, as the integral in the last term of the right-hand member
of the equation is of the same form as the original integral, but
has the index of its denominator less by unity, so may the same
process be repeated successively until finally n = 1, in which
case the formula fails to give a determinate result : but the
r $ x
integral becomes / - ., and we have, see Art. 14,
J x -\- tt
J x
.^itan- 1 -. (56)
i* a a
The method is known by the name of "Integration by Suc-
cessive Reduction."
4(1 INTEGRATION OF RATIONAL FUNCTIONS. [28.
Ex. 1. Integration of = 5-5.
(a? + a 2 ) 3
/* dx _ x 3 [ dx
J(**+a*) 8 ~ 4a 2 (# 2 + fl 2 ) 2 + 42./( < r J + a 2 ) 2
3 r x \ C dx
h 40 2 I2a 2 (,r 2 +a 2 ) + 2flVs* + a
X 3 3C 3 3C
~~i) + 80^ tan
28.1 Integration of ^
^
A process exactly parallel to that of the last article gives the
following formula,
/" dx _ x 1 2ra 3 T c?<r
J ( 2 -^ 2 )" = 2 (w -1 a 2 a 2 ^P 2 "- 1 + a 2 2w 2J (a?-a? n - 1 ' ^
1) a 2 (a 2
and by successive reduction the last integral becomes
dx
/f . fi v
the formula failing to give a definite result when n=l.
, r dx x \ C dx
> '
29.] Integration of
/x m dx C x dx m _ 1
(,r 2 + a 2 )" = J (a* + a*) X
And in the formula juetv = uv I v du
let dv = - - u = x m
v = - T -- - - z - z- , du = (m I) x m ~ z dx.
2 "- 1 '
2(n
x m dx x m ~ l m
I f x m - 2 dx
^2j (a? + a z ) m ~ 1 ' (
By which means the integration of the original function is
30.] INTEGRATION OF RATIONAL FUNCTIONS. 47
made to depend on that of another integral of precisely the
same form, but whose numerator and denominator are of lower
dimensions ; and thus, by successive and similar reductions, the
integral will be brought either into a fundamental form or to
that of Art. 27 ; the formula, it will be observed, failing when
n = 1.
//via fj y*
3-^ : here m = 2, n = 4.
/x 2 dx x 1 f dx
(x 2 + a 2 ) 4 6 (x 2 + a 2 ) 3 6 J (# 2 + a 2 ) 3 '
and the latter integral has been determined in Art. 27, so that
it is unnecessary to repeat it.
C 7^ //'X*
Ex.2. / , , 9 : herem = 3, n = 2.
J (x^ + a^y
p 3 dx x 2 f x dx
30.] Integration of ( ^_^ yi -
A process exactly parallel to that of the last article gives
x m dx _ x m ~ l m l C x m ~ 2 dx
a 2 -x z ) n - l = 2(/i-l)( 2 -^ 2 )"- 1 ~ 2n-2J(a 2 -x 2 ) n - 1 ' (
By which means the indices in both numerator and denomi-
nator are diminished, though the form is unchanged : and by a
similar reduction we shall arrive at an integral either of a fun-
damental form, or of the equation (57) ; the formula fails
when n = 1.
//w5 /7/y>
7-3 - 3-3 : here m = 5, n = 2 : therefore
(a 2 a? 2 ) 2
x 3 dx
C x 5 dx x 4 * f x 3
J (a 2 -* 22 == 2a 2 -^ ~ J a 2
48 INTEGRATION OF [31.
SECTION 3. Integration of Irrational Algebraical Functions.
fi-f*
31.] Integration of ^.
Since d.sin- 1 - ,
dx . , x ,f-i\
= sm- 1 -. (61)
x dx
And again, since d . cos - =
= cos- 1 -. (62)
It would appear therefore from (61) and (62) that
, x , x f dx f dx
sm- 1 - + cos- 1 - = /- i - /- ; = 0, (63)
a a Ja?a?z J IPvF*
which result is untrue, because sin- 1 - + cos- 1 - = - : we must
a a 2
therefore have recourse to the accurate process of definite inte-
gration ; and let us suppose that we integrate between the
limits and x,
f X dx = [sin- 1 -T= sin- 1 - , (64)
Jo (2_ #2)4 L J a
1 CLQC 00 \ , CO 7T
-/ = cos- 1 - ^cos- 1 ---; (65)
Jo (ffi x i\2 L JQ a &
therefore by addition
sin- 1 - + 008-!-=^, (66)
ft ft O
which is a correct result; (61) and (62) therefore are not simul-
taneously true in the forms of indefinite integrals.
-7
32.] Integration of
(2ax-
. j , x dx
Since a.versm- 1 - =
a ff)
^ ' 1 & //w\
= versm- 1 -. (67)
v> V 2\5 Ct
35-] IRRATIONAL ALGEBRAIC FUNCTIONS. 49
Or thus,
d(ax) .ax ,x
- = cos" 1 - = versm- 1 -
*
dx C d(a
- = /
{2ax-x?}* J {a 2 -(a-
f/y
33.1 The integration of - - .
x(x*-arf
d(-\
r dx r dx _ i r \ x >
J x(x z -a 2 )* * 2 / 2 \*~ "J / 2
' \ X*> \ *
= - cos- 1 -, by reason of equation (62),
CL X
= - sec- 1 - . (68)
a a
34.] The integration of -
Let a 2 + x 2 = z z , . . x dx = z dz
dx _ dz dx+dz
Z " X X+Z '
dx Cdx Cdx 4- dz
[dx r
= J^ = J
x + z
log(a? + z)
= log {x + (a 2 + #2)*}. (69)
/j
= log {x + (x z a 2 )*} .
(70)
dx dx
35.] Integration of , and of -
r
- ;
by reason of (70).
PRICE, VOL. II. H
50 INTEGRATION OF [36.
Again, ,
r J /* j i f d \
fj -ft //i' \ 1" >
' ' ' f / . / J. J t*/
y> \ (i~ -- y*^* ** / /7* \ * / /T
(72,
36.] Now the relations existing between some of these last
integrals deserve consideration ; for taking the definite integral
of (69) between the following limits, we have
(73)
(a 2
but the left-hand member may be put under the following form,
whereby its integral is determined by equation (61) : viz.
i r x d(x*f^l) _i_
>/IIlJo { a a_( <r/v /"^i)2}i
and that (73) and (74) are identical may thus be shewn :
sm
,- t .
(74)
T 4
Let : sin" 1
x
^-1
= 2,
X
e- z e z
-2 '
replacing sin(2r\/ 1) in terms of its exponential value as
given by equation (30), Art. 58, of Vol. Ij
x + (a 2 +
z =
sm
,
- 1
=log
(75)
37-] IRRATIONAL ALGEBRAIC FUNCTIONS. 51
which shews that the two results though different in form are
rx dg,
equal in value. Similarly might / -^ - - 2 be put in the form
Jo Q> +<#
, and be integrated according to Article
1 x
15. and the result shewn to be equal to - tan -1 - .
a a
- / - - - .
^l Jo a?-(xV-
37.] Integration of
(a + bx +
First let c be positive; then
-i-f
J^J
by reason of equation (70).
Again, let c be negative; then
f dx = J_ r
J (a + bx cx 2 Y ^~ c ^
i b \ 2 "J
~3^^ J
by reason of equation (61).
Ex.1, f - *? _
7
/rfa? /*
/y^a i O ^ _1_ 9A2 J
1 . C
= -p sin- 1 4 - - , (77)
= log {^ + 1
Ex.2. /* _ ^ _ = r rf^ + 1) =gin . 1 /
J -- J -- 2 * ^
H 2
52 INTEGRATION OF [38.
38.] Integration of - .
x (a + bx + cx^Y
dx f dx
r dx _ f
x
dz i
1> lf * = ~;
(a* 2 + 62 + c)*
whereby the integral is reduced to one or other of the forms
(76) or (77).
_ =
x (1
Ex. 1. f __ ^ _ = f
J x 1-2 x + 3*2* J /l
__ \
.z* 2 a?
= -!-,
l -
39.] The integration of (cPaP^dx, and of (a 2 + a? 2 ) 2 dx.
In these cases we shall employ the method of integration of
parts, see Art. 10, Theorem III, viz.
ludv = uv Ivdu. (78)
To determine / (a 2 a? 2 )* dx.
Let u (a 2 a? 2 )* dv = dx,
xdx
du -
Substituting which in the form (78),
xdx
du - v x.
(a 2
Then, adding and subtracting a 2 in the numerator of the last
quantity, and writing the fraction in two parts, and cancelling
40.] IRRATIONAL ALGEBRAIC FUNCTIONS. 53
in the second part (a 2 x 2 )^, which occurs in both numerator
and denominator, we have
(a 2 -
= x(a 2 -x 2 )* + a 2 f dx l - f(a 2 -
J (a 2 a? 2 ) 5 J
= x (a 2 a? 2 )* + a 2 sin- 1 - /(a 2 - tf 2 )* dx,
CL %/
... fa-*?*, = '<"-*>' + %ma-^. (79)
J A A CL
To determine /(fl*
Let u = (a 2 + x 2 )^ dv = dx
xdx
du = - v = x.
Then, following a process similar to that of the last integral,
- [J
J 2
r a*
J
dx
by reason of equation (69) ;
2 . (80)
Similarly it may be shewn that
o*)*}. (81)
40.] Integration of (2 ax x 2 )* dx, and of
= f{a z -(x-a
54 INTEGRATION OF [41*
which latter integral, if x a = z, becomes / (a 2 Z 2 }*dz, and
is therefore of the form (79), and we have
/, L . (x a) (Zax tf 2 )* a 2 . .x a
(Zax x*ydx = + TT sin- 1 - . (82)
A An
Again,
= J(z z -arfdz, if z - x + a;
-i- , . Q ,
2 + (z*-a z r }, by equation (81) ;
a 2 , i,
}. (83)
dx
41.] Integration of -
/_J_ =/.
dx
(84)
Hence * ' dx
f dx J_ f
* (a + bx + cx*)% c% J
{(-+B)
2c' 4c 2
Let therefore # + = , . . dx = dz
4<ac b 2
- = A 2 ; whence we have
4c 2
42.] IRRATIONAL ALGEBRAIC FUNCTIONS. 55
/ - = j ; and by equation (84)
1 1 z
(85)
(4<ac b 2 )
42.] Examples illustrative of the preceding methods.
-- f (m + nx) dx
JiiX. 1 . / -- -
J (a + bx + ca?^
i _ n &\ f
% c 'J
n, 9s i 2cm nb C dx
= -(a + bx + cx^ + - - /
2c J
c
the latter term of which has been integrated in Art. 37.
Ex . 2 . I * = \l ^
.If *_ if;r ,_
2 / * '
1 -1JL
= 2 S1 a 2
1 x 2
= sin- 1 .
2 a 2
Ex.3. - t a-(a-x}
C adx C(ax)dx
a versin- 1 (2ax
a
"O / / ** ~I~ rt/ \ _7 / \^ "I *^/ ^**^
HiX. 4. / I J ax = I
/a dx f x dx
{ + /
/rt2 _^ /y>2\5 J i ft"
\U ^^ tA/ J \U
INTEGRATION BY RATIONALIZATION. [43.
Ex
dx r dx
f dx r
. 5. / - = /
J ~.2/2 i ~,2\$ J
SECTION 4. Integration of Irrational Functions by
Rationalization.
43.] Many infinitesimal element -functions involving irra-
tional quantities may by a judicious substitution be transformed
into equivalent integral and rational functions, and thus in-
tegrated by the methods which have been investigated in the
first two sections of the present chapter ; the process of such
transformation is called Rationalization, and we proceed to in-
quire into the conditions requisite for its application.
p
To find the integral of x m (a-\-bx n }^dx, where m, n, p, q are
constants, integral or fractional, positive or negative.
(
dx = - - 1- (* ) n dz;
nb"
r ? o r ^t 1 -!
.-. jx m (a+bx n }dx = L iT I z v+9- 1 (z a) n dz. (86)
nb n
m + l . m+l i
If therefore - - is an integer, (z q a) n is of a rational
IV
, -1 fn+ l_]
form : and if - -- 1 is positive, (z* a) n may be ex-
ft
panded by the Binomial theorem, and each term of it having
been multiplied by z p+ i~ 1 may be integrated by means of
Art. 11. And if --- 1 is negative, the integration may be
/
accomplished by means of Section 2, and chiefly by the Reduc-
tion-formula of Arts. 27-30.
45-] INTEGRATION BY RATIONALIZATION. 57
44.] Again, as the elemental expression whose integral was
investigated in the last article may be written as follows,
,
mH --
np p
x i (b + ax~ n ydx, (87)
and as the form is the same as that of equation (86), it follows
that by substituting b + ax _ n _ ^
fjYl [ 1 ifl
the result will be rational if -- 1- - is an integer : and we
n q
shall be able to integrate according to known methods.
Hence we may by means of rationalization determine
*x m (a + bx n ydx,
(1) when - - is an integer, by substituting a + bx n = z q ,
(2) when - - + - is an integer, by substituting b + ax~ n z q .
ft G
45.] Examples of the two preceding articles.
Ex. 1 . / x (a + bxy dx.
p 3 ra+l
In this case m=l, n 1, - = - ; . . .= 2, and is integral.
Let a + bx =
</
z z a
' --T-
Zzdz
dx =
.-. \x(a + bx)^dx = yg / ( z ~
2s" **7 5 -v
r z a z i
= Piy- "5-;
_ 2 (a + bx}% fa + dx _ ")
~P~ '17 5j-
* For other methods of Rationalization, and indeed for a complete collec-
tion of integrals of all kinds, let me refer the reader to " Sammlung von
Integraltafeln," von Ferdinand Minding; Berlin, 1849.
PRICE, VOL. II. I
58 INTEGRATION BY RATIONALIZATION. [45.
Ex.2. /"-**L
J In*
( 2
In this case m = 3, n = 2, = ^; .'. - =2, and is
q n
integral.
T 10^ ft 2 i_ /v>2 ^_ <92 nn f ipt ^^ fi&\*
XJCL C* ~p ct/ <fr ** ^ ^'t i* y ^
zcfo
dx =
/y? dx f,
- - = (zz-tf
(a z + x?)* J
= ~ a *
. > 2 ^
Ex. 3.
77 5 /7i t 1 71
In this example w=2. n=2, -= s j .'. - 1-- = 1,
g ! n q
an integer.
p*&
/dx
, a 2
2 x f
a? 3 '-
a 2 a
Let -5 + 1 = z 2 , .-. x =
azdz
//*i2 /"/ /y> / /7 -5*
it/ t*c6 J. l ll ~
( fj ,\ i*2\U tt %/ 6
47-1 INTEGRATION BY RATIONALIZATION. 59
P
. C ia+x\<* fa-}-x\ s ~) .
46. | Integration of R < x n , 1 , I I . J- ax.
(_ ' \c + ex' \c + ex' )
where R is the symbol of a rational function.
Let I be the least common multiple of the denominators
of the fractional indices ; and let us assume
a + bx cz l a
~/ /y>
& W 7 7 }
c + ex b ez l
l(bc ae)z 1 - 1 ,
dx = -^TJ -^ dz.
(b ez l Y
p r
(a + bx\ v - <a + bx\~ s -
Also - = zi - = z s ,
^ c + ex * v c + ex '
all of which are rational ; and therefore
- d
= faiWdz, (88)
where RI denotes a rational function of z.
47.] A particular case is when a = e = 0, and b = c ; in which
case the integral becomes
/- - T
[jj/yi/i vt(J *y> L // <yi /'jsQ^
tv ^ ot/ *' J &/ r f t'f j I VVJ
and we must assume as above, if I be the least common mul-
tiple of the denominators q, s,
IT ~"~ 2
and then integrate.
Ex.1. f^^d,r.
Let x = 2 6 , .-. flfo? = Gz 5 ^, and
C\ x$ C\
IL^Ldx = 6 /i-
*/ 1 _ tV * J 1
-z* .,
dz
Other methods of rationalizing irrational functions by means
of substitution must be left to the ingenuity of the student.
I 2
60 INTEGRATION OF [48.
In many cases however when the irrational term is of one
or other of the following forms we may rationalize the expres-
sion by the substitutions indicated ; if there be involved
let x = y ~ a
x = b
SECTION 5. Integration of Irrational Functions by successive
Reduction.
48.] Our object in the present Section is, by means of Inte-
gration by parts, to make the integrals of certain irrational
functions depend on those of similar forms and lower indices,
and thereby to reduce them to forms whose integrals are funda-
mental or have been already determined.
A careful examination of the process of reduction of the ra-
tional forms which were integrated by this method in Articles
2730, and of one or two of the following formulae, will give a
clearer insight into the method than any general remarks and
rules, and therefore I proceed at once to examples.
n^
49.1 Integration of
In all cases we use the same form, viz. ludv = uv vdu.
x n dx C xdx
Now - - = /-
J
xdx
Let u = a?*" 1 , dv = -
du = (n-\}x n - z dx, v = (a 2 -^ 2 )*.
i r
x n ~ l (a z x*Y+(n 1) /a? n - 2 (a 2 x*y dx
f x
-(n-1) -
J
x n dx
5 I ] IKKATIONAL FUNCTIONS BY REDUCTION. 61
~ 2 dx
f x n dx . f x n
.-. n - = x n ~ l (a 2 x 2 ) 1 * + (n 1) 2 /
J (a 2 x 2 )* J (a 2
C x n dx x n - l (a 2 x 2 }^ n l 9 fae n -*dx
- -= -- - + - a 2 - -. (90)
J (a 2 a*)* n n J (a 2 -a?^
By means of which the integral will at last depend on,
if n be odd, f XdX x = - (a 2 -* 2 )* :
J (a 2 -^ 2 )*
f u C dx . x
it n be even, / = sm" 1 .
J ( a 2_,p2)i a
The formula, it will be observed, is always applicable when
n is odd ; but if n is even, ultimately, when n = 0, it becomes
infinite, and fails to give a determinate result.
, f x*dx a?(a?'x z ) 2a 2 f xdx
x * * / - 1 = -- Q - " "I o~ /
J 2_, r 24 6 J a 2_,p2
(2_, r 2)4
^fl 2 -^* 2 2
Ex. 2.
50.] Similarly may it be shewn that
/* a? w <te a?- 1
I _ _
^ 2 *
o I ~ MX
+ /7 1i ( 91 )
and the integral will at last depend on,
if n be odd, f Xdx = (a 2 + * 2 )* ;
J ( 2 + ^)*
/-
-- = log [x -\- (a 2 -f x 2 )^ } .
(*+*)*
nflH tt 9 1
51.1 Integration of
/x n dx C(ax)dx C x n ~ l
--=- p -- i -x n - l + a-
(2a?-^ 2 )* J (2ax-x 2 y* J (2ax
To integrate the first part; let
(a x)dx
u = % n - 1 dv =
(2aa? **)*
du = (n \)x n ~ z dx v = (2ax
62 INTEGRATION OF [52.
J (Vftri 2 ^
\ <W (<<*- * )
r n
C x n
/ -
J
,,
~ l dx
C x n ~ l dx
+ a
J (Zax-
r y,nlff r
= -x n - l (2ax-^ + 2a(n-l) - ^~
J (2ax-x^
, /" x n dx C x n ~ l dx
-(n-l) - + a - ;
J (2ax-x*y* J (2ax-x*y*
.-. nf xndx =-g
J ? ? 2 *
(2 aa? a? 2 )* J (2 a^p - a? 8 )*
/vn/?v> v n ~-l(9,r/r> r 2 ^ 2w 1 /* -r w - 1 //r
c6 tvc/' ct I (W 1*06 ^^ t6 ) <w /e- ^^ A I Uf C*tC /rvrfcv
- H / -. (92)
'2.r a? 2 )* w n J (2ax
By which process the integral will at last depend on
dx , x
- = versm" 1 -
a
Ex. 1. f X ' d
J -
)* 3 /* xdx
I f|
j (2 ax x 2 )^ -\- a versin" 1 - I
a?
versm
52.] Integration of
By a process exactly similar to that of the last article it may
be proved that
f x n dx _x n - l (2ax + x 2 )* 2n l f x n ~ l dx
I - -- . -- d I - _ (9o)
J (2ax+xrf n . n J (2ax + a?)*
54-] IRRATIONAL FUNCTIONS BY REDUCTION. 63
so that ultimately the integral depends on, see Art. 35,
/
X
53.] Integration of (a 1 x^Y dx , where n is odd.
On comparing /(a 3 x*y* dx with ludv, let
u = (a 2 # 2 )s dv = dx
du = nx(a i x^~ l dx v = x
. - . J(a? - x 2 )? d.v = x (a 2 x^ + n ((a? x^~ l x* dx
= x (a 2 # 2 )* + n I (a? a?)* ~ l { 2 - (a 2 - x 2 ) } dx
= x ax + no
. _-
n+l n+l
f(a?z 2 y*~ l dx,
2 f( a *-a?fi- l dx. (94)
By which means the process of reduction may be continued
until n= I, when the formula becomes infinite, and there-
fore fails to give the correct expression ; but in which case
we have ,, ,
l dx . . x
I - ; = sm- 1 - .
J a?x*'* a
Ex.1.
Ex
. 2. f (<*-
. ,
54.] Similarly may it be shewn that
((a^x^dx = ^ (g2 + f )S --^-a*[(a* + x^dx, (95)
J n + L n 1 J
64 INTEGRATION OF [55.
and the integral finally depends on
x
55. 1 Integration of - - , where n is odd.
/dx r
- - with the typical form / u dv,
(a? a?)* J
/
let u = (a 2 .r 2 )" 3 dv = dx
nxdx
du - v = x;
dx x C x z dx
a 2 a? 2 a 2
x C dx C t
= + n j 1 - W 2 J
f dx \ x
1
for n write n 2,
dx 1 x n 3 1
so that finally, when n 3,
r dx x
/n v* if* ty w
UtJu X c6
-^- | = 3-^2 -^- i +
/7-y>
56.] Integration of
x n (d
2
57-] IRRATIONAL FUNCTIONS BY REDUCTION. 65
xdx
Let dv = - u =
v = (a 2
-(rc+1)/-^
X
n+1
dx
*
for w write w 2, and divide by (w 1) a 2 ,
f dx (a 2 -^ -2 1 r dx
J x n (a?a?)% (n \}a^x n - 1 n \ a*J x n-2( a z_ a z^'
Finally, when n is even, the formula is applicable, and we have
dx (a 2
and when n is odd, the last integral is, see Art. 35, equa-
tion (72), - , 9 .i
C dx 1. a ( 2 # 2 )*
I ; i = ~ 1S -
57.] Integration of - -.
(a + to)*
/# n <& /*
- with udv,
* J
C
' 7
J
let
(a + to)*
u =. x n dv = (
3
<?M = nx n ~ l dx v = -
A/
x n dx
~ l dx 3n C x n dx
2b 36 ^(a + to)*
PRICE, VOL. II. K
66 INTEGRATION OP [58-
/ 3n\ r x n dx
"
x n dx 3x n a + bx* 3n ax n - l dx
'
afx n - l d
*-'(fl + te)
Which formula is applicable for all values of w; and at last,
when w = 0, ,
ax 3
2
_, .. C x 2 dx 3 x 2 (a + bxY 6 a C xdx
x< / == G~A "A /
3 a 3
f/"Q'y2 Q/y'y ^//y2
I <-*(// tjl.lt.fj (W / It
. jj _._ j,^ . J __ I
58.] Integration of -
(a + bx + i
f x n dx 1 /*(
(2cx+b)d* xn _^ b r x-*dx .
a + bx + cx 2 )* 2c >> (a + bx + cx 2 )*'
f (2cx + b)dx ... C
On comparing / - x n ~ l with ludv,
J (a + bx + cx 2 )* J
(2 ex + b) dx
let dv - ^^ u = x n ~ l
du = (n l)x n - 2 dx-,
v 2
* + Vd* M _
C(a
-2(n I)/
^
4-bx + ex 2 ) x n ~ 2 dx
/* T^n 2 ff r
-2(n-l)a -
J (a + bx + cx 2 )*
, . , /* x n ~ l dx . . C x n dx
-2(n-l)b - -- 2(n-l)c -
J (a -f bx + ex 2 )* J (a + bx + ex 2 )*
58.] IRRATIONAL FUNCTIONS BY REDUCTION. 67
Substituting which in (99) and adding and reducing,
/x n dx x n ~ l (a + bx + cx 2 )^ n 1 a C x n ~
' n i A/VI i //yi2\u n c n c J / fj _i_ /i/i
It*' ~f~ C/c/ ~f" \sdU I \ ~l "
~ 2 dx
(100)
so that the last integrals become
xdx (a + bx + cx 2 )* b C dx
b C dx
5- / - ; and
2cJ 2 *
r fty,
I - -, which has been integrated in Art. 37.
J (a + bx+cx 2 )*
Ex.1, f - ^-^ - ; here n = 2, a = 2, b= -2, c = l.
J (2-2x + x 2 )*
/x 2 dx x IC . i /*
- r = _(2-2# + #2)*_ /
2- 2 ^ 2 / -
3
but , = (8-8, + ^)* + / - * - -,
J (2-2x+x 2 )* J (2-2x + x 2 )*
r dx r d(x-i)
/ - - = /
/ (2-2a?+^ 2 )* ^{(^-l)
,
and
K JJ
68 INTEGRATION OF [59.
CHAPTER III.
INTEGRATION OF LOGARITHMIC AND CIRCULAR FUNCTIONS.
SECTION 1. Integration of Exponential and
Logarithmic Functions.
59.] Integration of a x dx } and of e mx dx.
Since d. a x = \og e a.a x dx,
.-. llog e a.a x da? = a x
1
'logea"^
= e*. (2)
Also since d.e mx = me mx dx,
.'. le mx dx = . (3)
J m
60.] Integration of x n e ax dx, where n is a positive integer.
On comparing lx n e ax dx } with udv,
Let u =. oc n dv = e ax dx
du = nx n ~ l dx v = :
a
.- . lx n e ax dx = lx n - l e ax dx. (4)
J a aJ
By which formula the integral will ultimately become
a
/a? 3 e ax 3 C
v*e ax dx / x^e^dx
a aJ
T S p ax Q r T 2 p ax O /*
I I o.r
a a \ a aJ
-^ _J_ J gOX I
a a 2 a 2 \ a a z j
a a a
62.] EXPONENTIAL AND LOGARITHMIC FUNCTIONS. 69
e ax dx
61.] Integration of
cv
/e ax dx C
with udv,
x J
dx
let u = e ax dv =
x n
du = ae ax dx v
Ce ax dx e ax a Ce^dx
i _ ___ , i i 1^1
* * I y,n [m 1^ X n ~^~ 71 1 J X n ~^~ '
By which formula the integral finally becomes
X
which does not admit of integration in finite terms, but may
be expressed in a series : for since
ax a 2 x 2 a 3 x*
e ax _ 1 i --- ( ___ | ___ i
1 1.2 + 1.2.3^
(6)
Ce x dx e x Ce x dx
Ex.1. / g- = +/
/ 30 Sv v vu
X
62.] The preceding are the integrals of the simpler expo-
nential functions ; other combinations however often admit of
being reduced to algebraic forms by means of substitution, and
thereby of being integrated by the methods of the last Chapter ;
of these some examples are subjoined.
r 2# ~\ r x x
Ex. 1. / ^ ~ dx - I ~ e dx
/%x ~\ r x x
^i^ = J^T^
Ex. 2. le^e x dx = e eX .
- /* e x x dx C C e x dx e x dx
/ 71 ^2 = / 1 T~T
i_U
1+a?'
70 EXPONENTIAL AND LOGARITHMIC FUNCTIONS. [63.
63.] Integration of x m (\Qgx) n dx.
On comparing (log x) n x m dx with udv,
let u = (log^)" dv x m dx
dx a?"* +1
du = n(losx) n - 1 v == - = :
x m + 1'
c (]c\(f f\ n v> m +]- n r
.-. (\ogx)x m dx = { 6 ; ; --- -^ (logx) n - l x m dx-, (7)
J v m + 1 m + lJ v
by means of which the last integral becomes
/^m+l
v m dx = - -.
m + l
Ex. 1. fx*(logx) z dx = ( lo gjf) 2 * 5 _ ?j"log^.^^
a? 6 (log <z?) 2 2 r r 5 log a? a? 6 -*
~~5~ ~5\ 5~ ~6J
Ex. 2. / log x.dx = # log x x.
64.] Examples of integration of various logarithmic functions.
/dx C
log ^ = /log2?.
3u J
_ (log a?)
2
fx\os.xdx C
Ex.3. / - 2 - - = /
^ J
a 2 dx C xdx
= (a 2 + x^ log g + a log
66.] INTEGRATION OF CIRCULAR FUNCTIONS. 71
SECTION 2. Integration of Circular Functions.
65.] Integration of the fundamental circular functions.
Since d. cos mx = m sin mx dx,
cosmx
!
smmxdx =
m
sin x dx = cos x. (9)
Again, since d. sin mx = mcosmxdx,
/sin mx
(10)
(11)
(12)
(13)
(14)
(15)
(16)
m
.. cosxdx = sin a?.
Again, since d . tan mx = m (sec mx) 2 dx,
r 1
.-. l(secmx) 2 dx = tan mx ;
J m
.-. (secx) 2 dx = 1 {1 + (tan x) 2 } dx = tan#.
Again, since d.cotmx = m (cosecmx) 2 dx,
/ N2 ^ f dx cot mx
(coscc mx) ax / . . ,
J (smmx) 2 m
/r dx
J (sin x)
r 1
Similarly, / sec mx tan mxdx = sec mx,
r 1
/ cosec mx cot mx dx = cosecm,r. (17)
J m
66.] And all the above formulae are of course true when for
x any function of x, say f(x), is substituted, provided that dx
is replaced by /'(#) dx. Thus
sin(mx + n)dx = /sin (mx + ri) d(mx + ri)
1
= cos(mx
m
I sin (# 3 ) x 2 dx = ~ / sin (x 3 ) d . x 3 t
cos (mx 2 + nx + p) (2mx + n) dx
= sin(w# 2 + nx+p).
72 INTEGRATION OF CIRCULAR FUNCTIONS. [67.
67.] Integration of other circular functions by means of
transformation into the fundamental forms.
/Csmxdx Cd.
tana'aa? =/- -= /
J COS X J C
COS X
cos a?
= log cos x "= log sec x. (18)
/f cos a? dx Cd . sn
cot a? a? = / = - = / :
J sin a? J sin x
= log sin a?. (19)
/dx Tsin x dx C d . cos a?
sin a? ~ J (sin a?) 2 ~ J 1 (cos a?) 2
= - log r , by equation (16), Art. 15,
<w A ^ COS 30
, /I cos a? \* , a?
= log 1= ) = log tan - . (20)
VI + cos x> 2
/rf!a? Tcos xdx _ C d . sin a?
cosa? ~J (cos a-) 2 ~J 1 (sin a?) 5
sin-
x
= log tan (I + 1) (21)
/c?a? /* (sin a
sin a 1 cos a? J si
sin a?) 2 + (cos a?) 2 .
dx
sin a- cos x
= / (tan x + cot a 1 ) c?a>
= log cos x + log sin x
= log tan x. (22)
/da 1 T (sin a?) 2 + (cos a 1 ) 2 ,
(sin a?) 2 (cos a?) 2 ~~ J (sin a-) 2 (cos a?) 2
= / { (sec a?) 2 + (cosec a 1 ) 2 } dx
= tana? cot a?. (23)
/ (tan a?) 2 da? = / {(sec a?) 2 l}rfa?
= tana? a?. (24)
/ (cot a?) 2 dx I { (cosec a?) 2 1 } c?a?
= cot x x. (25)
67.] INTEGRATION OF CIRCULAR FUNCTIONS. 73
r dx f dx
J a + b cos x J
x\* i . xy~\
S 2) -(* m 2> S
(sec^) dor
/ *v
a + b + (a b) (tan - )
^ '
2
(a) let a be greater than b, then the last expression becomes
. tan -
r dx _ __ 2 r
J a + bcosx ~~ a bj a
+ b
(27)
(/3) let a be less than b, then (26) becomes
f d * * L
J a + b cos x b aJb
rtn
d . tan -=
A
L_ log _ , (28)
. x
3m 2.
,(29)
Or, if a be less than 6, we may substitute as follows :
Let a = b cos a.
, dx I C dx
Then
C
/
J a
2bJ a+x ax
COS jj COS -
1 /
= 01- / I tan ~o ^ tan "~o~ T
26smy (,2 2 J
ft W
26 sin a J (.2 2 J
PRICE, VOL. II.
74 INTEGRATION OF CIRCULAR FUNCTIONS. [67.
[" dx 1 r, a + x , a x-\
I Z - = ~, = "1 lg sec o lg S6C O r
J a -f- b cos x b sin at. & J
0. X
1 ]0 cos ^-
> sin o a + x
COS -3-
c^p r </a?
sin.r ~J , . x
a + 2b sin -= cos
j?
(y \
sec^j
(/vt v 2 o?
tan ^ J + 2 6 tan -
T
(-fl)
If be less than 6, we may conveniently substitute b sin a for ,
and after reduction integrate as in the last example.
67.] INTEGRATION OF CIRCULAR FUNCTIONS. 75
C dx f cosxdx
J a -f b tan x ~ J a cos x + b sin x
1 f(bcosxasinx)dx a C sinxdx
bJ b sin x + a cos x bJ bsina
b sin x + a cos x bJ bsinx + acosx
log (6 sin x + a cos a?) a fbsmx + acosx
' sin a? + a cos a?
cos x dx
b 2 J bsmx + a cosx
log (b sin a? + a cos a?) a a 2
log (b sin ,2? + cos x) a
~
ax
(32)
sinmxcosnxdx = ^ /{ sin (m + )? + sin (m n)x}dx
1 * cos (m + w) a? cos(/w
mn
2 1. '
f cos mx cos nx dx = - /{cos (m + w)a7 + cos (m w)#} c?a?
1 r sin (m + n) x sin(m ri)x~}
~ 2(, m + n m n ) '
Similarly may the integrals of the product of three or more
sines and cosines be determined.
dx
, -( 8in l) 2
/ 30 \
(sec -j c?a?
3+(tan|)
2
L 2
76 INTEGRATION OP CIRCULAR FUNCTIONS. [68.
Ex. 2. / sin -5- cos dx ^ / (sin 2a? sin -^- ) dx
J O O AJ V. . O '
Ir cos 2.37 3 2a?~i
3 2x 1
= - cos T cos 2 x.
4 34
Ex. 3. /sin (wa? + a) cos (nx + J3) dx
1 /*/ v _
= - / (sm{(m-j-ft)a > + a + /3} + sm{(m w)a? + a /3} laa?
<j \
21.
cos{(m
m + n m
-w J'
f dx C
. 4. / - - -3 r - - -= /-
J a (cos <z0 2 + o (sm <^) 2 J
dx (sec x) 2 ,
(cos x) 2 + b (sin x) 2 J a + b (tan #) s
. tan x
tan
68.] Integration of (sinx) n dx and o.f (cosa?) w rfa?.
These integrals may be determined by the method of inte-
gration by parts :
r . r
(smx) n dx = / (sin a?)"- 1 sin # eta?;
On comparing the above with the usual typical form,
let u (sin a?)"- 1 dv = sinxdx
v
- cosx(amx) n - l + (n l)(smx) n -
= cos x (sin x) n ~ l + (n 1) /(sin x) n ~ 2 {1 (sin x) 2 } dx
= cos x (sin x) n ~ l + (w 1) / (sin #) n - 2 cfcr (n 1) /(sin #)" dx ;
/. . cos x (sin a?)"" 1 w 1 /* .
[sm a?)" ?a? = L_ _ + / ( sm -p)n-z ^. (35)
By means of which the integral is finally reduced to,
if w be even, I dx x,
if n be odd, / sin x dx = cos a?.
68.] INTEGRATION OF CIRCULAR FUNCTIONS. 77
Again, in a similar manner
/, sin a* (cos x) n ~ l n I/*
fit IV *J
and the last integrals become,
if n be even, \dx x.
if n be odd, / cos x dx sin x.
If n be odd the following method of integration is more
convenient :
Let n = 2m -f 1,
.*. l(sinx) 2m+1 dx = {l (cosx) 2 } m sva.xdx
= \ l m (cos a?) 2 + (cos a?) 4 . . . I d . cos x
J L A . 6 )
Also in a similar manner
(cosx) Zm+l dx = I (cos x) Zm cos x dx
= / (1 (sin x) z } m d . si
rc, m(m I), ) ,
= / < l m (sm a?) 2 H ^=5 - (sui a 1 ) 4 ... V a . si
wi (m
sn x
(38)
r . cos a? (sin a?) 3 3 T
Ex.1. /(sma?) 4 d# = - ^^ -+~-*nmi*)*df
j * QJ
cos x (sin a?) 3 3 f cos a? sin a- x
+
f cos # sin a? a?")
nr - + 2}
4 "4
cos x (sin a?) 3 3 sin a? cos x 3x
~T~ ~8~~ h T '
/* 5 /"
' J J
= / { 1 2 (sin a 1 ) 2 + (sin a?) 4 } c? . sin x
2. (sin a-) 6
= sm x 5 (sm a?) 3 + - ~~- .
78 INTEGRATION OF CIRCULAR FUNCTIONS. [69.
69.] Integration of (smx) n dz, and of (cos.r) M e?#, in terms
of sines and cosines of multiple arcs.
(sin^r)" and (cos#) n may be expressed in series of terms
involving sines and cosines of multiple arcs by the method of
Art. 59, Vol. I ; but since the general term admits of various
forms according to the form of n, the application of the method
will be better exhibited by means of examples.
Ex. 1. To integrate (sinx) 6 dx.
Employing the same abbreviation as in Art. 59, Vol. I,
let 2 v 1 sin x z :
z
561
z z
2 cos x 12 cos 4# + 30 cos 2x 20 ;
6 cos 4 a? -f 15 cos 2<r 10} ;
v 6sin4# 15 sin 2 x
Ex. 2. n
2 cos x = z + -
z
3 1
2 3 (cos <r) 3 = z* + 3z -\ |- -
z z 3
= 2 cos 3 x + 6 cos x
/(cos a?) 3 dx = ^2 / 1 cos 3 x + 3 cos x \ dx
J A J \,
70.] Integration of _, and of
C dx /*
/ ; - = /
J(smx n J
(smx)
dx
/cos x dx
= cos x
And, integrating by parts,
/cos x dx cos x 1 [ dx
COS T ~~~- - _ I _ .-
(sin#)" (w 1) (sin^p)"- 1 n 1 J(sin.r)"- 2
70.] INTEGRATION OF CIRCULAR FUNCTIONS. 79
f dao cos x n 2 /* rf#
' ' J(sin#) n ~ ~ ( IHsina?)"- 1 + rc 1 ./(sina?) w ~ a '
by means of which the last integral becomes,
/" c?<r cos a?
if w be even. / : - -5 = -- : = cot x,
J (sin xY smx
r dy* w
if n be odd, / - = log tan-, by equation (20), Art. 67.
J S1H OC &
Again, by a similar process
C dx _ sintf n 2 C dx
J(cos^) n ~ (n 1) (cosa?)"- 1 + n l7(cos-) n - 2 '
by means of which the last integral becomes,
f rf# cos x
if ft be even, / - - -= = -: = tan x,
J(cos^) z smx
I (1 t* I IT 1P\
if w be odd, / - = log tan (-7 + ^) by equation (21), Art. 67.
J COS 00 ^rfc <w'
In cases however wherein n is even, the integrals are more
conveniently found in terms of cotangents and tangents : thus,
Let n = 2m,
.-. / : - = /(
J (sin x} n J ^
I { 1 + (cot a?) 2 } m ~ 1 (cosec #) 2 rfa
,v(coto?) 8
.-l) 3-^
Similarly
= / (1 + (tan tf) 2 }- 1 rf . tan a?
- 3 + ... (42)
80 INTEGRATION OF CIRCULAR FUNCTIONS. [7 I .
= / { 1 + (tan a?) 2 } d . tan x
(tan a?) 3
= tan x H --- 5 .
o
71.] Integration of ( sin a?) m (cos a?) "da?.
The value of the above integral can easily be found when
either m or n or both are uneven positive integers; and when
m + n is an even negative integer.
(a) Let m = 2r + 1,
.*. i(sina?) w (cos a?)" da? = /(sina?) 2r+1 (cosa?) n da?
= / { 1 (cos a?) 2 } '' (cos a?) n sin a? da?
= /{I (cosa?) 2 } r (cosa?) n d.cosa?; (43)
of which expression each term after expanding (1 (cos a?) 2 } 7 '
may be integrated immediately.
(/3) Similarly may (sin #) w (cos a?) M dx be integrated, when
n is of the form 2r + l.
(y) Let m + w = 2r,
.'. /(sina7) m (cos^) n c?a? = /(tan#)" l (cos#) w+w cfo?
= / (tan a?) m (sec #) 2r da?
= /(tana?) m {l + (tana?) 2 } r - 1 d.tana?. (44)
Each term of which after expansion is immediately integrable.
Ex. 1 . / (sin a?) 3 (cos a?) 2 dx I (sin a?) 2 { 1 (sin a?) 2 } d . sin x
(sin a?) 3 (sin a?) 5
~3~~ ~~5 '
Ex. 2. / (sin a?) 3 (cos x)*dx = I { 1 (cos a?) 2 } (cos a?) 4 sin a? dx
~ /{( cos #) 4 (cos a?) 6 } d. cos a?
(cos a?) 5 (cos a?) 7
~~ ~ '
72.] INTEGRATION OF CIRCULAR FUNCTIONS. 81
Ex. 3. r (Sm ^! dx = /(tan #) 2 (sec #) 2 dx
J cos <* j
(tana?) 3
~~8 '
72.] When neither of the three above-mentioned conditions
as to m and n is fulfilled we must have recourse to integra-
tion by parts, and proceed as follows :
/ (sin x} m (cos x) n dx = I (sin x) m cos x dx (cos x) n ~ l ;
on comparing which with the typical form udv,
let dv = (sin x) m cos x dx u = (cosa?)' 1 " 1
(sin #)" l+1
v = - - du (n 1) (cos#) n ~ 2 sin,r6?,r;
. . / (sin x) m (cos x) n dx
- 1 n \[ f . n , //IK .
1 -- ^^ /(sm^)' n+2 (cosd7) w - 2 c?a?; (45)
which is an useful form when m is negative and n is positive.
Also similarly
(sin #) m (cos <r) M cfo?
/(
. . . , //IA ,
1 -- - /(cosa7) n+2 (sma7) M - 2 ^; (46)
n-\-\. n + L
which is useful when n is negative and m is positive.
Also the last term of the right-hand member of (46) may be
written in the form
/(
(cos x) n (sin #) m ~ 2 (cos a?) 2 dx
= /
r r
= I (cos x) n (sin x} m ~ z dx I (cos x) n (sin x} m dx ;
substituting which in (46) and reducing, we have
/(sin#) m (cosa?) n cfo?
(COS OC ) (Sill OC ) f KYL ~~~ JLi . . o_? /A ^\
1 / (cos x) n (sin x) m ~ l dx. (47)
Similarly may other formulae be constructed, but the example
PRICE, VOL. II. M
82 INTEGRATION OF CIRCULAR FUNCTIONS. [73.
to be integrated will usually by its form suggest various modi-
fications by which it may be transformed into some known
integral.
/*(COScT)* (COS#) 3 _ /*.
Ex.1. . - '-^dx . - -- 3 (cosx) 2 dx
J (sin x)*- sin a? J
(cos.2 1 ) 3 C sin x cos x x
sm^
r ri<r> r,
Ex
/fj fVt
7--TT = /
(sm x) 5 cos x J
=/
(sm x) 5 cos a? J (tan x) 5
Ktan#) 2 } 2
7T- ./ o?.tan.y
(tan <r) 5
= / { (tan x)~ 5 + 2 (tan ^)~ 3 -j- (tan x)~~ 1 } d. tan x
~4(tan#) 4 ~(tan#) 2+ S tan<z> -
73.] Integration of (tan x) n dx, and of (cot x) n dx.
r ' r
I (tan x} n dx = / (tan ^)' ! ~ 2 (tan xydx
= I (tan #) n - 2 { (sec <r) 2 1 } d#
[ n-2 /^
J v'
( ran xj /.. . , /^o\
= rr l(tanx) n ~ z dx. (48)
/i ~~ A /
Similarly,
/ (cot x) n dx = - 4^ / (cot #) n ~ 2 dx ; (49)
which formulae give definite results for even values of n, but
fail when n is odd ; in which cases however, by equations (18)
and (19), Art. 67, ,
/ tan x dx = log sec x
cot x dx log sin x.
74.] Integration of x n cos x dx.
jx^cosxdx = a? n sinx nlx n ~ l $
l)x n - 2 cosxdx. (50)
75-] INTEGRATION OF CIRCULAR FUNCTIONS. 83
By means of which the last integral becomes
/ cos x dx = sin x, or / sin x doc = cos x.
Similarly may it be shewn that
lx n $mxdx x n cosx + nx n ~ 1 sia.xn(n ~\.')lx n - 2 smxdx. (51)
Ex. 1. x^cosxdx = a? 3 sin ^ + 3 a? 2 cos a? Gxsmx 6cos#.
Similarly may formulae be constructed for determining
/#" sin #<&?, and x n coskxdx. (52)
And hence we may integrate infinitesimal elements of the forms
x n (sin x) m dx, x n (cos x) m dx ; (53)
for if (sin#) m and (cosx) m be expressed in terms of the sines
and cosines of the multiple arcs, by means of Art. 59, Vol. I,
then each term of the integral will be of one of the forms (52),
and may be integrated accordingly.
75.] Integration of e (cos x) n dx, and of e ax (sinx) n dx.
/pax (QQQ w} n n C
[cosx^e^dx =- + - /(cos x) n ~ l sin x e ax dx
_ e ax (cos x} n n ( (cos x) n ~ l sin x e
a a\ a
- /{(cos x) n (n 1) (cos x) n ~ 2 (sin a?) 2 } e ax dx\
) n (n 1) (cos x) n
.'. /(cos x} n e dx
n(n
2
.
( COS *) e dx ' ( 54 )
Similarly may a formula be found for
f ' n ax
J
/C\ * \ rt ft-**
fjClX fif\c* sy* ( ft f*f\Q W I, */ QTVl /V> | J, pllJ?
, . o 7 t' Vy\Jo / I M* l-<wii3 w \^ e* ollJ. oi/ ^
?"* (COS a?) 4 tt^ 1 = ^ , + -; 5 .
4 + a 2 4i + a 2 a
M
84 INTEGRATION OF CIRCULAR FUNCTIONS. [76.
76.] Integration of e^cosnxdx, and of e ax smnxdx.
/cos nx e^ n C .
;os nx e * dx = h / sin nx e ax dx
a aJ
n C$aa.nxe ax n C 7 ")
+ -<- Icwnxe^dx V ;
a (, a aJ j
e'" (a cos nx + n sin nx}
= * , ^2 ' ( 55 >
Similarly - _ e"* (a sin wo? cos wa?
/ sin /z^ c dx g > ("")
These results may also be obtained as follows, by expressing
sin nx and cos nx in terms of their exponential values :
C 1 /*
je^cosnxdx = le ax {e nx " / ~ l + g-^v-ij fa
\ r
o" / 1 i~ ^ / tvci?
^ J
. + n\
a 2 4- w 2
Or thus: Let Si =
S 2 =
1 = le
M<n--/-
a + nV 1
gOd? / - / -
= -T - -(cosw#+ v 1 sinwa?) (a wv 1):
a 2 + w 2
and therefore, equating possible and impossible parts,
S 2 =
78.] INTEGRATION OF CIRCULAR FUNCTIONS. 85
77.] Integration of f(x) sin~ l xdx, f(x)tan- l xdx, &c.
Integrals of these forms must be determined by integration
by parts; the method is best exhibited by examples such as
follow :
/*. . , f xdx .
Ex. 1. Ism- 1 xdx = xsm~ l x
= x sin- 1 x + (1 a? 2 )*.
"R-v- O leiTi~l'*> I 1T1~1 w // cin~l
_LjA. <w. I olil Ob ~ I iolll iv w olll t
,, _ ^dx
Ex
.3. tf ,
J 1 + x*
r _ 1 /" _i ^
~7 an J an ^i+a?
-(tan- 1 a?) 2 .
-.-. / Wi?7 ^ I ttc
Ex. 4. / - = /
J (1 + ^)* ^(1 +
atan" 1 ^ _
(1 + * 2 )* (1 + 2 )1 + ^
c?a?
78.1 Integration of
+ 6 cos x) dx
/c?a? _ f(a
(a + bcosx) n J (a + b cosa7) n+1
/* adx , /* 1
= / r + b I cos a? ax r 7
/* ac?a? , r sin a? , /* 6(sina?) 2 rfa?
= J(a + 6cosa7) M+1+ l( + 6cos^)^ +1 ~ (ro+ V(^
r ^ T6 2
a j (a + 6costf)" +1 ~ (n + 'J (a
,
X
86 INTEGRATION OF FUNCTIONS [79.
but
dx = I : ; -T3 &K
_ , 2 _ 2 __ 2
a V(a4-6cos#)" +2 a
substituting which, we have
b sin .2?
/-o. i\ /* ^ /*
~ 'j (a + bcosx) n+1 J (
+ bcosx) n+1 (a + 6 cos a?)"'
for w write w 2 ; therefore
(a + b cos a-)"- 1 7 J (a + b cos #) ' V (a + b cos )-
(2n 3) a
1)(6 2 a 2 ) (a + 6 cos a?)"- 1
w 2
f (
By which means the integral becomes reduced to
dx
/* c?.r
-aJa + ^cos^"- 2 '
J a
a + bcosx'
the value of which has been determined in Art. 67.
79.] Many of the algebraical functions which have been inte-
grated in Chapter II may by substitution be transformed into
circular functions, and in some cases have their integrals deter-
mined with greater facility ; and by a reverse process many of
the circular functions which have been integrated in the present
Chapter may be transformed into algebraical functions. The
method is best exhibited by the following examples :
1 f dx
' Ja* + x*'
Let x = atanfl, .-. dx = (sec0) 2 e?0;
Ex.
79-] BY MEANS OF SUBSTITUTION. 87
f dx Ca(sec0) 2 d0
'' Ja 2 + x 2 ~ J a 2
(sec
1 a?
= - tan" 1 -.
a a
EK. 2. f.
Let # = ataia 6, .'. dx = a(sec0) 2 d0,
f d * . [*(**WdO _ 1 f .
' J(a* + a*) ~J 2w (sec0) 2 * ~ a 2 "- 1 ^
which last integral is of the form (36), Art. 68, and may be
integrated by the reduction-formula therein given.
Ex. 3.
(a 2 -;
Let x asin0, .-. dx = acos0d0,
/dx fa cos dd
/a 2 -*? 2 )* "
<
"**
= = sin" 1
a
T* f dx
E*.4. j-
Let x = atan0, .'. dx = a($ec0) 2 d0,
f dx f
- - = sec0d0
J (ft2 j_ v&\2 J
/"
-Jl
= /-
sec + tan , , .
sec d0
sec + tan
sec 0) 2 + sec tan
tan + sec
= log (tan + sec 0]
' x
,
,
WC7
88 INTEGKATION OF FUNCTIONS BY SUBSTITUTION. [79.
~RY ^ l(n z r z \* dr
lj\. c>. I \U i* ) l*O/.
\J
Let x = asintf, .'. dx = acosOdO,
r i r
J J
g f sin 6 cos e
= a | - + 2
/* g
J (a*-o
Let x =. a sin 0, .. dx = a cos c?0,
Ex.6. .
2)i
(a* -
which, according as w is odd or even, is by Art. 68, equation
(35), equal to
cos0(sin0) n - 1 n 1 ... 8
cos "-
w w(w-2)
~ n(n 2)(n 4) C(
(w-l)(w-3)...4.2
. . 5^ ; =r COS V ,
n(n 2) (w 4). ..5. 3
or to
cos0(sin<9) n - 5
n( 2)( 4)
\(59)
and replacing in terms of x, the results are identical with
(90) in Art. 49.
8o.] IMPORTANCE OF DEFINITE INTEGRALS. 89
CHAPTER IV.
ON VARIOUS PROPERTIES OF DEFINITE INTEGRATION.
80.] THE last two Chapters contain an account of almost all
the known methods for finding indefinite integrals ; very few
indeed they are, and they may be reduced to two or three
general heads : so that most of the labour consists in trans-
forming given expressions into other and equivalent forms, of
which the integrals are known. Should any one be urgent to
inquire why the known integrals are so few, the reply is easy :
we have no means of expressing them ; our materials fail : it is
not because the Calculus as a system of rules for integrating
and disintegrating (or differentiating) fails, but it is because
the materials, on which it has to operate, fail. A word or two
will shew how this is. When differentiation is performed on a
given function, in most cases it changes the nature of it, and
reduces it from a more complex and transcendental to a more
simple form : thus log x is by differentiation changed into
(ae)~ 1 dx > that is, into an algebraical form; sin" 1 a?, tan" 1 a?, ....
similarly give rise to algebraical expressions : in the reverse
process therefore of integration the simple functions are changed
into more complex ones ; algebraical functions will become
logarithmic and circular. In order then that logarithmic and
circular functions should generally be integrated, there must be
other transcendents higher than they are, and of which they
are the typical infinitesimal-elements : but such functions do
not as yet generally exist ; and until they have been discovered,
studied, and had their values calculated and tabulated for given
values of their variable subjects in the same way as logarithmic
and circular functions have been treated, it is vain to seek for
indefinite integrals of the (at present) highest transcendents.
Many instances of our want of other and higher transcendents
will occur in the sequel.
In most future cases of the application of our Calculus, the
solution of a problem will depend on a definite integral : if the
indefinite integral can be found, the definite integral can be
immediately obtained ; see Art. 5 ; but as all indefinite integra-
PRICE, VOL. II. N
90 THEOREMS OP DEFINITE INTEGRALS. [8 1.
tion cannot be performed, we are obliged to have recourse to
artifices of series, of approximation, and of other kinds, and
from them to infer pregnant properties of the definite integrals.
Certain general Theorems have been already investigated in
Chapter I, Art. 8, and we proceed now to add others ; and herein
to lay the foundation of the most useful applications of the
Integral Calculus, and to point out the direction in which lies
the most hopeful prospect of advancing the boundaries of the
science.
Whenever therefore in the sequel we meet with the expres-
sion " cannot be integrated," let the exact force of it be borne
in mind ; it is not meant that the infinitesimal element-function
to which the expression is applied is not the element of some
finite function, for doubtless such a primary function exists,
and it may be a question of time only when functions will have
been examined with accuracy sufficient to have their values
tabulated and their properties understood : but it is meant that
such an infinitesimal function is not the element of any circular,
logarithmic or algebraical function which has already been the
subject of complete analyzation ; and thus that the integral can-
not be expressed in terms of the ordinary functions or symbols
with which we are familiar. Many instances of this incomplete
state of our science will occur in what follows.
SECTION 1. Further researches into the Theory of
Definite Integrals.
81.] In order to have a clear notion of a definite integral, be
it remembered that the symbol on the left-hand member of the
following equation is only an abridged form of the series of
which the right-hand member consists when the parts into
which x n XQ is divided are infinitesimal, and therefore when
the number of terms is infinite; viz.
/
/
... + (x n # n _i) F'(#-I) ; (1)
in which equation Xi, x%, ... x n -\ are the values of x correspond-
ing to the points of division of x n x$ ; and F'(#) is finite and
continuous, and does not change sign, between the limits
a? n and -r .
8 1.] THEOREMS OF DEFINITE INTEGRALS. 91
Now in (1), #1 XQ, # 2 %\, x n n-\ are quantities of the
same sign ; therefore, by Preliminary Theorem III of Vol. I,
the sum of the series is equal to
(X n XQ) v'{x + 6 (#n tfo) } ', (2)
where 8 is the symbol for some undetermined positive proper
fraction. Hence
/
Jxn
)diB = (x n XQ) v'{x + 0(x n X Q )} ; (3)
and therefore, if F(#) be the indefinite integral of f'(x)dx,
F (x n ) - F (a? ) = (x n X Q ) r' {X Q + 6 (x n X Q )}. (4)
Suppose that the difference between the limits, viz. x n X Q) is
infinitesimal, then, if XQ be finite, (4) becomes
F (d? B )
that is, the sum is reduced to the first term of series (1), and
this evidently ought to be the case.
In continuation of the four Theorems on definite integrals
given in Art. 8, and which the student is recommended to read
again, the following are the simplest examples of substitution :
/
JXn
r xn ,
+> XQ
fx n -a
= I F (x) dx.
p'(x)dx = v'(z)dz. (7)
Hence also in the general case of / /{<(#)} dx; let <f>(x) = y }
so that x = \lr(y), dx = ^'(y)dy, and
Thus also . Xn , rax n +b
I F\ax + b) dx = - / F'(^) dx ;
J&Q *"JaxQ+b
ft*. /*!
f F' (a?) dx = (x n x ) I F'{X O + (x n x )x } dx.
&Q /0
N 2
92 THEOREMS OF DEFINITE INTEGRALS. [8a.
82.] THEOREM V. The superior and inferior limits of a
definite integral will mutually change places by changing the
sign of the integral.
x =
( l 'n
F'
. *o
= -{F(^O) F (#)}
/*t
Y'(X) dx ;
A
rxo /*
.-. / v(x)dx= v'(x)dx. (8)
JiF . fjSm.
THEOREM VI. A definite integral of which x n and x are
the limits is equal to the sum of a series of similar definite
integrals, provided that the extreme limits are the same^nd
that the several intermediate limits are continuously additive.
r-^n
Let / F'(#) dx be the definite integral under consideration ;
Jx
and let x n XQ be divided into n parts, to the several points of
division of which let a?i, x z ,... #_! refer; and let XQ, x, x% } ...x n
be such that y'(x) does not change sign within any two con-
secutive points of division (the correctness of the argument will
not be injured by its changing sign at a point of division) ;
then since ^
(9)
and also since
F
^
/ Y'(X] dx =
Jx
f'(x}dx
/ F v^v ^*<-^ * \P^n) ~"~ *^ \^N. 1 /
Jn-\
therefore by addition
r*i , r*2 r* .
~rf ( -yi\ /T'y* I I T? i '*>\ fi v* I _J_ / TJI / *y\ // /y> ^ IP f *Y* \ _ _- i> / O^A
I Jr ^*y Wt6 T I ^ \*/ ^^ i * T I ^ \*"/ *^ \*w/ ^^ ^ v^O/
= (***' (ae)dx. (11)
A definite integral therefore taken between assigned limits
may be resolved into many others of the same form, if the
82.] THBOKEMS OF DEFINITE INTEGRALS. 93
extreme limits are the same, and the intermediate ones are
continuously additive.
If then be a value of x intermediate to x n and X Q)
fao = f F/ O) dx + 1 *'(#) dx -
Jx J(
Now one of the conditions for calculating a definite integral,
(" x n
such as / F'(#) dx, is, that F'(#) must not change sign between
Jx
the limits ; if however we have to integrate F'(#) dx between the
limits x n and XQ, and F'(#) is such that it changes sign at a
value of x, say at , between these limits, then we may resolve
the integral into two others of the same form, one of which
has for its limits and XQ, and the other x n and : instances
frequently occur in geometrical and mechanical applications.
Also if F'(#) changes sign at many points between x n and XQ
fx n
a similar mode of resolving / F'(#) dx into several other defi-
Jx
nite integrals must be adopted.
Also if be a value of x lying outside of and beyond x n , and
if F'(#) dx does not become infinite or discontinuous or change
sign between X Q and , then, since by (7)
r r* n r(
I F'(#) dx = / F'(#) dx + / *(x) dx,
Jx Jx Jx n
= f p'(a?) dx + f X \'(x) dx ; (13)
A /
and as such a method admits of being extended to any other
values of x outside of x n x , provided that the requisite con-
ditions are satisfied, it follows that the enuntiation of Theorem
VI may be enlarged so as to include all values of x.
Hence also follows a Theorem of great importance ;
THEOREM VII. If ju be an arithmetical mean between x n
and XQ, and if F'(#) have the same values and the same sign at
equal distances from p on either side of it, that is, if F'(JU x)
/>
= F'0/. + a?), then the two definite integrals / F'(#) dx and
/# J*i
?'(x)dx are equal; and from (12) we have
-
/**!, />
/ v\x)dx = 2 v'(x}dx.
JXQ JXQ
(14)
94 THEOREMS OF DEFINITE INTEGRALS. [82.
Wherefore to find the required integral it is necessary to cal-
culate only one of the two equal definite integrals. Thus for
example
/u /*5 r~ ~"|5
cos xdx = 2 cos x dx 2\ sin#
5 Jo L Jo
= 2
Jo
Hence also follows
THEOREM VIII. If n be an arithmetical mean between x n
and XQ, and if v'(x) has the same values, but of different signs,
at equal distances from /x on either side, that is, if F'(JU, x)
/jU, /"*
F'(/Z + x) ; then the definite integrals / F'(#) dx and / F'(#) dx
Jxn Ju,
neutralize each other, and
ix = 0.
Thus, for example,
/ cos x dx = \ sin x\ =0
Jo L Jo
r5 r -il
/ sin x dx cos x\ =0
'-f J i
[ +X 1-22
./ ao
Hence also if R symbolizes, a rational function
/ R{sin#, (cos<r) 2 } cosxdx = 0, (15)
Jo
a Theorem of great importance in subsequent investigations.
Also if for x we substitute x n + x x, then
I i
^ 1 Tji / nn i l_ yi /y>\ fj W ^1 fi\
J-^o
the only effect of the substitution being to reverse the order of
the elements which are in number and value unaltered.
Hence also if XQ = 0,
r x n r x *
I f'(x)dx = / v'(x n x)dx.
Jo Jo
Sometimes also the preceding considerations enable us to
83.] THEOREMS OF DEFINITE INTEGRALS. 95
determine the value of a definite integral, although the indefi-
nite integral may not be previously determined ; thus
rl rl
I ^OOS 2?^ CLtJT ~"~ / ( S1H T*"^ (If*
Jo Jo
because the values of the elements are equal when taken through
the quadrant; therefore
rl 1 /-I
/ (cos x) 2 dx = q / { (cos a?) 2 + (sin x) 2 } dx
Jo Jo
1 /!
= ^ / dx
~ 4'
r x
83.] Hitherto in determining / F'(#) dx, v'(x) has been con-
Jx
sidered finite and continuous for all values of x between x n and
x Q , and the values at the limits have been considered to be finite
also: but if the necessary conditions are not fulfilled, or if the
limiting values are infinite, we are unable to affirm that the defi-
nite integral has a finite value ; and nothing that has been said
enables us to attach any intelligible meaning to it.
r +1 dx
Thus, for example, consider / , in which the quantity
J-i x
(a?)" 1 becomes infinite when x = 0, that is, when x has a particu-
lar value included between + 1 and 1 ; then, since by reason of
equation (12) ^
/ / -\- I
J-l X /.j X Jo X
= 00+00,
the integral assumes an indeterminate form.
Similarly the following integrals take infinite forms,
/ + r- -i +oc
e*dx = \e x \ = e x e~ x = o
30 L J X
f'^dx r. ~\ +x
I logo? = oo + oo = oc
-' <K
That the result of the integrals in (17) is indeterminate may
thus be shewn. Instead of integrating between the limits given
in equation (17), let the superior limit of the former integral be
zero less an infinitesimal, and let the inferior limit of the latter
96 THEOEEMS OP DEFINITE INTEGRALS. [84.
be zero increased by an infinitesimal ; thus suppose i to be an
infinitesimal, and fj. and v to be two finite but undetermined
positive constants ; then
f- = ['*'- = log(- M i)-log(-l) =
J _ i 30 *' 1 *^
_l X X
which is a value absolutely indeterminate.
And if the limits of the definite integral involve infinity,
either positive or negative, we must replace the limits by quan-
tities differing from such infinities by an infinitesimal. Thus
we shall replace as follows :
f **()& = ( lki jf(x)dai. (19)
J 00 J __ L
vt
As to the superior limit being infinite, it is important to
/**
remember that according to the definition of / F'(#) dx given
Jx
in Art. 6, the sum includes F'(<T O ) and excludes F(&n) ; the form
therefore of such definite integrals indicates that there is an
infinitesimal difference between the last value of F'(#) dx and
that corresponding to the superior limit.
84.] Similarly if F'(#) becomes infinite for many values of x,
say %i, x^ ... x n _i, lying between x n and X Q , then if i be the
symbol of an infinitesimal and p-i, v\, /u 2 , v z , ... be positive and
undetermined constants
r
/
Jx
dx
i ^ii
F'# dx
-x\+ v\i
and if the limits are + oo and QO , then
ro: 2 /nji fx H
I P'(O?) dx + ... 4- / p'(ff) dx, (20)
'x+ vi Jx_+v_i
/
J t
dx - T?'(X} dx. (21)
The definite integrals thus deduced may be either finite, in-
finite, or indeterminate, according to the values given to the
84-] THEOREMS OF DEFINITE INTEGRALS. 97
arbitrary constants pt, v, ____ If in this last case all the arbi-
trary constants are replaced by unity, the definite integral takes
a particular value, to which M. Cauchy has given the name of
principal value. Thus the principal values of the integrals given
in equations (20) and (21) are
rx^i
/ F'(#) dx -(- ...
Jx\ + 1
... + f* ?'(x}dx, (22)
dx I F'(#) dx
\ +
dx
i
= T v'(x] dx. (23)
/ --
/ + QO ^/ji
- given in equation (18) is
oc #
log ( -) ; but which = log (1) = 0, when ju = v = 1 ; and is
therefore the principal value.
The preceding remarks also explain such apparent contra-
dictions as are involved in the indefinite integrals of some posi-
tive infinitesimal elements being negative : thus for instance
/ (cot #) 2 dx = I { (cosec a?) 2 1 } dx
"""" "~ COt< X "~~~" uU
which is entirely without meaning, unless the values corre-
sponding to the limits are introduced.
Or thus again, - +Xndaf 1 x
which is a negative expression, though all the infinitesimal ele-
ments of it are positive. But (<zO~ 4 is infinite when x = 0, that
is, when x has a value intermediate to the superior and inferior
limits : we must therefore divide the integral into two parts, viz.
- vi dx 1 1
1
efe? 1 1 1 1
which is equal to + oo when i = 0.
The subject however is too difficult to be pursued further at
PRICE, VOL. II. O
98 EXAMPLES OF DEFINITE INTEGRALS. [85.
this present part of the Treatise, but we shall have to return
to it hereafter, and then we shall exhibit other properties fol-
lowing from the above view of definite integrals which is due to
M. Cauchy.
SECTION 2. Examples of Definite Integrals.
85.] In the first place we shall give some examples of definite
integrals which are deduced immediately from the indefinite
integrals of the preceding Chapters.
/I i- y,n + l -il J
Ex. 1. / xdx = \ = T .
Jo Lw + lJo *-M
Ex. 2. [* e~ x dx =[-6-^1 =1.
Jo L Jo
r x dx ir, ,^-1" TT
Ex. 3. / -s- T: = - tan- 1 = .
Jo a + a? a\_ J ^
r 1 ! x m , r x z x z x* x m -\ l
.4. / ^ - dx =Lp+ + + -;-+... +-
Jo 1 x L 234 mJo
Ex.
o x
111
Ex.5.
Ex. 6. [ a (a i -x*)*dx= r^Ca^
Jo L^
.,
J-oo (x
1 T 77 7T"| 7T
= 6l2 + 2J = 6'
T 00 re-"* (i sin d^ a cos &r)
Ex. 8. / e~ aA cos bxdx =
J
(00
cos a? dx = ;
85.] EXAMPLES OF DEFINITE INTEGRALS. 99
r r i
but / cos x dx = sin x = sin oo ;
Jo Jo
since = 0.
/"* re~ ax ( a sin bx b cos bx}~\ x
Ex. 9. / e~ ax sm bx dx = 2 p
jo L -f- o Jo
r
Hence if = 0, 6 = 1, / sin x dx 1 ;
Jo
but / sin x dx = cos x ;
cos oo + 1 = 1,
cos oo = 0.
x_ _ 2n3 f
(w-l)(l + ^ 2 ) M - 1 + 2n-2J(
_ 2^3 f 00 <to
= 2^2 Jo (l + a?2)n-l
dx
-5. ..5. 3.1
3) (2^ 5). ..5. 3. ITT
Ex.11. --
l dx
f*<x
= nl e- x x n ~ l
Jo
= (-l)(n-2)... 8.2.1
= n(-l)(-2)... 3.2.1. (24)
* See Example 3, Art. no, Vol. I.
o a
100 EXAMPLES OF DEFINITE INTEGKALS. [85.
r xndjc - r <yn ~ i ( i ~ j?a )^ rc i [ xn ~ 2dx i 1
Jo (I_<p2)4 w ft J (!_#*)* Jo
n (i_
If therefore w be even,
f l x n dx (n l)(n-3)...3.l f 1 dx
(w-l)(-3)...3.1r . I 1
sin" 1 ,??
(25)
.(.-2)...4l
(n l)(ro 3)...3
( 2)...4.2 2'
And if w be odd,
x n dx (n l(n 3..A.2r l x dx
r 1
n(n 2)...5.3
(n-l)(n-8)...4.2 T_ ^
(-2)...5.3 L l J J
(!)( 3). ..4.2
( 2)...5.3
(26)
The remark made at the end of Art. 6 is of great importance
in reference to examples such as this and Ex. 5, viz. that the
value of the infinitesimal element corresponding to the superior
limit is excluded, while that corresponding to the inferior limit
is included in the definite integral ; for were this not the case,
x n
as - becomes equal to ao, when x = 1, the integrals
(1-a?*)*
would not satisfy the conditions, which the theory of such sum-
mation requires : but as the limit unity, being the superior
limit in the above examples and that which renders infinite the
infinitesimal element, is not reached, the definite integrals are
correct.
x n
Again, since - - is, for all values of x between and 1,
(1 - a? 2 )*
y>n 1 # n + l
intermediate to -- and - , therefore
(1 # 2 )* (1 # 2 )*
P x n dx C l x n - l dx r l x n+1 dof
I - is intermediate to / - - and / -- ;
^o (1 a? 2 ) 5 J (1 x 2 y* J (1 x 2 Y
86.] EXAMPLES OP DEFINITE INTEGRALS. 101
Hence -- .
is intermediate to
2 2.4...(w 2)
2.4...(ro 4)(n 2) , 2.4...(
an
8)(n
* 2 274...(n 4)(n 2)
_ 2.4. ..(ro 4)( 2)
~ 3.5...(n 3)(n 1)
w I
quantity > ^, < 1 j ; (27)
if therefore w be a very large number we have the following
approximate value of TT,
2.2.4.4.6.6 ......
= 1.8.8.5.5.7 ...... '
a result which was first discovered by Dr.Wallis.
86.] We may often conveniently by means of definite inte-
gration expand a function in powers of its subject variable.
The following instances are most useful :
Ex. 1. Since by the Binomial Theorem
1
JQ J. j- <V JQ
(29)
Ex. 2. Again^ by the Binomial Theorem
_ .^ 1 _ /y>2 _t /y4 . ^6 | /y>8 ___
o -L cv ~1~ Cv w ~T~ cv *
/**
I -
/ i
'O - 1
r , T r ^ 3 ^ 5 ^ 7
L tan J,= L*"T + T"T
g3 y **
= ^ - + - --- ^ + ... (30)
102 EXAMPLES OF DEFINITE INTEGRALS. [86.
Ex. 3. Again, since
1 x* 1.3 1.3.5
- + * f *
C* dx f* r_ a? 2 1.3 1.3.5 ( -) ,
. ' . / - = / \ 1 + -TT + ^ r 8* + K -. ; x 6 + ... > o-r
Jo (1_#8)* ^o I 2.4 2.4.t>
a; 3 1.3# 5 1.3.5 a?*
8m * := * + 2-3 + 274 T + 27476 y + - (31)
Hence, if x 1,
TT 11 1.31 1.3.51
The series however converges too slowly to be of any use for
calculating TT.
Also let x = -,
' 6 2 '
Here again it must be observed that although in (32) the
superior limit is 1, which renders - - infinite, yet the
series is correct: the reason being that the definite integral
does not include the value corresponding to the superior limit.
Ex. 4. In the case where x is greater than 1 in series (30),
that is, when tan -1 x is intermediate to T and ~ , it is better
4 li
to expand as follows :
Since
^ tan- 1 a?
2
r l 1 1 l
= --- (- 55 - = + -= ..
L x 3x 3 5<r 5 7x 7
1
x
7T_1 J_ J_
2 > ~*~ *~>3 R 5 + "
87.] TAYLOE'S SERIES. 103
Let x = I,
7T 111
' 4 =1 -3 + 5-7 + - (35)
87.] The method of definite integration also yields a simple
proof of Taylor's Series^ and one which exhibits the remainder
in the convenient form of a definite integral.
Let T?'(x + h z) be a function of z which does not change
sign, and is finite and continuous, for all values of z between
z = h and z = ; then
i- -i A
(x + h z)dz = F(# + A z)
L . Jo
= p(# + A) *(a?); (36)
and suppose in addition that the several derived-functions of
F'(# + A z) up to the rath do not change sign, and are finite
and continuous for all values of z between z = h and z = 0, and
at the limits ; then by a series of successive integrations by parts
we have
/h |- -i A rh
T/(x + h z)dz \ ZT?'(x + hz) \ + I -$"
Jo >/o
rh 2 s
H- / *""(# + h - z} YY~
and so on for n integrations, until
h h h 2 h 9
*(x + h-z)dz = J(a>) l + !"(x}^ +F"'(*) T - I - 5 + ...
and replacing the left-hand member in terms of its equivalent
from (36), we have
h .n ( .]^_ n, ( h 3
n (z + h-z)^ * . ,,dz. (38)
104 TAYLOR'S SERIES. [87.
Hence it follows that the sum of all the terms after the nth
is expressed by the definite integral
fr + *-* (39)
and we can shew that this is equivalent in value to the expres-
sion (18), Art. 120, Vol. I, viz. to
h n
(40)
1.2.3...W
for the definite integral (39) is equal to a sum of terms each
of which is of the form
z n ~ l dz
where z varies continuously from to h ; and as in all the terms
? n (x-\-h z) is of the same sign, by Preliminary Theorem III,
Vol. I, the sum is equal to the sum of all the factors of which
gn 1 fa
r-T- - is the type multiplied into some mean value of
1.2.3. ..(n 1)
the factors of which v n (x + h z) is the type, as z varies from
to h; but such a mean value is obtained, if 6 be a positive
proper fraction, by writing h for h z; and therefore
z n ~ l C h z n ~ l dz
= '"<*+"'> ora.
and therefore
h h 2
'
Whereas then in this latter form of the remainder there is an
indeterminateness arising from 6 being an undetermined proper
fraction; in the former expression, wherein the remainder is
given in the form of a definite integral, there is no such inde-
terminateness ; but the function may be of a form that does not
admit of indefinite integration, and in that case we are obliged
to have recourse to an approximation towards its actual value,
and thereby the result is perhaps only approximately correct.
88.] MACLAURIN'S SERIES. 105
88.] We may also in a similar manner prove Maclaurin's
development of a function of x, and put the sum of all the
terms after the wth in the form of a definite integral.
Let p'(# z) be a function of z which does not change sign,
and is finite and continuous for all values of z between and x,
then r-x r- -\z=x
i v'(xz)dz = v(xz)
Jo L _b=o
= p(ff) p(0). (42)
And suppose also that all the derived-functions of F'(# z) up
to the nth do not change sign, and are finite and continuous
for all values of z between and x, then by integration by
parts we have
(X r- -iX fx g
?'(xz}dz = \zv'(xz} + / v"(x z) = dz
-v L Jo Jo
T
n\
and replacing the left-hand member from (42), we have
F(*) = F(0) + F'(0) + F"(0) + F"'(0) + ...
which is Maclaurin's Series. Hence it follows that the sum of
all the terms after the nib. is expressed by the definite integral
/X ~n \ rty
"(*-*) oo ,7 1V (45)
I..S...( 1)
which may be written in the form, if 6 be a positive and proper
fraction,
(X 2 n ~l f7? T n
..i.....(.-D = " ( " ) onr^ i (46)
in which case it becomes identical with that given in equation
(19), Art. 120, Vol. I; but as 6 is an undetermined fraction,
and as there are in general no means of determining it, equation
(45) is the more correct form for the remainder of the series to
be expressed in.
PRICE, VOL. II. P
106 SERIES FOR APPROXIMATING [89.
SECTION 3. On methods of approximating to the value of
a Definite Integral.
89.] When we are unable to find a definite integral either
by indefinite integration, that is, by reversing the rules of the
Differential Calculus, or by any of the means of definite inte-
gration which will be explained hereafter, we may often expand
the infinitesimal element -function in terms of ascending or
descending powers of its variable, and taking the definite in-
tegral of each term separately thereby approximate to the value
of the original integral.
This process is known by the name of Integration by Series,
and the correctness of the method rests on the following theorem :
Let F'(#) dx be the infinitesimal element-function ; and sup-
pose that F'(#) admits of being expanded in a convergent series
oftheform MO + Ml + 2 +...++... (47 )
and suppose that R is the sum of all the terms after the mth ;
then, as the series is convergent, R becomes infinitesimal when
m becomes infinite : bearing which in mind we have
r*n r*
?'(x}dx = I Undx + u\dx + ...
4 ^r
C x * r x n
... + u m dx+l ndx. (48)
/*
Now / ndx = (o? n MO) x some value of R intermediate to
JXQ
those corresponding to x n and X Q ; but since R becomes infini-
tesimal, when m becomes infinite, so will also its mean value;
and therefore
becomes infinitesimal and must be neglected. Hence
rx r% r%
I v'(x)dx = u dx+ u\dx + ... (49)
and the same process is also true for indefinite integration, viz.
/ F'(#) dx = I UQ dx + / HI dx + ... (50)
The principle involved in these remarks includes also the
90.] TO A DEFINITE INTEGRAL. 107
cases of integration investigated in the last Section, and there-
fore justifies the process which has been therein applied.
90.] Suppose then that we have to determine
/*
I T(W\ n v
I J \<^) *,
and that /(#) is capable of expansion by means of Maclaurin's
Theorem, so that
- + /'- 1 (0) 1 OQ*"/- T. +/ r (^)l-^
.-. f *'
the last term of which must be neglected when the series is
convergent ; and we have
fx n r- ~>2 i3 -\x n
) f^ d * = [/w-+/(0) + ^>r^s + -I,
) x l + ... (52)
Cx. rl v px.
Ex.1. / ^ f "(1-^3)-
^ (i-^? 3 ) 5 Ai
1 1-3
1.3 a? 7
r l
f 2
/x fx f v>% v>4> *> 6
.- <b =j( O-T + O-lis-
r ^ 3 a? 5 a? 7
~r3 + ro"~o"^
"
1.3 5 . 1,2.3.7
A series which enters extensively into the mathematical theory
of Chances.
p 2
108 SERIES FOR APPROXIMATING [91.
91.] Bernoulli's Series for approximating to a definite integral.
r x
Let / f(x) dx be the definite integral whose value is required ;
J*t
then, integrating by parts, we have the following series ;
jf(x) dx = xf(x) If (a) x dx
lf'(x)xdx = /'(ar) -f"(x}dx
[T* ft& ^\^n
qn f ( nn\ f f vt\ 1 _ f ( v\ (^~&k\
*j \<L ) .. f.j \j,) -f ^ ^ ~-y i,t*; ... K V""/'
which series may be derived as follows from Taylor's Series :
, h h 2
1 1.2
J,r-l for
VJ 19 fr ~\\ ^ ' 1 2 r '
**' V' *^ A.f ...
where F(X) and all its derived functions up to the rth are finite
and continuous for all values of x between x and x + h.
For h write x,
2 3
r-l r-1 XT ~ l r fi X *
P) 1.2...(r-l) ( 1.2. ..r'
and since is a proper positive fraction, 1 6 is also positive
and less than unity; representing it therefore by the general
symbol 0, we have
F(#) = F(0) .
For F (x) write //() c?^,
.-. T/(X) = /(X)
F"(a?) = /'(a?)
93-] TO A DEFINITE INTEGRAL. 109
3C r ~\ x *
... ( Y~ l f r ~ l (Qx)Y~9, ( 55 )
omitting F (0), because it disappears in the definite integral.
92.] The following also is an useful series for approximating
to the value of a definite integral : since
/**" /
J
and since by equation (14), Art. 1 19, Vol. I,
/ \ *>
I VI 1P{\\
a / nn \ ir//v, \ I / /v> ~, \ '//v, \ , \ W / "//) \ I
F \XQ) -f- ...
.2
(57)
(58)
and writing /(a?) for F'(#)
^~^ o)} ^ (59)
the right-hand member of which rapidly converges, if x n X Q be
small; and if x n X Q is infinitesimal, we have, taking two terms,
) dx = (x n -x )f(x ) + ( ^~ )2 /(^o). (60)
X Q
93.] Again, suppose x n x to be finite, and to be divided into
n parts, each of which is equal to i, so that x n XQ = ni ; then
r*o+{ rx +2i r Xn
=l f(x)dx + i f(x}dx+...+ /(a?)te,
-^o ^o+i /*^+(-l)<
and replacing the definite integrals by their values in (60) we
have
Fm
I f(x)da = i {/(ad) -f/<* +*0 + +/(*o-f (-!)*)}
^ -2
-I)i)}, (62)
and a nearer approximation may of course be made by in-
cluding more* terms of (59).
110 APPROXIMATION TO A DEFINITE INTEGRAL. [94.
94.] Again, if the infinitesimal element-function be of the
form/(<r) x F(#) dx, the integral of which is to be taken between
XQ and x n) and if f(x) does not change sign within these limits,
and if x\ y x%, ... x n -\ are the values of x corresponding to the
points of division of x n XQ, then by Preliminary Theorem III,
Vol. I,
p(a? ) (x\ x Q ) +f(xi) F(#I) (a?g a?i) + . . . +/(o?-i) F(a? n _i) (x n a? w _i
x 2 x l ) + ... + i>(x n -i)(x n a? n _i)} x some
mean value of f(x)
/"*
n XQ)} I F(a?) dx,
J X
wherein is a proper positive fraction. Wherefore
(63)
....
96.] SUCCESSIVE INTEGRATION. Ill
CHAPTER V.
ON SUCCESSIVE INTEGRATION OF AN EXPLICIT FUNCTION OF
ONE VARIABLE.
95.] IN the preceding Chapters methods have been investi-
gated for determining either exactly or approximately a finite
and continuous function of #, whose infinitesimal element-
function or whose first- derived function is given : that is, the
object has been to find ?(#), F'(^) dx having been given. I
purpose now to extend the methods to the discovery of F(#),
when v n (x)dx n , that is, the wth infinitesimal element-function,
is given.
It is plain that such a process requires n successive integra-
tions of the same kind as those investigated above ; and as
each integration brings in an additional4;erm, either as an arbi-
trary constant, or as a function of the limits of integration, so
by the whole process will n additional terms be introduced;
and the final integral is not to be considered complete unless
it contains these ; we shall at present find it more convenient
to consider them as arbitrary constants.
96.] Suppose F(#) to be a function of x finite and continuous
for all values of its subject- variable within the range for which
we consider it; and suppose its derived functions to be v'(x),
f"(x), ... F n (#), and to be subject to like conditions: then, as
explained in Art. 7,
jv'(x}dx = F(a?) +c n , (1)
Y"(x)dx = * + _!, (2)
fv*(x)dae = v n ~ l (x) + c i; (3)
and therefore from (2)
/ / F"(#) dx dx = / p'(a?) dx -\- c n _x x
F(#) + c n _i# + c n . (4)
112 SUCCESSIVE INTEGRATION. [97.
Similarly
// j*'"(x) dxdxdx = *(d?) +c B _2^g + c*-\x + c rt ; (5)
/n
dx n for
/ / ... dxdx, when the latter series involves n symbols of inte-
gration,
x
n-l
_l_ r* /v J_ r< /A\
c 2 , ... c n being n arbitrary constants. Hence
a? n x n ~ l
\ x n ~^-
e x dx n = e^ -j- Cj ^ = }-... + c n _
1.2... (n 1)
1 1
' Cl 1.2.8. ..(n-l) 4
97.] Suppose f(x) to be a given function of x, finite and con-
tinuous for all values of x between XQ and x ; and suppose it to
be the nth derived function of v(x), so that
dx n '
=**>*
and suppose XQ and a? to be the limits of integration, and F(#)
and all its derived functions to satisfy the requisite conditions
within these limits ; then integrating (7) we have
= ( X f(z)
Jx
dz + G! ; (8)
Hence again, taking the same limits,
f(z) dz dx + Ci (x XQ} + c 2 ; (9)
Jx n "fa
97-] SUCCESSIVE INTEGRATION. 113
but the double integral in the right-hand member is easily
reduced to a single integral. For since
\ X (xz} m f(z} dz = m( X (x-z} m - l f(z} dz,
/ M <v\WZ 1 -f { &\ //^ _ m f 9* -T\ W& f ( *y\ ft <y a*
i {^ ~~~* * / / \ / ^*" "^^ l\ / / \ / l *"' >
whence integrating with respect to x, and between the limits
Xo and x, we have
/"* /%*)-*/(*) dzdx = - f\x-z) m f(z} dz ; (10)
'-,, 4/M WC" //V
and therefore, if m = 1,
n# r*'
f(z)dzdx I (xz)f(z)dz;
and therefore (9) becomes
^ ' ^^ / / /y> ^\ -f / ^\ /Tf ^ j ^i / /v) _ l^^tj''* I I I ^
7 o ~" f \** * / / \ / '**' "i ^1 v *' ^^ ^O/ "i 2 ^ \ /
Similarly
^ ra ~ 3 F(^) _ r x (xz) 2 j
dx n ~ J x 1.2
and therefore ultimately
... + c n _i (a? a? ) + c w ; (12)
by means of which process the multiple integral becomes ex-
pressed in terms of a single integral. Also, since (12) is a
definite integral, the constants which are apparently arbitrary
admit of the following determined values. Returning to the
integrations by means of which they were introduced we have
= / (x z)f(z)dz + Ci
(x
and as these equations are true for all values of x which
* For a complete discussion of the process of differentiating with respect
to a quantity under a sign of integration see Chapter X.
PRICE, VOL. II. Q
114 SUCCESSIVE INTEGRATION. [98.
satisfy the requisite conditions, they must be true when x = XQ ;
in which case Cl = F -i (a?o)
C =
C n =
Therefore (12) becomes
-'i) +
>x * V */
which series is identical with that investigated in Art. 119, Vol. I,
and has the sum of all the terms after the nib. expressed in the
form of a definite integral.
98.] Thus far as to the general expressions of Successive
Integrals : let us however consider the subject as it is pre-
sented to us in the light of the Calculus of Operations, and
according to the principles of Chapter XIX, Vol. I. By the
Calculus of Operations, be it remembered, we deduce results
involved in the laws of succession and relation which functions
are subject to; now successively derived functions are subject
to the commutative and distributive laws, as is proved by the
Differential Calculus ; and so are also successive integrals of
functions of one variable, as is manifest from the preceding parts
of the present volume ; and these alone, thus far, have we dis-
cussed. But Integration is the reverse process of Differentia-
tion ; it is an undoing of Differentiation ; Differentiation is Dis-
integration. We pass therefore from one process to the other
by changing the sign of the index, or other symbol, which in-
dicates the number of operations which have been performed
on the subject; and therefore as d n expresses differentiation
performed n times, so does d~ n express integration performed
n times : but, for reasons given in Chapter I of the present
volume, a more suggestive symbol for integration is /, so that
/ is equivalent to d~ l , and therefore
d~ n = ///- (to n symbols)
(15)
99-] SUCCESSIVE INTEGRATION. 115
because the operation is subject to the index law. Hence also
(// \ ~ n
-?-) = d-*dar
dx>
I fl \-n fri
(i) = /* < 16 >
We proceed to make a few applications of these formulae, taking
care to select such cases as satisfy the required laws.
Ex. 1. Since
a)
= m"sm
TT\
- ),
dx n 2 /
.*. / sin (mx + a) dx n = sin(mx + a n^ } ;
and if n = 1 in this last formula,
/ sin (mx + a) dx = sin(m,r + a ^ j .
Ex. 2. Similarly
/ n I / TT\
cosmxdx n = -cos (mx n^ ),
Icosmxdx = cos(m# -).
J m \ 2 7
Ex.3.
where $ = tan- 1 ( ) ; see Ex. 6, Art. 52, Vol. I.
iT W?)
99.] Again, taking Leibnitz's Series for ' n , where u and v
QiOG
are explicit functions of x, and making n negative, we have, as
in Art. 367, Vol. I,
/n f"n ff u fn+I
uv dx n u I v dx n n-j- v dx n+l
'
Lt 1
Let n = 1,
//* <fe r 2 <^ 2 w r s
uvdx ulvdx -j- I vdx 2 + -r-^ I vdx 9 ... (18)
Q 2
116 SUCCESSIVE INTEGRATION. [99.
which series must apparently be continued to an infinite num-
ber of terms unless the derived-functions of u should vanish;
the limit of their sum may however be determined as follows :
j f*
Considering -r- to refer to u only, and I dx to affect v only,
and on the right-hand member of the equation separating sym-
bols of operation from their subjects, we have from (17)
( n u v dx- = ( ( n dx - n ~ /'" + kr+ 1 + rc(rc + l) *L
J U dxJ 1.2 dx 2
d
... y uv
uv
dx n
uv; (19)
and taking therefore symbols of operation only
F
A' = c /f ..' (20)
remembering that the symbol in the left-hand member refers
to the integral of u v, whereas those on the right-hand side refer
to either u or v. And in the symbolical form the limit can
easily be expressed by means of the general expression for the
limit of Maclaurin's Series.
In (19) let n 1 ; then
fdx
luvdx = uv, (21)
l + -r- dx
dxJ
or, dx=dxl + dx (22)
In (18) let v = I ; then
//* du C z d 2 u f 3
udx = ut dx ;-/ dx 2 + -=-s / dx 9 ...
J dxJ dx z J
x du x 2 d 2 u x 3 _
U I~~dxr2 + dx*T^3~
which is Bernoulli's Series for the calculation of an integral;
see equation (53), Art. 91.
100.] SUCCESSIVE INTEGKATION. 117
100.] Suppose that v = e ax ; then / v dx = - e ax ; so that
(19) becomes
J_
a n
/*"
I
J
ue ax dx n =
1 d \ n
adx)
d ~ n
and as -7- refers to u only, we may write
and therefore
d \~ n
n id \~ n
ue ax dx n = e ax (a + -j- j u; (24)
/*"
* \ ue ax dx n ; (25)
Let n = 1,
.-. (j-+) = e-^jue^dx. (26)
- /* *
Again, in (21) let ?; = e a , u = x n , .*. \vdx ae a ',
/- - f ^ ") -1
e a ^ n rfa? = a e a < 1 + a ^- j- a? n
?
= ae{a? n anz n - l + a z n(n l)x n ~ 2 ...}; (27)
du
Again, in (18) let w = e^; .-. = ae*, ...
(juOQ
... (28)
dx
1 + a I dx
118 GEOMETRICAL APPLICATIONS. [lOI.
CHAPTER VI.
APPLICATIONS OF THE INTEGRAL CALCULUS TO GEOMETRY.
RECTIFICATION OF CURVED LINES ; AND
KINDRED SUBJECTS.
101.] As the general problem of indefinite integration is,
given v'(x)dx, to find v(x), of which ~$'(x)dx is the differential,
so the form which it presents in reference to the properties of
plane curves is, given the general value of the trigonometrical
tangent of the angle between the tangent to the curve and a
fixed line (the axis of x}, to find the equation to the curve ;
because if y = F(#) be the equation to a curve, F'(#) ( = -jf- j is
the tangent of the above-mentioned angle : and with regard to
definite integration, the limits being X Q and x n , and y and y n
being the corresponding values of the ordinates, the problem is
to construct the curve between these values, that is, to deter-
mine the relation y = F(#), for all values of x and y between
these limits. Of this process the following are examples :
Ex. 1. In Vol. I, Art. 190, Ex. 4, the defining property of the
equitangential curve is found to be
dy y
-f- ; (1)
UX /% .2\i
whence dx = - v - y > ^ ; (2)
y
It is required to find the equation to the curve.
In fig. 2, taking x and y to be the current coordinates to the
curve, and therefore to refer to any point P, and observing that
y = a, when x = 0, we have
r* r"
I dx = -J.
1 01.] GEOMETRICAL APPLICATIONS. 119
by equation (72), Art. 35 ;
... x = a \og a + (a2 ~ y2)2 -(d*-y*)*', (4)
y
which is the equation to the Equitangential curve.
Ex. 2. Suppose the defining property of a curve to be
dy __ (2ax
dx x
and the origin to be on the curve, and the tangent to the curve
at the origin to be the axis of y, and the limits to be the
coordinates to any point on the curve, then
/w :
Jo Jo
=f;
Jo f
dx (5)
dx
[~\y r~ i x'
y = (2ax x z Y +versin- 1 -
Jo L tt
y
.-. y = (2 ax a?)* 4-aversin" 1 - ; (6)
Q
which is the equation to the Cycloid, whose highest point is origin.
Ex. 3. To find the curve whose subnormal is constant.
This defining property, expressed mathematically, is
4 =
y dy = adx,
and taking the origin to be on the curve, and the limits of inte-
gration to be the coordinates to any point whose coordinates
are x and y, ~ y
I ydy = al dx
^o Jo
y 2
-2=
y 2 = 2 ax;
the equation to a Parabola whose latus rectum is 2 a.
120 GEOMETRICAL APPLICATIONS. [lOI.
Ex. 4. Find the curve whose normal is of a constant length a.
By equation (42), Art. 186, Vol. I, we have
dx y
d\t
and taking the origin on the curve, and -- to be positive,
f *" = /
Jo (a z y 2 )* *o
= HI
rf + a = x
Q? y 2 = (x a
the equation to a Circle whose radius is a. (8)
Ex. 5. Find the equation to a curve which cuts all its radii
vectores at a constant angle.
This property, expressed mathematically, is
rd6
w = c
dd _ dr
c r '
If therefore = 0, when r = a, we have
rdi9 _ f r dr
Jo c ~ J a r '
-:i- w:
- = log r log a ;
e
.-. r = ae c ; (9)
the equation to the Logarithmic Spiral.
We shall hereafter return to these and similar problems,
because others require more means of integration than we have
at present at command. We proceed to other applications of
the Integral Calculus.
102.] RECTIFICATION OF PLANE CURVES. 121
SECTION I. Rectification of Plane Curves referred to Rectangular
Coordinates.
102.] The Integral Calculus enables us to determine, either
exactly or approximately, the length of a plane curve in terms
of the coordinates of its extremities, and thus to compare its
length with that of a straight line ; whence arises the name of
Rectification.
Let y =/(#), or F(#, y) = c, be the equation of a plane curve
referred to rectangular coordinates, and let it be required to
determine the length of the curve between the points (XQ, yo)
and (x n , y n ) ; that is, to determine the length of a straight line,
along which if the curve be made to roll (not to slide) the ex-
tremities of it will coincide with those of the curve. Now,
adopting the notation of Vol. I, Art. 185, let ds be an infini-
tesimal length-element of the curve, then the required length
is the integral of ds between the specified limits ; but
ds = {dz 2 + dy z }*, (10)
and therefore
the required length = {dx 2 + dy 2 }^, (11)
the integral being taken between the given limits.
Let s represent the length of the curve ; then if the equation be
y = /(#),
dy = /'(#) dx
.-. ds=
s= {l + (y) 2 }, (12)
And if the equation to the curve is of the form
* = f(y)
doe = f'(y) dy
(13)
As the radical expression in (10) involves an ambiguity of
sign which is continued in (12) and (13), and as s is an absolute
length, we must choose that sign which the circumstances of
the problem require; that is, ds and dx or dy must be taken
with the same or different signs according as x and y increase
or decrease when s increases.
PRICE, VOL. II. R
122 RECTIFICATION OF PLANE CURVES.
103.] Examples of Rectification of Curves.
Ex. 1. The Circle; see fig. 3.
Let the centre be the origin, and let the beginning of the
arc AP, whose length is to be determined, be at A ; then, if
OM = x, as s increases, x decreases ; and let the length of the
arc AP W be required, where OM n = x n :
x z + y z = a 2
xdx + ydy =
dx__ dy _ ds
y x a '
ado:
ds =
/** adx
.-. the length Ap n = /
Ja (tf-
(
r i #"1*
= flcos- 1
Ja
= acos- 1 . (14)
a
/* a
Hence also length of Quadrant AB = / -
Ja (2_
adx
a? 2 )*
i #"1
= a cos- 1
Ja
TTtt
.-. Perimeter of Circle = 2ira. (15)
Hence also if OM O = X Q , and if the length of the arc P O P M is
required, then
/"* a dx
Arc P M P O = / ^
r T x ~\ Xn
= a cos~ l -
L aj x
->- -\
(16)
Ex. 2. The Parabola ; fig. 4.
Let the arc whose length is required be measured from the
vertex; and let it be op n ; let OM n = # re , M n v n = y n) and the
equation to the parabola be
1 03.] RECTANGULAR COORDINATES. 123
y =
dy dx _ ds
2a
p
.-. length of arc OP W = /
Jo
. , .n.
~^~ ~2^~
as appears by equation (80), Art. 39; and as y n may be the
ordinate to any point on the parabola, let us write for it the
general value y, so that the length of the arc of the parabola
beginning at the vertex is equal to
and if for y we write 2 (#)*,
the length = (aw + # 2 )* + log " . (18)
2
Ex. 3. The Cycloid.
(a) Let the highest point be origin, see fig. 5 ; and let the
arc be measured from the vertex ',
i ^
y = (&ax x z y* + a versin" 1 -
dy dx _ ds
(2 a -a?)* a?*
j
as = I } ax ',
x
/x n /2a
(
_ V x
dx
= 2(2<w? B )*; (19)
and as x n may be the abscissa to any point on the curve, we
may write for it the general value x ; and then
s = 2(2cw?)*
s 2 = Sax. (20)
B 2,
124 RECTIFICATION OF PLANE CURVES.
/ 2a /2\*
( j dx
r 4~i 2a
= 2(2#)*
L Jo
= 4a; (21)
therefore the whole length of the cycloid is 8 a, that is, four
times the diameter of the generating circle.
(/3) Let the starting point be origin ; fig. 6.
OM = x, MP = y ; OM n = x n , M n p n = y n ,
V i
x a versin" 1 - (2ayy 2 ) ls
a
dx dy ds
OB =
{(2a)-(2a- yj| )} ; (22)
"/ 2 a \*
Ex. 4. The Tractory ; see fig. 2.
Let the required arc be measured from A, and the ordinate
to its extremity be y n ; then
dy dx ds
y ~ (a z -yrf ~ a '
. , /** a dy
.'. required arc = /
= f-alogyl
L J
= a log ( ) ; (23)
and writing for y n the general value y, we have
(24)
1 03.] RECTANGULAR COORDINATES. 125
Ex. 5. To determine for what values of m and n the curves
expressed by the equation
a m y n = x m+n
are rectifiable.
m m + n
a n y x n
.-. a n dy = - x n dx,
on comparing which with equation (86), Art. 43, the condi-
tions requisite for integration by rationalization are, that either
n n 1 , , , ,
TT or -p. -- 1- 77 should be an integer.
2m 2m 2
From the former of which conditions we have
m + n 357
--_ = -, or = 3 , or = B ,...;
and from the latter
m+n 246
= P r = 3> or = 5>--
Ex. 6. To determine the length of the arc of the Catenary,
measured from its lowest point to any point on the curve ;
see fig. 7.
a r - --->
y = 2\ ea + e Jf
1 r * -X
dy = - -i. e a e a
1 r X ^
ds -j e a + e a | dx ;
and taking a general value a?, which will refer to any point on
the curve, for the superior limit
/* 1 r x _*-.
- -{ e a + e a \dx
<* *- >
[n r x x > T*
H''-"'}].
(28)
126 RECTIFICATION OF PLANE CURVES. [103.
the same result as that found in Ex. 4, Art. 244, Vol. I. Hence
5 2 _ y2_ fl 2. (29)
the arc therefore measured from the lowest point is the side
of a right-angled triangle, of which y (= MP) is the hypothenuse
and a (= Mn) is the other side, that is, AP = np.
Ex. 7. It is required to find the whole length of the Hypo-
cycloid whose equation is a? + y* a* ; see fig. 10.
The equation to the curve may be put into the form, see
Vol. I, Art. 166,
' x = a (cos 0) 3 -j
y a (sin 0) 3 /
.-. dx = 3a(cos0) 2 sin0d0
dy 3a(sin0) 2 cos0d0,
.-. ds 2 = dx z + dy 2 = 9 a 2 (cos 0) 2 (sin 0) 2 d6\
ft
length AB = 3 a sin 6 cos 6 dd
Jo
3a
. . whole length of curve = 6 a.
Ex. 8. On the lengths of Elliptic arcs ; fig. 8.
Let it be required to find the length of an arc of an ellipse,
beginning at B the extremity of the minor axis.
Let CM = x, MP = y : then the equation to the ellipse being
5 + F- 1 < 30 >
//2 1LJLJ _- ^2 2
ds 2 = -
if a 2 e 2 = a 2 -b 2 ; (31)
taking for the superior limit a general value x which refers to
any point on the curve.
There is no known method of integrating (32) and expressing
it in terms of the more common formulae of the Integral Cal-
1 04.] RECTANGULAR COORDINATES. 127
culus, such, that is, as arise from finite algebraical, or circular,
or logarithmic functions : we are therefore obliged to have re-
course to expansion ; and observing that x is always less than a,
X GOD
and therefore that - and a fortiori is less than unity, we have
a a
(a
2
) _ a / e
* " 2 -* * 2
_
47^ ~ 2.4.6 a 6 ~ "
.*. arc BP
* dx C e*x* e*x* 1.3.e e a? 6
"" " J
. . the length of the quadrant of an ellipse
a dx
" ; (
But by equation (25), Art. 85, if n be even,
dx (n \)(n 3). ..3.1 IT
/a /yiH fJi
<As U/<A/ \rv Jty^/i/ -* y . . . ^-^ . /OK\
a n (a2 _,^ = (n-2)...4.2 2'
a adx C e 2 x 2 e 4 ^ 4 1.3.
~
2.4. a 4 2.4.6. a 6
and therefore the perimeter of the ellipse
11. 3\ 2 . 1/1.3.
104.] Although it is impossible to express the length of an
arc of an ellipse in terms of any ordinary function, yet we can
from the differential equation (32) deduce certain properties
which deserve consideration.
The equation to the ellipse when expressed in terms of the
eccentricity is y , = (1 _ e3) (a2 _^ (8g)
Let T be the acute angle contained between the axis of x and
128 RECTIFICATION OF PLANE CURVES.
the tangent to the ellipse at the point (x, y) ; then, as dx is the
projection of ds on the axis of x,
ds = secrdx (39)
s = secrdx', (40)
ds
the integral being taken between given limits, and -r- being
CL-X
affected with a sign which will yield a positive value of s. Also
since from (38)
dy (\-(?}*x
tanr = - -- =
asinr ,.,.
x = (41)
(1 (ecosr) 2 }*
dx = a(1 - e2)COSr ^; (42)
{l-(ecosr) 2 }*
and therefore from (40)
s = (l-e 2 )/ -, (43)
* and r being so related as simultaneously to increase and de-
crease ; and * being the length of the arc contained between
the points at which the tangents of the ellipse are inclined to
the axis of x at angles r and T O .
105.] Again, if x and X Q are the superior and inferior limits
of x, and s be the arc of the ellipse between the corresponding
points, we have
f'f*-*+\*j
s = / L = - 5-) dx. (44)
To simplify this, let
x = a cos $ (45)
dx = a sin d$ ; (46)
then taking s to be the arc of the ellipse contained between the
points to which and $ correspond, we have
s = a {l-(ecos(/)) 2 }<fy>. (47)
**
Now between < and r corresponding to the same point on an
ellipse there is a remarkable relation: substituting in (41) the
value of x given in (45), we have
I06.] RECTANGULAR COORDINATES. 129
cos${l (ecos r) 2 }* = sinr, (48)
1 - (cos r) 2 (cos <f>) 2 + e 2 (cos 0) 2 (cos r) 2 = 0, (49)
e 2 sin r cos r
.*. a.(e z cos d>cosr) =
. -
{l-(ecosi-) 2 }*
= -{l-
a (1 - e 2 ) dr
TO {1 (ecosr) 2 }^
= |e 2 cos < cos r + a {I (e COST)-}* dr. (51)
J TO JTQ
But by (47) the last term of the right-hand member of this equa-
tion is equal to the length of the arc contained between points
to which and $ correspond when they are equal respectively
to r and TO ; therefore if <r be the length of the arc contained
between points on an ellipse determined by c/> and $ , and s be
the length of the arc contained between points determined by
s = ae 2 {cos $ cos T cos $ cos T O } + a-, (52)
.*. s <T = ae 2 {cos < COST COS$ O COST O }. (53)
Let x and x be the abscissae to the extremities of s, and
and the abscissae to those of o- ; then
x = cos <f> ~j = a cos r ->
r f t> (54 ^
r = a cos $ J f o = cos T O J
e 2
.-. 5 0- = {z A'O^O}; (55)
that is, the difference of two elliptic arcs is expressed as a func-
tion of the abscissas of their extremities. The discovery of this
Theorem is due to Fagnani ; and the geometrical interpretation
is the following :
106.] Since in (45) we have assumed x = a cos $, it follows
that if a semicircle be described on the major axis of the ellipse
as a diameter, and a radius be drawn from the centre at an angle
</> to the major axis, the point on the circle whose abscissa is x is
that whose ordinate cuts the ellipse at the point whose abscissa is
a cos $, and whose ordinate b sin $, as is plain from the equa-
ds
tion to the ellipse. And since from (47) -j- is positive, s and <f>
Cl(p
PRICE, VOL. II. S
130 RECTIFICATION OF PLANE CURVES. [107.
simultaneously increase, therefore in (52) <r is measured in a
direction along the curve towards the extremity of the minor
axis and from the major axis ; similarly because * and r simul-
taneously increase and decrease, s in (52) is measured in a
direction from the minor axis towards the major axis.
Thus in fig. 9 let p P O be the arc whose length is s, p and P O
being the points to which r and T O correspond: viz. P O T O O = T O ,
PTO = T; and let RON = $, R O ON O = $o> where (p = T, < T O ;
and from u and R O let the ordinates R N, R O N O be drawn, cutting
the ellipse in Q and Q O ; then the arc QQo = &; and OM = x,
^o; ON - , oN =:fo; and therefore from (55)
PP QQo = {OM XON OM XON }. (56)
tt
If PO be at B, that is, if the arc be measured from the extremity
of the minor axis, then r = 0, <r = 0, <o = 0, = a, and Q O is
at A : in which case
e 2
BP AQ = OMXON, (57)
CL
the abscissse of the points p and Q being connected by the equa-
tion (49), which in terms of x and is
a 4 -a 2 # 2 -a 2 2 + e 2 tf 2 2 = 0. (58)
107.] Hence we easily deduce a geometrical interpretation
of the right-hand member of (57). From o draw a perpen-
dr
dicular oz on the tangent at P, then, if OP = r, PZ = r -^-; and
CIS
since from (38)
r 2 = a 2 (l e
rdr = <?xdx,
rdr , dx
.'. --j- = e^x-j-;
ds ds
i t /KQ\
and from (08) ON = .= a
e z e 2 dx
OMXON = xaf
a a ds
dx
= e*x~r
ds
= PZ;
.-. BP AQ = PZ ; (60)
which is tiic geometrical form of Fagnani's Theorem.
I08.] POLAE COORDINATES. 131
If the points P and Q coincide, then = x, and from (58)
.-. BP AP = a b; (61)
that is, the difference of the arcs into which the elliptic quadrant
is divided is equal to the difference of the semi-axes.
Further researches into the properties of the definite integral
which expresses the length of an elliptic arc would be un suited
to the present stage of our Treatise. Some properties of such
arcs have been discovered, and proved on the geometrical infini-
tesimal method, by the late Professor Maccullagh of Dublin,
and are contained in Vol. XVI of the Transactions of the Royal
Irish Academy : and some others are proved by the same pro-
cess in Salmon's Conic Sections, p. 296, 2nd ed., Dublin, 1850 :
these however are but slight contributions to a subject of great
extent and difficulty. A student, desirous of fuller knowledge,
must refer to
(1) Legendre, "Theorie des Fonctions Elliptiques," Paris, 1825-28.
(2) The Collected Edition of Abel's Works, edited by M. Holmboe,
Christiania, 1839.
(3) Jacobi's " Fundamenta Nova Theorise Functionum Ellipticarum,"
Koenigsberg, 1829.
(4) Mr. Leslie Ellis' Report " On the recent progress of Analysis"
to the British Association, and printed in the Report, 1847;
and which, with other valuable information, contains a com-
plete historical account of the problem.
(5) The last Chapter in Gregory's Examples on the Integral Cal-
culus, 2nd ed. ; of which Mr. Leslie Ellis is the author.
(6) The volumes of the Mathematical Journals of Crelle and Liouville.
SECTION 2. Rectification of Plane Curves referred to Polar
Coordinates.
108.] If the equation to a curve be given in the form
F (r,0)=0, (62)
then, by Vol. I, Art. 220, equation (12)
ds = {dr 2 + r* d6 2 }*, (63)
and therefore s = J{dr 2 + r 2 d6 2 }* ; (64)
S 2
132 RECTIFICATION OF PLANE CURVES. [109.
the integral being taken between the limits assigned by the
conditions of the problem.
And if the equation to the curve admits of being put into the
f rm r = f(0),
dr = f(0) cie,
s= I'{(f(0))'+(f(0))*}*d0. (65)
\
And if the equation to the curve be put into the form
= /(r),
dO = f(r) dr, '
s = r{l + r*(f(r))*}*dr. (66)
Jr
109.] Examples in illustration.
Ex. 1. To find the length of the spiral of Archimedes measured
from the origin.
r = ad,
dr = add,
s = f a (1 + 2 )* de
Ex. 2. To determine the length of the Logarithmic spiral.
r = a 6 ,
dr = log a .a 8 dO
= r log . dd,
s = I ^ ' V^&^ J rf/ .
log a
(r r ).
log a
a result which immediately follows from the fact that the curve
cuts all its radii vectores at a constant angle, and therefore that
the difference between any two radii vectores is equal to the
projection of the length of the curve between the corresponding
points on a line to which it is inclined at the constant angle.
110.] POLAR COORDINATES. 133
Ex. 3. To find the length of a circle, the extremity of the dia-
meter being the pole.
dr = 2a sin dO,
re
s = 2a dO,
Jo
the arc being measured from the extremity of the diameter ;
.-. s = 2a0,
7T
and therefore, if = -,
A
the semi-circumference = net.
110.] If the equation to the curve be given in terms of r
and p, then, by Vol. I, Art. 222, equation (23),
( r a -j^
the integral being taken between limits assigned by the problem.
Ex. 1. To find the length of the involute of the circle between
any given points on it.
r 2 = a 2 -\-p z ,
rdr
r rd,
~ Jr n a
and if s begins at the point where the involute leaves the circle,
r = a ; and r % _ fl 2
s = ~2^~-
Ex. 2. It is required to find the whole length of the hypo-
cycloid whose equation is x* + y* = a* ; see fig. 10.
The equation in terms of r and p is
3jo 2 = a 2 r 2 ,
(T 3^7* diT
= 6 a.
134 RECTIFICATION OP [ill.
The limits of integration of this problem deserve attention.
In the fig. OA = OB = a, oc = - ; now if * be measured from A,
ds
-j- is negative from A to c, and is positive from c to B ; at c
Off
therefore it changes sign ; we must not then integrate between
limits which include r = ^, but, in accordance with Art. 81,
jo
Theorem VI, divide the interval into two parts, and integrate
from r = to r = , and multiply by 2 to determine the
<i
length AB.
SECTION 3. Rectification of Non-plane Curves.
111.] The infinitesimal length-element of a non-plane curve
or of a curve in space, as determined by (2), Art. 288, Vol. I, is
ds = (dx 2 + dy 2 -f dz 2 )*, (69)
whence, by integration between the given limits, the length of
the arc of the curve may be found.
If the equation to the curve be given in the form
# =/(*), y = <t>(z),
dx f'(z) dz, dy <j)'(z) dz,
) 2 + 1}* dz. (70)
If the equations be given in terms of another variable, say <,
and of the forms
y =
then dx = f'(<$>) d(j), dy = p'(0) dfy, dz = \j/'((f))
<t>0
Ex. 1. To determine the length of the helix between two
given points.
Taking the equation to the curve as found in Vol. I, Art. 295,
equation (32),
x = acos<, y = asin</>, z = katy,
dx = a sin dty, dy = a cos d$, dz = ka dtf),
ds 2 = dx 2 + dy 2 + dz 2
= a 2 (1 + k 2 ) d(j> 2 ,
112.] NON-PLANE CUE YES. 135
s =
If therefore the arc begins at the point \vhere Z Q = 0, then
a result which also follows immediately from the geometrical
generation of the curve.
Ex. 2. To determine the length of the curve formed by the
intersection of two right cylinders, of which one is parabolic
and perpendicular to the plane of yx, and the other is cycloidal
and perpendicular to the plane of xz.
Let the equation to the director- curves of the cylinders be
y 2 = 4c#,
z = aversin- 1 - + (2 ax xrf,
8
ds* = dx* +
=/
Jo
x x
dx
SECTION 4. Determination of the Equations of Curves when
Relations are given between the Length and the Coordinates.
112.] In the last three Sections we have expressed lengths
of curves contained between given points in terms of the co-
ordinates of those points ; we proceed now to investigate the
inverse problem, and to find the equations of curves when a
relation is given between a length and the coordinates to its
extremities.
Let the given relation be
ds = /'(a?) dx,
136 THE PROBLEM INVERSE TO [lI2.
.-. (f'(x)} 2 dx 2 = dx 2 + dy 2 , (73)
dy = {(/'(*)) 2 -l}*<te, (74)
y = f{(f())*-l}**i. (?5)
thus (74) is the differential equation to the curve ; and if the
integration indicated in (75) can be performed, the integral
equation to the curve can be found.
Ex. 1. s 2 = 4>ax,
.'. s = 2 ax*
ds = (-} dx,
\'
- dx 2 = dx 2 + dy 2
dy = (- -) dx
x
a x
-dx,
(axx 2 )*
/ x a x
-dx
- (ax x 2 )*
[a ,2x n i~\ x
^versm- 1 (- (ax x 2 ) 2
A a J
a , 2x n i.
= versm" 1 \- (ax x 2 ) 5 ;
iV ft
the equation to a cycloid, whose vertex is the origin, and the
radius of whose generating circle is .
Ex. 2. s 3 = ax 2 ,
s = a* a?*,
113.] THAT OF RECTIFICATION. 137
let - a* k* , and let the limits of a? be a? and If, ; therefore
y
Q / ^ 2
y = - / (&*
AJk
the equation to the hypocycloid, Vol. I, Art. 179, equation (41).
Ex. 3. z s z = a 2
ds=
y L a 2
T* dy
x = a
Ja (2_ fl 2
the equation to the catenary : the problem being the inverse
of Ex. 6, Art. 103. *
SECTION 5. On Involutes of Plane Curves.
113.] For the determination of the involute of a plane curve
it is necessary that the length of the curve between given points
should be capable of being expressed in terms of the coordinates
of those points ; and it is thus only of rectifiable curves that the
involutes can be determined.
Let AH, see fig. 11, be a part of the curve whose involute is
to be found. Let ON = , NH = -q, and let the equation to AH,
which is the evolute, be f(f . ,g.
and let the element of the arc be do-; let pn be the tangent at n,
* For other examples see a memoir by Tortolini in Art. 29 Crelle's Journal,
Vol. XXVI, Berlin, 1843.
PRICE, VOL. II. T
138 INVOLUTES OF PLANE CURVES. [l 14.
whose extremity p is the generating point of the involute ; and
therefore pn is the radius of curvature of the involute. Let
Pn = p, then, by Vol. I, Art. 243, equation (40),
d<r = dp,
.'. p = <rc; (77)
c being a constant, the value of which depends on the position
of the generating point with respect to the point on the evolute
from which Pn is at first drawn. Thus in the fig., if AH = <r,
and np = An, then <r p, and c = 0; and if pn be longer than
An, then c is the excess of length ; that is, if a string of the
length pn is wound round An, and ultimately becomes a tangent
at A, c is the length of the remnant of the string.
Let OM = x, MP = y, then, since -7- = tan HTN,
sinnTN cos nTN 1
therefore from the geometry of the figure,
X = ON NM = ,
da
dr\
11 = N n n R = n p-r-
r da-
in which equations p must be expressed in terms of and rj ;
and and 57 having been eliminated from them and (76), the
resulting equation will contain x and y only, and be that to the
required involute.
114.] From (79) by differentiation we have
7 ^^ 7 ^t
dx = d d pd.-f- = pd.-f-
da da- ,
< 8 )
dy = dt] dr) p-d.-^- = pd.
do- da-
dy
dx
da
But by Art. 237, Vol. I, (21) and (22), the numerator and de-
nominator of the right-hand side of (80) are proportional to the
direction-cosines of the tangent of the evolute ; and therefore we
conclude that it is perpendicular to the tangent of the involute.
II5-] INVOLUTES OF PLANE CURVES. 139
Again, squaring and adding the two equations (80), we have
But if p is the radius of curvature of the evolute at the point
(, 17), then, by Vol. I, Art. 236, (19),
ds d<r
P ' P
but dor = dp,
and therefore by means of (34), Art. 262, Vol. I,
p 3 ds d z s (dx d 2 y dydtx) ds z (dxd*y dyd^x)
~ = ***
115.] Examples of involutes.
Ex. 1. To find the involute to the catenary, the generating
point being in contact with it at its lowest point.
_rl e -i^
By equations (28) and (29), Art. 103,
a f ^ -
a- = - J. e a e a k
*w L ->
/T " 1 - -i // 2 , , 2
I/ -^ 71 ^^ C* Lt y
dor r) dcr f]
d-n ~~ a' d ~~ a'
| 2 = |{;^-5) +e -:- (l -y}
the equation to the tractory, the form of which is evident from
fig. 7.
T 2
140 INVOLUTES OF PLANE CURVES. [H5-
Ex. 2. To find the equation to the involute of the cycloid,
the generating point being in contact with it at its vertex.
Let the cycloid be placed as in fig. 12 ; and let ON = , Nn = rj ;
OM = #, MP = y; then the equation to the cycloid is
f = aversin" 1 h (2ar? rj 2 )*,
a
df d-n da
(2-77)* 77* (2a)*
.-. o- = 2(2ar,)* = P ,
y = 17-21,, x = f -2 (2017-17*)*,
therefore by substitution
# = a versin" 1 - (
the equation to a cycloid in an inverted position, as OPD in the
figure, and lying below the axis of x.
Ex. 3. To find the involute of a point.
Let the coordinates of the point be = a, r; = b ; and let
c = the length of the string which is attached to the point, and
whose extremity generates the involute -, then
c 2
*" )
which is the equation to a circle whose centre is at the point,
and whose radius is equal to c.
Ex. 4. To find the equation of the involute of the semi-
cubical parabola whose equation is
27 ar? = 4 3 ,
the length of HP being longer by 2 a than the arc AH. Fig. 13.
drj d da dcr
1 1 6.] INVOLUTES OP PLANE CURVES. 141
3 (3 a)
3 (3 a)*
. . p = a- + 2 a, by conditions of problem,
3 (3 a)*
therefore by equations (79)
3v 2
= 3x + 6a, and = -^-,
the equation to a parabola, situated as in the figure.
116.] On involutes of curves referred to polar coordinates.
See fig. 14. Let AP be the curve whose involute is to be de-
termined ; and let its equation be
r=f(p). (82)
Let PP' be the tangent at p, p' being the generating point of
the involute. Draw from the pole s, SY perpendicular to PP',
and SY' perpendicular to pV, which is the tangent to the invo-
lute at P'. Then
SP = r, SY = p, SP' = /, SY' = p',
and our object is to find the relation between / and p'. Let
ds represent an elemental arc of the original curve ; and, since
PP' is the radius of curvature of the involute at P', let PP'= p ;
then, by equation (40), Art. 243, Vol. I,
.-. dp - ds, (83)
p = s c ; (84)
and from the geometry
r' 2 =p*+p' 2 , (85)
r 2 = /a + p'a_2 p y. (86)
and after eliminating r, p, s from the equations (82), (84), (85),
(86), there will remain an expression involving /, p', which will
be the equation to the involute.
142 INVOLUTES OF PLANE CURVES. [ll6.
Ex. 1. To find the equation to the involute of a circle.
Let centre of circle be pole : then, if a = the radius, its equa-
tion in terms of r and p, is
r = p = a,
whence (85) becomes r' 2 p 2 = a 2 ;
which is the equation to the involute of the circle.
Ex. 2. To find the equation to the involute of the logarithmic
spiral.
Let a be the constant angle at which the curve cuts all the
radii vectores ; then its equation is
p = r sin a. (87)
Therefore, see fig. 15, if PP' is equal to the length of the curve
from the pole to the point p, and if PP' = p, by Art. 110, (68),
("r
= /
cos a
From (86), completing the square,
(p'-p'Y = r 2 -r' 2 +p 2
= r 2 p 2
= r z (cos a) 2 ,
. . p' = p' + r cos a ;
and substituting for p from (88),
p' = r sec a r cos a
= r sin a tan a;
.-. from (85) r' 2 = p' 2 +p 2
= p' 2 + r 2 (sina) 2
= r sec a. (88)
= p' 2 (cosec a) 2 ,
. . p' = r sin a ;
the equation to a logarithmic spiral, similar to the original one,
that is, which cuts all its radii vectores at a constant angle the
same as that of the original spiral. From (88) it is evident that
PSP' is a right-angle, and therefore SP'Y'= SPY = a : the involute
therefore is also the locus of the extremity of the polar sub-
tangent.
1 1 7.] QUADRATURE OF SURFACES. 143
CHAPTER VII.
QUADRATURE OF SURFACES, PLANE AND CURVED.
SECTION I. Quadrature of Plane Surfaces. Rectangular
Coordinates.
117.] WE proceed to another of the most useful applications
of the Integral Calculus, viz. to the method by which we can
express, either exactly or approximately, a plane or curved su-
perficies in such a form that it may be compared to the area
of a square : hence arises the name Quadrature ; and we shall
first consider the most simple case, and investigate the area of
a plane superficies contained between the axis of x, two ordi-
nates parallel to the axis of y and at a finite distance apart, and
a curve whose equation is given.
Let y =/(#) be the equation to the curve POPP; see fig. 16;
OM O = #O, OM n = x n ; and let/(#) be finite and continuous, and
be of the same sign, for all values of x between XQ and x n ; our
object is to determine the area of p M M n p w .
Take any point E within the boundaries of the area, and let
the coordinates to E be OR and y ; take EF and EG infinitesimal
increments of y and x, so that EF = dy, EG = dx ; then the area
of the element = dy dx, and the area of the superficies required
is the sum of all such infinitesimal elements : the summation
being performed according to the principles of the Integral
Calculus, and the limits being given by the geometrical con-
ditions of the problem.
Let the other lines be drawn as in the figure ; and let us
consider x to be constant, and sum the elements with respect
to y from the axis of x to MP, that is, integrate dx dy with
respect to y from y = to y = /(#), dx being a constant factor
throughout the process * : the result of such an operation will
be the area of the differential slice PMNQ, whose sides are paral-
lel to the axis of y, because x is the same for all the elements,
* For the future we shall call such an integration the y -integration, and
similarly the integration with respect to x, the ^-integration.
144 QUADRATURE OF PLANE SURFACES. [1*7-
and which is of the breadth dx and of the length MP or f(x),
and therefore , . ,
PMNQ = f(x)dx; (1)
and as this area is expressed in general terms of x, it is the
type of all similar elemental slices ; and therefore the sum of all
such between assigned limits is the required area.
In accordance then with the principle of symbolization which
has been hitherto employed in the treatise,
(*x r f(nc)
area p M M n p n = / "/ dydx; (2)
Jx Jo
the meaning of which symbol is, that /(#) and are respectively
the superior and inferior limits of / dy, viz. y, which is therefore
equal to f(x) ; and that x n and XQ are respectively the superior
and inferior limits of x in \f(x) dx : therefore if A represent
the area required,
(JC /*f(<%)
J Q dydx, (3)
-i/(*)
tl \ fl V*
y <*
Jo
Integrals of the form (3), wherein integrations have to be per-
formed, one on the back of the other, and subject to certain
relations, algebraical or geometrical, are called multiple inte-
grals, and to the general consideration of them we shall proceed
in a subsequent chapter. The specific form (3) is called a double
integral; the order in which the integrations are performed is
the same as that in which the differentials are arranged, and
the reverse to that of the signs of integration : thus in (3) the
y- integration precedes the x- integration, though the integral
signs are in a reverse order: the reason of the arrangement
being, that the process symbolized by / dx is performed on the
back of, and so includes, that represented by / dy,
If the superficies, whose area is to be determined, is of the
form o P P, ( M W of fig. 4, then the inferior limit of x is 0, and we
nave /*, rf(x)
A = / / dydx. (5)
Jo Jo
1 1 8.] RECTANGULAR COORDINATES. 145
Let it not be supposed that any inaccuracy of result arises
from the circumstance that the differential slice is an imperfect
rectangle at the point p where it meets the curve ; for though
the real value of PMNQ is intermediate to
f(x} dx and f(x + dx) dx,
yet the difference between these two, viz. {/(<# + dx) f(x)} dx,
is equal to f'(x) dx 2 , and is therefore an infinitesimal of a higher
order, and must be neglected.
We proceed to give some examples in which the above for-
mulae are applied ; but in all cases especial care must be taken
that the limits of integration do not include any value of the
variable which makes the element-function to change sign, as
it may be that the sum of the elements on one side of such a
critical value will exactly neutralize that of those on the other
side, and the result will be nugatory.
118.] Examples of quadrature of plane surfaces.
Ex. 1. To find the area contained between the axis of x, an
ordinate, and the parabola whose equation is
Let the extreme abscissa (see fig. 17) = a, and the extreme
ordinate = b, ^ b * = ima,
and the equation to the parabola is
area OAB
therefore the parabolic area OAB is equal to two- thirds of the
rectangle OABN.
PRICE, VOL. II. U
146 QUADRATURE OF PLANE SUEFACES. [ll8.
Ex. 2. To find the area of a quadrant of a circle.
x 2 + y 2 = a 2 ,
n(a2-,r2)i
dydx
_
a .
.
TTtt
.'. area of circle = ira 2 .
Hence also, see fig. 18, if AC = c, and CB = b, and therefore
area of segment BCB'A = 2 area of BCA
fa r(a2
= 2/ /
Jac^O
% f* (a*-
Jac
-no , , a c
Ex. 3. To find the area of an ellipse.
^2 2/2
a 2 ^ b 2 ~
area of ellipse = 4 area of quadrant
= 4
r o a
. b TTd'
= 4
a 4
1 1 8.] RECTANGULAR COOKDINATES. 147
Ex. 4. To determine the whole area included between the
curve and the asymptote of the cissoid of Diocles ; see fig. 19.
y i
(2ax-xrf
for which value of y it is convenient to have a specific symbol,
and therefore we shall express it by Y, so that it may be distin-
guished from the y which is the ordinate to the area-element.
Hence, as OA = 2 a,
fZa /*Y
whole area = 2 I dy dx
JQ Jo
2 a
(see Ex. 1, Art. 51)
ax
r
= 2\
L
i 3a 2 x~\ 2a
\axx' 2 y* + -^- versin- 1
A aJo
therefore whole area = three times area of base-circle.
Ex. 5. To find the whole area of the cycloid.
Let the vertex be the origin ; see fig. 5 ; then the equation to
the curve is x ^
y = a versin- 1 |- (2 ax x z )* }
which expression, as the limit of the definite integral, we shall
represent by Y : then
whole area = 2 OABP
pZa ft
= 2 I dydx
Jo Jo
pa , x
= 2 1 \a versin- 1 - + (2 axx^ \ dx
JQ v- tt J
= 2 fa? { a versin- l ^ + (2 ax xrf } f(2 ax xrf dso\
a (2x-a) versin- 1 -?
aJo
therefore the area = three times the area of the generating circle.
The value of the indefinite integral shews that if x |, the
148 QUADRATURE OF PLANE SURFACES. [ll8.
area of the segment of the cycloid does not involve the length
of a circular arc, or any circular transcendent. Hence if, in
a 3% a 2
fig. 20, OM = -, the area of POM = ^ = the triangle QMA.
6 o
.,. -no? TTO ..
Hence also, 11 x = a, OPSC = 2 + , cs = a + ; therefore
4
a 2
OQDSP = 2 ; and segment OPS = ^, and does not involve any
A
circular transcendent.
Ex. 6. To find the area included between the tractrix, the
axis of y, and the asymptote.
The differential equation to the curve is
Then, fig. 2, taking y to be the general value of the ordinate to
the curve, /* r y
whole area = / / dy dx
Jo Jo
f,<
=
Jo
but yda? = dy(a 2 y-)*; and when #= oo, y = Q; x = 0, y=a;
C Q
.'. whole area = / (a 2
Tid*
By similar processes let it be shewn that
Ex. 7. The whole area contained between the asymptote and
the witch of Agnesi is four times the area of the base-circle.
x z y 2
Ex. 8. If the equation to the hyperbola be = 1, the
a 2 o 2
area included between an ordinate, the axis of x, and the curve is
/*!/ ///) f If* ti **\
" 'it 1 1 l> I H I
-2~ -T^IT + TJ-
Ex. 9. If the equation to the rectangular hyperbola be xy = k 2 ,
the area included between two ordinates, the axis of x, and the
curve, is
I 1 9.] RECTANGULAR COORDINATES.
Ex. 10. The whole area of the companion to the cycloid is
twice that of the generating circle.
Hence the area of the cycloid is trisected by the base-circle
on its axis, and the companion to the cycloid ; see fig. 20.
Ex. 11. The whole area of the loop of the curve whose equa-
tionis
included between xa } and x 0, = .
V
Ex. 12. The area included between the axis of x, two ordi-
nates, and the logarithmic curve, y = a x , is
a x n a x
log e a
and that included between the curve, the asymptote, and the
axis of y is ; -- , since XQ = oo , x n 0.
ige a
Ex. 13. The area OAMP of the catenary in fig. 7 is equal to
the rectangle contained by OA and the arc AP, and therefore is
equal to twice the triangle PIIM.
119.] In all the above examples of integrating dy da?, the
^/-integration has preceded the #- integration, and we have by
this process first determined the general value of a differential
slice of infinitesimal breadth dx, contained between parallel or-
dinates, and by the summation of these determined the required
area. In most cases however the order of differentiation is in-
different, though of course if the order be changed the limits
must be altered : this we shall exemplify in a few cases.
Thus, to determine the parabolic area OAB, fig. 21, where
OA = a, AB = b, and the equation to the bounding curve is
if we first perform the ^-integration, y being the same for all,
?/ 2
we sum the elements along the line PK, that is, from x a^
to x = a; and thereby obtain the area of the slice PQLK con-
tained between two parallel abscissae separated by the distance
dy ; which slices must again be summed with respect to y, the
limits of integration being b and 0. Hence
150 QUADRATURE OF PLANE SURFACES. [l2O.
na
dx dii
^
= / \x\ dy
Jo i JsL y , y
ab
= ab
2ab
the same result as that found in Ex. 1, Art. 118.
Or, again, the equation to the tractrix being, equation (4),
Art. 101,) >-"4
y
and x being the general value of the abscissa to the curve, the
area included between the curve, the axis of x, and the axis of y
=n
Jo Jo
TTCL*
120.] If it be required to determine the area contained be-
tween two ordinates corresponding to x n , XQ, and between two
curves whose equations are
y =/(#), y = <t>W>
the former being the equation to the upper, and the latter that
to the lower curve ; then, as is plain from fig. 22, the y-integra-
tion must be first performed, and between the limits f(x) and
<(#) ; the result of which will give the area of the slice PP'Q'Q;
and the subsequent definite <r-integration will give the sum of
all such slices between the assigned limits ; and this will be the
required area. Thus
121.] RECTANGULAR COORDINATES. 151
C*n /*/(*)
area = dy dx. (6)
J* e ./*{)
If however the superficies, whose area is required, be of a form
such as that delineated in fig. 23, it is more convenient to resolve
it into slices whose bounding lines are parallel to the axis of x,
that is, first to perform the ^-integration, for in such a case the
equations to the curves will give the limits of integration : the
equation to the curve AQPB giving the superior and that to
AQ'P'B giving the inferior limit; which manifestly they do not,
if the ^-integration be first performed ; and in this case, if the
equations to the curves are
and if the ordinates to A and B are y n and y ,
area =-. / / dxdy. (7)
Sometimes also it is necessary to divide a problem of quadra-
ture into two or more parts, and to integrate each of the double
integrals in the order which its form and limits render most
convenient : such division however must be left to the ingenuity
of the student, the principles of the calculus being of sufficient
breadth to include all such cases.
121.] Examples illustrative of the preceding principles.
Ex. 1. To determine the area included between the parabola
whose equation is y z = 4>ax, and the straight line whose equa-
tion is y = fix ; see fig. 24.
The coordinates to the point B, determined by elimination
between the given equations, are OA = ^, AB = ; therefore
area OPB
A
4a
4cz i
/p f8()'
= / / dy dx
A> Jpx
= I *
/o
3
8 a 2
>\TS R<Y>\ fJr
) AJtt/ t \AJtAf
4a
X* ,
152 QUADRATURE OF PLANE SURFACES. [l2I.
Ex. 2. To find the area contained between an hyperbola, its
transverse axis, and a central radius vector; fig. 25.
Let the coordinates of the point p n to which the radius vector
is drawn be # n , y n : and let the limits of x be symbolized by
x,, and x ; then the equation to OP is
x n
x = y = xo,
yn
and the equation to the hyperbola is
therefore area o A ? = / / dx dy
Jo Jxn
ab / y n + (ft 2 + ?)*) x n y n
snce { 2 + y 2 } = *
Now the order in which the integrations have been performed,
and the limits of them, deserve attention ; as the superficies
p n OA admits of being resolved into slices by lines parallel to the
axis of x, the limits of which are given by the equations to the
straight line and the curve, we have performed first the ^-inte-
gration, and subsequently the y- integration ; but the order
could have been reversed, only subject to other conditions : viz.
if we had integrated first with respect to y, the limits would
have been the ordinate to the straight line and zero, for all
values of x from o to A, but at A, and thence on to p, ( , the supe-
rior and inferior limits would have been respectively the ordi-
nate to the straight line and the ordinate to the hyperbola.
122.] RECTANGULAR COORDINATES. 153
The definite integral must have been broken into two parts cor-
responding to these limits.
Similarly let it be shewn that
(1) AP being the catenary, in fig. 7, whose equation is
a r x - _*i
itt 1 pQi I p & L
i ft ft
andoA = a, ON = -T-, area APN = {51og e 2 3}.
~X TO
(2) The area included between a parabola whose equation is
y z ^ax } and a straight line through the focus inclined at 45
1 fi
to the axis of x is a 2 (2)*.
o
122.] A quadrature may often be elegantly and conveniently
determined by means of a substitution, and chiefly by putting the
equation to the bounding curve into simultaneous equations by
the introduction of a subsidiary angle, according to the method
of Art. 166, Vol. I.
Thus, for example, to determine the area of the cycloid whose
starting point is the origin, and whose equations are (see fig. 6)
x = a(0 sin0), y = a(i cos 6).
Let the ordinate to the curve as the limit of the ^-integration
be symbolized by Y : then
whole area of cycloid = 2 area OAB
fira rv
= 2 I dydx
A) Jo
rira
= 2/ Yte;
Jo
and replacing Y and dx in terms of 0, and observing that =
when x = 0, and that = IT when x = Tra } we have
whole area = 2 /""a 2 (1 - cos 0) 2 dQ
= 3ira 2 .
PRICE, VOL. II.
154 QUADRATURE OF PLANE SURFACES. [123.
Again, to find the area of the hypocycloid whose equation is
222
See fig. 10. Let x = a (cos B)*, y = a (sin 0) 3 ,
whole area = 4 area of AOB
ra fY
4/ I dy dx
Jo Jo
fa
= 4 / Y dx,
Jo
where Y is the ordinate to the curve : then replacing Y and dx
77
in terms of B, and observing that = when x = a, B = ^ when
x = 0, we have
whole area = 4/ 3a 2 (cos0) 2 (sm0) 4 d0
J t
rl
= 12 a 2 {(sm0) 4 -(sin0) 6 }e?0
Jo
integrating by means of equation (35), Art. 68.
If the area to be determined and the bounding curve be re-
ferred to a system of oblique coordinate axes, say to a system
of axes whose angle of ordination is &>, then the area of the
element is dx dy sin (a, and
area required = dy dx sin w,
and the process of integration is precisely the same as that in-
vestigated in the preceding articles : it is therefore unnecessary
to discuss it separately.
SECTION 2. Quadrature of Plane Surfaces. Polar Coordinates.
123.] Let it be required to find the area contained between
a plane curve and two radii vectores separated by a finite angle ;
see fig. 26.
Let AP O PQP H be the curve whose equation referred to polar
coordinates is r = f(0}, (8)
and let it be required to determine the area of P O SP M .
POLAR COORDINATES. 155
Let SP O = /*O, SP W = r n , p sA = 0oj v n sA=d n , and let E be
any point within the bounding lines ; draw through E the radius
vector SEP, and also a consecutive one inclined to SP at an infi-
nitesimal angle d0 ; from s as centre and with SE as radius draw
the small circular arc EG, and also another arc at an infinitesimal
distance from it : then, if the polar coordinates to E are r and 0,
EF = dr, EG = rd0, and the area of the element = rdrdQ, the
element being ultimately an infinitesimal rectangle ; then the in-
tegral of rdrdd, with respect to r, between the limits and/(0),
will give the area of the triangular slice SPQ, and the integral
of all such triangular slices between and H will give the area
of the required superficies, and therefore we have the following
symbolization, , 6 ^ ,, /(e)
area sp P n = / / rdrdQ. (9)
J0 Jo
Performing first the r- integration, the superior limit being f(0)
or the radius vector of the curve, and the inferior limit being 0,
we have ^ p
area SP O P M = ^J "{f(0)} 2 d0, (10)
and replacing f(0) by its value r given in (8), r referring to the
curve, j re n
area sp p n = I r 2 d0.
The /--integration therefore gives the area of the sectorial slice
SPQ, which is manifestly equal to r 2 d0; and the whole re-
quired area is equal to the integral of this expression.
And let it not be supposed that any inaccuracy of result arises
from the fact that the element of the area is not rectangular
at the superior limit of the r-integration, that is, at the point p ;
for if two infinitesimal arcs RP, QT are described from s as a
centre with radii SP and SQ, then, if SP = /* =/(0), sQ = r + dr
= f(0 + d0), the area SPQ is intermediate to SPR and SQT ; that is,
,. r*d9 , (r + drYdO ^ ...
is intermediate to ^- and - , the difference between
* <*
which is an infinitesimal of the second order, and must there-
fore be neglected. Hence -
infinitesimal sectorial area.
r 2 d0
fore be neglected. Hence is the correct expression for the
X 3
156 QUADRATURE OP PLANE SURFACES. [124.
124.] Examples illustrative of the preceding.
Ex. 1. To find the area of a sector of a circle ; see fig. 27.
Let the radius of the circle = a, and the arc AB subtend at
the centre an angle a, then
/a ra
area BSA =11 rdrdd
Jo Jo
-i*
1
= q SA x arc AB.
Ex. 2. To find the area of a portion of a circle cut off by
equal chords drawn through a point in its circumference; see
fig. 28.
Let radius of circle be a, and let BSA = B'SA = a; the equa-
tion to the circle is
r = 2 a cos 0;
area BSB' = 2 area BSA
= 2/7
Jo Jo
a /*2acos0
rdrdS
= 2 a 2 {a -f sin a cos a}.
Ex. 3. To find the area of a loop of the lenmiscata whose
equation is
r 2 = a 2 cos 20.
/ /*a(cos20) J
Area of loop = 2 rdrdd
Jo Jo
coa20d0
o
2
a
~2'
Ex. 4. To find the area of the loop of the folium of Descartes ;
see fig. 63, Vol. I.
The equation referred to rectangular coordinates is
x 3 3 axy + y 3 = 0,
POLAR COORDINATES. 157
3 a sin 6 cos 6 3 a tan 6 sec
~ (sin0) 3 + (cos0) 3 l + (tan0) 3 '
let this value of r, which is the superior limit of the first inte-
gration, be represented by r,
nr
rdrdd
,
2 * (tan 0) 2 (sec 0) 2 dO
"
_ 2 r i "I*
ll + (tan0) 3 J
_ So 2
Similarly let the student find the following results :
Ex. 5. If the equation of the cardioid be r = a (1 -f cos 0), the
Swa 2
whole area = ~ .
A
Ex. 6. If the equation to the curve be r = a sin 3 0, the area
2 2
of each loop is -=-^- ; and the area of all the loops = -j- .
Ex. 7. If 4a = the latus rectum of the parabola, the area
contained between two focal radii inclined at d n and to the
least distance is equal to
125.] In all the above examples the r- integration has pre-
ceded the 0-integration ; the effect of which order has been that
the area is resolved into triangular elements with a common
vertex at the pole s ; and the sum of these is determined by the
^-integration ; and the areas, which are ordinarily subjects of
investigation, admit of such resolution : but if the 0-integration
had been first performed, r being constant, it would have deter-
mined the area of a circular annulus, the radii to whose bound-
ing circles would have been respectively r and r + dr, and the
subsequent r-integration would have given the sum of all sue 1
158 QUADRATURE OF PLANE SURFACES. [126.
annul! ; but the areas, which are commonly the subjects of such
processes, do not conveniently admit of being thus resolved, and
the equations of the bounding curves do not commonly yield
convenient values of limits; and therefore, although theoreti-
cally the order of integration is indifferent, yet we choose that
which is practically most convenient, and make the r-integra-
tion precede the 0- integration. The circle, I would observe,
when the centre is the pole, is adapted to both orders with the
same facility, because the limits of the two integrations are
constant.
126.] We proceed to the investigation of areas whose limits
are of a more complex character than those considered above.
Ex. 1. To find the area of a circular annulus, the radii of
whose exterior and interior bounding circles are a and b.
(2r r-a
I rdrdd
-u Jt
= Tr(a 2 -b 2 ).
Ex. 2. To find the area contained between the conchoid of
Nicomedes, its asymptote, and any two given radii vectores ; see
fig. 29.
Let SA = a, AB = PQ = b ; SP = r, BSP = Q ; therefore the equa-
tion to the curve is
also SQ = a seed.
Let 6 n and be the superior and inferior limits of ; therefore
-f
J0Q J(l S
i r e
= H / (2fl6sec
tsecfl+6
area = / / rdrdd
i sec 9
Ex. 3. To determine the area contained between two suc-
cessive convolutions of the spiral of Archimedes ; fig. 30.
Let the general form of the equation to the spiral be r = a<j>,
127.] POLAR COORDINATES. 159
< being the whole angle through which the radius vector has
revolved; and let SA, SB, sc, severally be the values of the radius
vector after n I, n, and n + I complete revolutions, so that
SA = 2(w 1) ira, $B = 2mra, sc = 2(/i + l)7r; let PSA = Q - }
therefore SP = {2(n \}-n-\-6}a, 8?!= {2mr + 0}a } which values
it is convenient to represent by r and r x ; the problem is to
determine the area of APBiBPiCiC, which is expressed by the
following definite integral :
pir /*r,
area = rdrdd
pir /*r,
= /
J* Ji
therefore the area generated in the first revolution of the radius
vector is 87r 3 a 2 ; and hence that generated in the nth revolution
is n times that generated in the first.
If the equation to the curve be given in terms of r and
p, that is, if a differential equation be given, instead of finding
the equivalent expression in terms of r and 0, and then inte-
grating as in the last articles, it is more convenient to pursue
the following course :
Let r dr dd be integrated first in respect of r ; and supposing
the limits of r to be the radius vector of the curve and 0, we
have, 9 and being the limits of 0,
1 C 6
area - / r 2 d0.
*/*
But, by equation (24), Art. 222, Vol. I,
r*M= rpdr (11)
(r 2 - jo 2 ) 2
1 /*' rpdr no .
.'. area = -= I - , (12)
<*Jr ( r 2 p 2)2
r and r Q being the values of the radii vectores of the bounding
curve corresponding to and ; in which expression p must
160 QUADRATURE OF PLANE SURFACES. [127.
be replaced by its value in terms of r, and the r- integration
then performed.
Ex. 1. To find the area contained between the involute of
the circle and two limiting radii vectores ; see fig. 31 .
The equation to the curve is
r 2 p 2 = a 2 ;
c e r
.-. area ASP = / rdrdO
Jo Jo
= !.[>>**
(r 2 a 2 ^
Ex. 2. To find the area contained between an epicycloid and
its base-circle during one revolution of the generating circle ;
see fig. 42, Vol. I.
By equation (9), Art. 219, Vol. I, the equation to the curve is
~ (7
area contained between pole and curve
-a+2& prdr
r(r 2 a 2 )^dr
a + 2b C a
L
a + 2b C a
a J a
which is easily integrated by substituting z 2 for r 2 a 2 ; and we
have
area contained between pole and epicycloid
G/
and as the area of the circular sector which is included in the
above expression is -nab, the area included between the circle
7 o
and the epicycloid is -
Cv
128.] POLAR COORDINATES. 161
128.] The method of the present section is also immediately
applicable to the following problem :
To find the area contained between a curve, its evolute, and
any two limiting radii of curvature.
In fig. 32 let OPQB be the plane curve on which p and Q are
two consecutive points, the coordinates to P being x and y, and
PQ being an infinitesimal arc and therefore equal to ds; let pn
be the radius of curvature at P, and be represented by p : then
the area of the infinitesimal triangle PHQ is equal to
PQXPn = P ds . (13)
and as the required area is the sum of all these, we have
1 C
area = ^ pds, (14)
in which p and ds must be expressed in terms of a single vari-
able, the limits of integration being assigned by the conditions
of the problem.
Ex. 1. To determine the area contained between a parabola,
its evolute, the radius of curvature at the vertex, and any other
radius of curvature.
Let the equation be y 2 = 4>ax;
2 3
then p = -(a + xy ,
a?
/a + a?\* 7
ds = I 1 ax :
V x '
. . area = / dx
TS JQ -v>2
O^
and therefore the area contained between the curve, the evolute,
and the radii of curvature at the vertex and at the extremity of
56
the latus rectum is equal to =-= a 2 .
15
Ex. 2. To find the area contained between the cycloid, its
evolute, and two given radii of curvature.
In fig. 32 let o be the starting point : then
11 ,
x =. a versm" 1 - (2ay y 2 )*,
a
PRICE, VOL. II. Y
162 QUADRATURE OF [129.
p = 2(2ay)*,
t 2a \*,7
tef/-^
. . area beginning at o = 2 a I - -
^o 2a- 2 *
= 2a { (2ay y 2 )* + aversin- 1 - j;
and therefore area OB'B = 27ra 2 .
SECTION 3. The Quadrature of Surfaces of Revolution.
129.] In fig. 33 suppose APQ to be a plane curve, and to
generate a surface of revolution by revolving about a line ox in
its own plane, A'P'Q' being its position when half a revolution
has been performed ; and let the equation to A p be y = f(x) ;
let OM = a?, MP = y, FQ ds', p and Q will, in a complete revo-
lution, describe circles whose radii are respectively y and y -f dy,
and therefore the paths traversed severally by P and Q are 2iry
and 2ir(y + dy): supposing the curve to be continuous and
without points of inflexion between p and Q, the element PQ
will describe a circular band whose breadth is ds, and the cir-
cumferences of whose bounding circles are %Try and 2"n:(y + dy) ;
the area therefore of the convex surface of the band is inter-
mediate to
j
and
neglecting therefore the infinitesimal of the second order, the
convex surface of the infinitesimal band is equal to
and therefore, as it is an infinitesimal band-element of the
G11T 1 ] Qf"*f* /
surface = 2nyds, (15)
the limits being given by the conditions of the problem.
If y = f(x) be the equation to the generating plane curve,
dy = f'(x) dx ; therefore
ds =
and surface = 2 TT ff(x) {1 + (/'(a?)) 2 }* da?; (16)
which is the form convenient in most cases; other processes
will occur in the sequel.
130.] SURFACES OF REVOLUTION. 163
130.] Illustrative examples.
Ex. 1. To find the surface of a sphere.
The equation to the generating curve is a? + y 2 = a 2 ,
.'. yds = a day,
ra
surface of sphere = 4nra dx
JQ
Hence also a zone of a sphere contained between two planes
perpendicular to the axis and at distances x n and X Q from the
centre is equal to ,,
2ira dx = 2ira(# n XQ);
see Vol. I, Art. 24.
Ex. 2. To determine the surface of the paraboloid of revolution.
,
as =
x
surface
= 4nr a* /
JQ
r. 3-1*
[<+*>]
3
Ex. 3. To find the area of the surface described by the revo-
lution of a cycloid about its base.
x = a (6 sin 6),
y a (1 cos 0),
a
ds = 2 a dO sin ^ ;
whole surface
= 2-n- y ds
64 2
Y 2
164 QUADRATURE OP L 1 Si-
Ex. 4. To determine the area of the surface described by the
revolution of the tractrix about the axis of x.
The differential equation to the tractrix is
dy_ _ y
dx (at yi^'
. . y ds = ady ;
.'. whole surface = 2-nyds
r
2Tra
J
o
dy
Similarly let the student prove the following results ;
Ex. 5. The whole surface of a prolate spheroid, the equation
to whose generating ellipse is
f 2 _i_l! - 1
2 + 6 2 "
and whose eccentricity is e, is 2irb 2 + - sin" 1 ^.
Ex. 6. The area of the surface generated by the revolution
of a logarithmic curve y = e x about the axis of x is equal to
IT {y (1 + y 8 )* + log (y+(l +
Ex. 7. The whole area of the surface generated by the revo-
lution of a cycloid about its axis is Sira^yn ~ ) .
131.] If the line about which the generating plane curve
revolves be the axis of y, then, see fig. 34, if o M = x, M p = y,
PQ = ds, the convex surface of the band generated by PQ in one
revolution is equal to %-nxds, and as this is an infinitesimal
element of the required surface,
whole surface = %TT xds, (17)
the limits of integration being assigned by the problem.
Ex. 1 . To determine the surface of an oblate spheroid.
Let the equation to the revolving ellipse be
and its eccentricity be e ; then
132.] SURFACES OF REVOLUTION. 165
whole surface
fy
= 47T/
Ex. 2. To determine the area of the surface generated by a
given length of the catenary revolving about the axis of y, when
the equation is a
y =
Hence by Ex. 6, Art. 103,
a
r*
.-. surface = 2ir x ds
Jo
= 2TT\xssdx\
ta 2 / *
xs -^-(e* + e
X \ ~\
~ a - 2) j
_X
e ~
a(y a)}.
132.] If the curve whose equation is y =/(#) generates a
surface by revolving about, not one of its axes of reference but,
an axis parallel to, say, its axis of as, at a distance a from it, and
in the plane of the curve, then the surface generated
= 2 77 a [*ds + 2ir \ f(x) ds,
Jx JgQ
x and %o being the abscissae corresponding to the extremities of
the generating curve ; and therefore if s be the length of the
166 QUADRATURE OF CURVED SURFACES. [133.
generating arc, and s' be the area of the surface generated by
the revolution of it about its axis of #,
surface required = 2 no, s + s'.
Suppose now that the generating curve is a closed figure,
such as that drawn in fig. 39, and admits of being divided into
two equal and symmetrical parts by a line EEC which is its axis
of x, then, if AB = , and the equation to EPC is y-=f(x), (EC
being its axis of x,} the surface generated by the revolution of
EPC about OX is
{a+f(x)}ds,
and that generated by the revolution of EP'C about the same
line is r Xn
2-rr {a-f(x)}ds;
JXQ
therefore the surface generated by the revolution of the closed
figure EPCP' is rx n
4ntal ds ;
J*t
which is equal to 4<iras; that is, to 2-na x length of the gene-
rating curve ; therefore the area of the surface generated by
the revolution about an axis in its own plane of a closed curve
which is symmetrical with respect to a line parallel to that about
which it revolves, is equal to the product of the length of the
curve and the path described by a point on the line of symmetry.
Hence if a circle of radius a revolves about an axis in its own
plane at a distance c from its centre, the surface of the generated
ring = 4>Ti 2 ac.
SECTION 4. Quadrature of Curved Surfaces .
133.] Let the equation to the surface whose area is to be
determined be
F (x, y, z) = 0, (18)
and let P, fig. 35, be a point on it, whose coordinates are o M = x,
MN = y, NP = ^T. Through P let planes PSLN, PRJN be drawn
parallel to the planes of yz, xz respectively ; and also let two
other planes severally parallel to them be drawn, and at infi-
nitesimal distances dx, dy ; so that NL = dy, N j = dx, and PSQR
is the intercepted infinitesimal element of the surface ; then the
coordinates to Q are x + dx, y + dy, z + dz: and let us imagine
1 34.] QUADRATURE OF CURVED SURFACES. 167
the whole surface by a similar process to be resolved into similar
infinitesimal elements : then the area of one of these having
been expressed in general terms, the area of the surface will be
given by the integral sum of such elements.
For convenience of notation let A represent the required area
of the surface, and dA the area of the element PRQS; and as a
tangent plane to a surface at a given point contains not only
the point but also an infinity of other points immediately con-
tiguous to it, so when dA is infinitesimal it will be coincident
with the tangent plane at p, and therefore the angle between it
and any other plane is equal to the angle between the tan-
gent plane and that plane ; in accordance with the notation of
Art. 279, Vol. I, let a, 8, y be the direction-angles of the normal
to the tangent plane.
Now the projection of d\ on the plane of xy is the rectangle
Similarly, if dA be projected on the planes yz and zx,
dy dz = dA cos a -
! , (20)
dz ax = Acos/3 J
whence, squaring and adding,
dA 2 = dy 2 dz 2 + dz 2 dx 2 + dx 2 dy 2 ,
the sum of which infinitesimal elements between assigned limits
will be the area of the surface ; and as dA involves the product
of two infinitesimals it is a double integral, and we have
rr , i
A- I I J //9/2 fj <yt I.. //a2 ///y2 _! ///y2 /7^/2 1 2 / O.O \
"^ III ^*y **<* T W<fr WcC- -|~ t*cfc *^V f y ^<w* /
which is the general value of the area, and will assume various
forms according to the particular surface.
134.] If v(x,y,z) = c be the equation to the surface, and if
dv \ / ef F \ / C?F \
) = ?' (d-y) = v ' (TZ) = W > (23)
Tj2 + V 2 + W 2 _ Q 2 . (24)
then, by Art. 279, Vol. I, equations (15),
u v w
cos a = , cos 8 = , cosy = :
Q Q Q
168 QUADRATURE OF CURVED SURFACES. I 1 35'
and therefore from (19) and (20)
A = ff-dydz (26)
(27)
(28)
either of the formulae being used, according as it is best suited
to the equation of the surface and to the assigned limits.
135.] If the equation to the surface be given in the form
z f(x, y), then by (22) we have
(|) 2 }W
and ( -T- ) and ( -r- ) having been determined by means of the
y
equation to the surface, we may substitute, and integrating
between the assigned limits find the required area.
The formula (29) may also be deduced from (28) ; for if
F(o?,y,2r) =f(x,y)-z = 0,
idz\ idz\
u = , v =(-=-), w = 1;
\O9' ^dy>
and therefore (28) becomes
Now after substitution by means of the equation to the surface
the quantity under the double signs of integration becomes a
function of x and y : let us therefore consider what the effect
of the successive integrations will be, and whether by a par-
ticular order the problem may be simplified.
Suppose the surface to be closed, and to be such as is con-
tained in the octant delineated in fig. 36; then, since PRQS is
the element of the surface, the effect of a y-integration, x being
constant, will be, the summation of all elements similar to PQ
from L to K, that is, from y = to y=Mn; or the aggregate of
the elements is the band LPK; and as the area of the band will
be expressed in terms of x, and is therefore the general value of
all similar bands, the effect of a subsequent r- integration will
QUADRATURE OF CURVED SURFACES. 169
be, to sum all such elemental bands of which the surface is com-
posed, and the limits of this latter integration must be x = 0,
and x = OA. If therefore MK, as determined by the equation to
the surface, = Y, and OA = a, then, bearing in mind the order of
integration as indicated by the arrangement of the symbols,
C a /' Y T / dz \ 2 / dz \ 2 ") ^
area of the surface = / / > 1 + ( -i- ) + I -r- ) r dy dx. (31)
Jo Jo I \dx> \dy' J
If the ^-integration be performed first, the effect will be to
determine the band GPRJ, and the limits of integration will be
HJ = x and 0; and the subsequent y- integration, with the limits
OB = b and 0, will sum all such bands contained between parallel
planes, and will give the area of the surface. In this latter case
the formula becomes
/ & T x
..(
The above is an outline of the general method of finding the
area of such surfaces : the limits of integration will of course
vary according to the conditions of each problem.
136.] Examples illustrative of the preceding formulae and
principles.
Ex. 1 . The surface of the eighth part of a sphere.
Let the surface delineated in fig. 36 be that of the octant of
a sphere : then, o being the centre,
2 2
=
x /dz
z* V dy
_ a _
+ + ~ == ~ =
and taking formula (31), Y = (a 2 x )*;
C a T Y a dy dx
surface = / / -
^0 ^0 a 2_ <2 ,2_
.. whole surface of sphere = 4-rr a 2 .
PRICE, VOL. II. Z
170 QUADRATURE OF CURVED SURFACES. [136.
Ex. 2. A sphere is pierced by a right circular cylinder whose
surface passes through the centre of the sphere, and the dia-
meter of whose generating circle is equal to the radius of the
sphere; it is required to find the area of the surface of the
sphere intercepted by the cylinder.
Let the cylinder be perpendicular to the plane of xy : then
the equations to the cylinder and to the sphere are
_ ax _
and using formula (31), Y = (# <z- 2 ),
C a r ady dx
surface = 2
o Jo (a 2 x*
2 ,
dx
ft C a . , (ax
= 2a sin- 1
A) (2 _
= 2 a I sin*
Jo
= 2 a a? + a sin- 1
138.] CUBATURE OF SOLIDS. 171
CHAPTER VIII.
CUBATURE OF SOLIDS.
SECTION I. Cubature of Solids of Revolution ; and of
similarly generated Solids.
137.] The Integral Calculus also enables us to determine the
volume, or the quantity of space, contained within given bound-
ing surfaces, and thereby to compare it with the volume of a
cube ; whence arises the name " Cubature of solids." We shall
first investigate the most simple case of the volume contained
within a surface of revolution, and between two given planes
which are perpendicular to the axis of revolution.
Let y =f(x) be the equation to the plane curve bounding the
area, by the revolution of which the solid is generated ; and let
the axis of x be that of revolution; see fig. 33; then, as the ele-
mental area PQNM revolves about ox, it generates a circular'
slice whose thickness is M N = dx, and of whose circular faces
one has a radius MP = y, and the other has a radius NQ = y + dy ;
therefore the volume of the slice is intermediate to
Tiy^dx and TT (y + dy) 2 dx ;
whence, neglecting infinitesimals of the higher order, as is neces-
sary, the volume of the elemental slice is equal to Tty^dx, and
therefore, if x and X Q are the distances from the origin of the
extreme faces,
rx
volume required = 1:1 y z dx (1)
J
138.] Examples in illustration of the preceding formulae.
Ex. 1. To find the content of a right circular cone, whose
altitude is a, and the radius of whose circular base is b.
The equation to the generating line is y = -x;
Z 2
372 CUBATURE OF SOLIDS. [138.
b 2 r a
volume of cone = TTZ x 2 dx
a 2 JQ
itb 2 a
~3~ ;
therefore the volume of the cone is equal to one-third of that of
the cylinder of equal altitude and equal base.
Ex. 2. To find the content of a paraboloid of revolution whose
altitude is a, and the radius of whose base is b.
The equation to the generating curve is
b 2
y* = -x;
. . volume of paraboloid = / x dx
a Jo
2
therefore the volume of the paraboloid is one-half of that of the
circular cylinder on the same base and of the same altitude.
Ex. 3. To find the volume generated by a circular arc re-
volving about the diameter.
x 2 + y 2 = a 2 ,
and let the abscissae to the extremities of the arc be x n and XQ ;
then
Mf n
the required volume = 77 / (a 2 x 2 )dx
Jx
Hence also the volume of a spherical segment
ra
= 7T/ (a 2 x 2 )dx
Jx
+
,
ra
and volume of sphere = STT/ (a 2 x 2 )dx
Jo
= -7T 3
3 17
Ex. 4. Let the student shew that the volume of a prolate
spheroid = ^
1 3 9.] SOLIDS OF REVOLUTION. 173
Ex. 5. The volume formed by the revolution of the cycloid
about its base. /f . a .
x = a (Q sin 6),
y = a (1 cos0) ;
(T
(1 cos d) 3 dO
.,-
Ex. 6. The volume generated by the revolution of the cycloid
about its axis is 020
Tttfl- }
1 \ 2 3/'
139.] If the plane area revolves about the axis of y, then, as
is manifest from fig. 34,
volume = Trlx^dy, (3)
a? 2 dy being expressed in terms of a single variable by means of
the equation to the bounding curve, and the limits being as-
signed by the geometry of the problem.
Ex. 1. To determine the volume of an oblate spheroid.
Volume
Ex. 2. The volume formed by the revolution of the cissoid
about its asymptote.
The equation to the bounding curve is
= 2irf ^(6 2 -
Jo o*
ni& - _
2a-x'
see fig.37; OM = a?, Mp = y, oA=2a, AM = 2a x: therefore the
content of the differential circular slice PQQ'P' is Tt(2a xfdy,
f2a
volume = 2 IT Qaocfdy
JQ
fZa
= 2v (Zax)(2a
J
174 CUBATUKE OF SOLIDS. ['4-
Ex. 3. The equation to the Witch of Agnesi being
9 4 n ~(L &*
y* = 4a 2 ,
the volume of the solid formed by its revolution about the
asymptote (the axis of y) is 47r 2 a 3 .
140.] The surfaces which bound the volumes investigated in
the preceding articles are all surfaces of revolution, and are
therefore generated by circles whose planes are parallel, and
whose radii vary according to a law assigned by the relation
which exists between the ordinate and abscissa of a given curve :
and the preceding method of cubature consists in our summing
by the Integral Calculus the circular slices into which the solid
admits of being resolved : a similar method therefore is applic-
able to solids whose bounding surfaces are generated by curves
or lines moving according to other given laws : the following
cases exemplify the process.
Ex. 1. To find the volume of the elliptical cone, of a given
altitude, which is generated by a varying ellipse moving parallel
to itself and perpendicular to the axis of x, along which its
centre moves, and the semi-axes of which are the ordinates to
two straight lines intersecting at the origin and lying respect-
ively in the planes of xz and of yx.
c b
Let the equations to the lines be z = x, y = x; then,
(f (I
as the area of an ellipse whose semi-axes are a and /3 is ira/3,
the volume of the elliptic elemental slice of the cone contained
between two planes perpendicular to the axis of x, and at a dis-
tance dx apart is ^^ c
ryit /Jfyt .
* l*it. ,
a 2
volume of cone = ^ I a? 2 dx
nbc C x
= ~r *
a- Jo
x being the altitude of the cone.
Ex. 2. Let the generating ellipse move as in the last problem,
and let its semi-axes be the ordinates of parabolas respectively
in the planes of xz and yx whose equations are
141.] CUBATURE OF SOLIDS. 175
volume of element = 4nr(ab)^xdx,
volume of solid = 4 (#)* TT / x dx
Jo
the bounding surface is the elliptic paraboloid.
Ex. 3. The volume of the ellipsoid may by a similar process
4
be shewn to be equal to ^-nabc.
o
Ex. 4. Two quadrants of circles being described from the
origin of coordinates as centre, and in the planes of zx and zy,
a variable square moving parallel to the plane of xy, and having
the ordinates of the quadrants for sides, generates a surface
called the Groin ; it is required to determine its volume.
The equations to the director-circles are, see fig. 38,
x 2 + z 2 = a 2 ,
y* + z 2 = a?,
and volume of square element of solid = (a 2 z 2 )dz,
C a
.*. volume of solid = / (a?z 2 )dz
Jo
3
and as the mode of generation may be continued into the other
1 J
seven octants, the volume of the whole groin = a 3 .
o
Ex. 5. A surface is generated by a rectangle moving parallel
to the plane of xy, one of its sides being the ordinate of a given
straight line passing through the origin and being in the plane
of yz, and the other being an ordinate of a semicircle which
passes through the origin and whose diameter is coincident with
the axis of z : it is required to find the volume of the solid.
Let the equations to the director-lines be
y = az, x 2 = 2azz 2 ;
iraa 3
then volume =
2
141.] If the generating plane area has not the axis of revo-
lution for one of its containing sides, but is bounded by two
176 CUBATURE OF SOLIDS. [142.
curves whose equations are y = f(x) and y = /(#), then the
elemental circular slice is equal to
Tr{(v(x)y-(f(x)y}dx,
and the volume required = TT / {(F(#)) 2 (/(#)) 2 } dx, (4)
the limits of integration being given by the geometry of the
problem.
Of the formula (4) the following is an useful result. Suppose,
as in fig. 39, that the generating plane, which for convenience
sake we will call A, is such as to be divisible into two parts per-
fectly equal and symmetrical by a straight line parallel to the
axis of revolution ox, in the same manner as the closed figure
EPCP' is divided by EC which is parallel to ox: then, if the
equation to E PC be y =f(x), when EEC is the axis of x, and if
AB = , the equations to EPC and EP'C with respect to ox as the
axis of x are severally
= a
and therefore by (4) the volume of the solid generated by the
revolution of EPCP' about ox is
f
}*-(a-f(x)}*}dx
= 4 Tra f(x) dx
= 2-nax A, (5)
that is, the volume of the solid is equal to the product of the
revolving area and the circumference of the circle whose radius
is the distance between the two axes.
Thus the volume generated by a circle of radius a about an
axis in its own plane at a distance b from its centre = 2Tt 2 a z b.
The volume generated by an ellipse revolving about a tangent
at the extremity of the major axis is 27r 2 o 2 6.
SECTION 2. Cubature of Solids bounded by any Curved Surfaces.
142.] First, let position be determined by means of a system
of rectangular coordinates fixed in space ; see fig. 40 ; and let E
be a point within the space whose volume is to be found, and
the coordinates to E be x,y,z; let F be another point within
the space, and infinitesimally near to E, and the coordinates to
142.] RECTANGULAR COORDINATES. 177
F be x + doc, y + dy, z + dz; then the volume of the infinitesimal
rectangular parallelepipedon of which E and F are two opposite
angles is dx dy dz : and the aggregate of all such between the
limits of integration assigned by the problem is the required
volume. If therefore v represents the required volume
v = j jdxdydz, (6)
and the volume depends on the integration of this triple integral
between limits assigned by the geometrical conditions of the
problem.
Let us consider the effects of such successive integrations,
when performed one on the back of another, and the relations
between the limits of integration and the geometrical data ; and,
to fix our thoughts, let us suppose the volume to be such as
that delineated in fig. 36, and contained within three coordinate
planes.
Now dx dy dz is the volume of the infinitesimal parallelepi-
pedon, one angle of which is at E, whose coordinates are x,y,z;
the ^-integration therefore, x and y not changing value, has
the effect of determining the volume of a prismatic column,
whose base is dx dy, and whose altitude is given by the equa-
tions to the bounding surfaces : thus, if the volume be such as
that delineated in fig. 36, and the equation to the surface be
z = /fa y), the limits of the ^-integration will be f(x, y} and ;
and the volume of the elemental column, whose height is N p, is
f(x, y} doc dy : or if the volume to be determined be that con-
tained between two surfaces whose equations are z=fi fa y),
z=f z (x, y}, then the volume of the elemental column is
{/i fa y) ~/2 fa y~) }dxdy;
and similarly whatever the bounding surfaces may be.
Suppose that we next integrate with respect to y ; the integral
expressing the volume is now a double integral, and of the form
v =Jjf(x,y)dydx, (7)
f(x, y) being a function of x and of y, which is introduced by
the limits of z. Taking then figure 36 to express the normal
case, since /(a?, y} dy dx is the volume of the elemental column,
the sum of all such determined by the ^-integration, when x is
constant, is the elemental slice LPKJB contained between two
PRICE, VOL. u. A a
178 CUBATTJRE OF SOLIDS. [143.
planes parallel to the coordinate plane of yz, and at an infinite-
simal distance dx apart. In the case therefore of the volume
being such as that of the figure, the limits of y are M K and ;
MK being the y to the trace of the surface on the plane of xy,
and which may therefore be found in terms of x by putting
z = in the equation to the surface. Or if the volume be con-
tained between two planes parallel to that of xz and at distances
y n , yo from it, y n and y being constants; they are in that case
the limits of y ; and similarly must the limits be determined if
the bounding surface be a cylinder whose generating lines are
parallel to the axis of z ; in all these cases the result of the
^-integration is the volume of a slice contained between two
planes at an infinitesimal distance apart, the length of which,
viz. that parallel to the axis of y, is a function of its distance
from the plane of yz ; and therefore if the length be expressed
by F(a?), r
v = lir(x)dx, (8)
and the volume will be the sum of all such differential slices
taken between the assigned limits. Thus suppose in fig. 36 the
volume contained in the octant to be required, and OA = a, then
the limits of x are a and : or suppose the required volume to
be contained between two planes at distances x n and # from
the plane of yz, then x n and X Q are the limits of x. The fol-
lowing examples however of such cubatures will serve best to
clear up any difficulties which arise from inadequate explanation
of general principles.
143.] Examples of cubature.
Ex. 1. Find the content of a rectangular parallelepipedon,
three of whose edges meeting at a point are a, b, c.
Let the point at which the edges a, b, c meet be the origin,
and let the axes of x, y, z severally lie along them ; then, if v be
the volume, /. ~ 6 ~ c
v = / / / dzdydx
JQ Jo JQ
>
cdydx
= I bcdx
= abc.
I43-] RECTANGULAR COORDINATES. 179
Ex. 2. To find the volume of a right circular cylinder whose
radius is , and whose axis is the axis of x, and whose alti-
tude is b. 2.92
y 2 + z 2 = a 2 ;
and therefore for the volume in the first octant the superior
limit of z is (a? y z )%, which I will call z, and the inferior limit
is ; the limits of y are a and 0, and of x, b and 0.
.. v = 4 x volume in first octant
z
dzdydx
b
JO
'11 fft% n-~\--! n% y
in 1
dx
/& ~2
i O O
Jo & &
Ex. 3. To find the volume of the ellipsoid whose equation is
Hence the limits of integration for the volume in the first octant
' f fjp& 77 ^
are : for z, c < 1 2 ~ 1^ [ > wn i c ^ I ^U ca ^ z > an ^ > ^ or y>
\. a o^ y
a ) ' wn i n I will ca U Y ? an ^ ; for x, a and ;
mz
rf!sr <fy dx
ft/, ra /*Y
/ / (^-
Jo Jo
r a ry(v*y 2 )* Y 2 . . y~]
/ o + IT sm
Jo L 2 2 YJ
Sc r a
-r
6
8C
A a
180 CUBATURE OF SOLIDS. [H3-
Zircb f* ,
v = 5 / (a 2 x 2 ) dx
a* Jo
_ -*T0C I 9 X
~^~ ~ 8 1
4
.-. volume of a sphere = swa 3 .
o
Ex. 4. To find the volume of the Cono-Cuneus of Wallis.
Fig. 41. OA = CB = CE = a; OC = AB = C; and the equation to
the surface is, see Vol. I, Art. 315, equation (89),
C Z 2 2 _ 2/ 2( fl 2_ a ,2).
therefore for the volume in the first octant the limits are : of z,
(cP x^y, which I will represent by z, and 0; of y, c and 0;
of x, a and 0. Therefore
volume required = 4/ / / dz dy dx
Jo JQ ^0
- 4>JT y - (a 2 - # 2 )* dy dx
Ex. 5. To determine the volume contained within the surface
of an elliptic paraboloid whose equation is
y 2 z 2
X _ I _ 4^
a b
and a plane parallel to that of yz, and at a distance c from it.
Hence for the volume in the first octant the limits of inte-
/ b \*
gration are: for z, \4ibx -- y 2 ) , which I shall call z, and 0;
for y, 2 (a a?)*, which I shall call Y, and 0; for x, c and ;
.'. volume required = 4/ / / dz dy dx
Jo Jo Jo
^0
naxdx
o
I45-] RECTANGULAR COORDINATES. 181
144.] In all the above examples, as well as in the general
explanation of the effects of the triple integrations performed
one on the back of another, we have integrated with respect to
z, y, x successively : but the order in which the integrations are
performed is indifferent; at least theoretically, and so long as
the conditions which are requisite for correct integration are
satisfied ; as, for example, the infinitesimal type-element must
not change its sign, and must not become infinite, between the
limits of integration of any of the variable quantities ; but if
these conditions are not fulfilled, we may obtain different results
by changing the order of integration, as M. Cauchy has shewn,
and as we might a priori expect ; great caution must be taken
therefore as to the order in which the integrations are performed
and as to the corresponding limits. Thus, for example,
FT
j-\ j-i
IT
j-i J-\
dydx = -
+ 1 ^y2 fjf^i
t dxdy TT ;
that is, the two results, instead of being identical, differ by Sir;
the reason being that intervenes between 1 and + 1, and
that the quantity to be integrated becomes oo when x = y 0.
145.] Frequently it is convenient to resolve the part of the
plane of xy which is intercepted by a surface into infinitesimal
area-elements in terms of polar coordinates, so that instead of
the base of the elemental prismatic column whose volume is
given in equation (7) being dxdy, it is rdrdQ, see equation (9)
Art. 123; and thus the general triple integral expression of the
volume is ,.,,
v = \ \\dzrdrdQ, (9)
in which however it is almost necessary to perform the integra-
tions in the order indicated by the arrangement of the differen-
tials ; see Art. 117 and Art. 125 ; and the effects will be the
following : the 2- integration will give the volume of a prismatic
column of altitude z, and standing on a base rdrdQ; the sum
of all such columns by the r-integration, being constant, will
be the volume of a sectorial slice, and if the limits of r are r
and 0, the edge of the slice will coincide with the axis of z, and
the depth of it will be r ; and if the limits be other given quan-
182 CUBATURE OF SOLIDS. [145.
titles, it will be a part of such a sectorial slice : and the final
^-integration will give the sum of all such sectorial slices, and
therefore the volume required by the conditions of the problem.
The following examples illustrate the formulae.
Ex. 1. The axis of a right circular cylinder of radius b passes
through the centre of a sphere of radius a, b being less than a ;
find the volume of the solid bounded by the surfaces.
Let the plane passing through the centre of the sphere and
perpendicular to the axis of the cylinder be that of xy ; then
the equations of the surfaces are
or, in terms of polar coordinates, the equation to the cylinder is,
r = b.
For the volume therefore contained in the first octant the
limits are : of z, (cPxPy^ or (a 2 /* 2 )*, which I shall repre-
sent by z, and ; of r, b and ; of 0, ^ and ; therefore
r b
= 8/ r
JQ JO Jo
Ex. 2. Through the centre of a sphere of radius a passes the
surface of a right circular cylinder the radius of whose director-
circle is half that of the sphere : find the volume of the solid
bounded by the surfaces.
If the centre of the sphere be the origin, and the plane
through it and perpendicular to the axis of the cylinder be that
of xy, then the equations of the sphere and the cylinder are
= ax
146.] POLAR COORDINATES. 183
and if the coordinates to an element of the solid are z, r, 6, the
equations to the surfaces may be expressed as follows :
r = acosd;
/*f r a cos 9 /*(a2~r2)4
.-. v = 4/ / / dzrdrdd
JQ JQ Jo
Cl ra
= 4/ /
J o Jo
~~ / (1~-
3 JQ
4 3 fTT 2
~3l2~3
Ex. 3. To determine the volume contained within a surface
of revolution, and two given planes perpendicular to the axis of
revolution.
Let the axis of z be the axis of revolution : then, by Vol. I,
Art. 317, equation (99), the general equation to such surfaces is
and if r be the distance from the axis of revolution of a point
on the surface, 2 __ // \
and if z and ZQ be the distances from the origin of the two
bounding planes, . g -
v = / / / rdrdOdz,
JZ Q Jo JO
r= {/(#)}* being the limit of the r-integration, and the order
of integration being that indicated by the arrangement of the
differentials ; . z
.' v = ,J {f(z)}*dz,
the same result as that obtained above in Art. 137, equation (2).
146.] Again, let position be determined by means of a system
of polar coordinates fixed in space, and somewhat analogous to
those employed in plane geometry ; see fig. 42. Let E be the
point whose position's to be determined. Take a fixed point o
for origin ; and through it draw a plane which we will call the
fundamental plane, and which, to fix our thoughts, it is conve-
nient to consider horizontal; through o draw oz perpendicular
184 CUBATURE OF SOLIDS
to it, and any line ox in it; join OE, and let OE, the radius
vector, = r, and the angle ZOE = 6 ; let ON be the projection of
OE on the fundamental plane; ON is called the curtate radius
vector, and is symbolized by p; let the angle of inclination of
the two planes passing through oz and OE, and oz and ox be
<j>, that is, N o M = $ ; the quantities r, and < are called the
polar coordinates to the point E, and whenever they have given
values, the position of the point E is determined. They are
manifestly equivalent to a system of spherical coordinates ; and,
according to the usual method of determining position on the
surface of the earth, if o be the earth's centre, oz the polar
axis, the fundamental plane that of the equator, and the plane
zox that which passes through Greenwich, then < is the longi-
tude and 6 is the co-latitude ; so that the radius vector describes
a meridian when 6 varies and < is constant, and describes a
parallel of latitude when 6 is constant and < varies.
To compare the above system with a system of rectangular
coordinates. Let the fundamental plane be that of xy, and in it
let oy be drawn perpendicular to ox : let OM, MN, NE be the
coordinates to E ; then, since
OM = x -| OE = r i
MN = y\ N = ' y do)
TJ w r rm? ft r
-> ' 1H \ & U U I/
#ON = <f> J
.*. z = rcosO "I x = />cos< = rsin0cos</> "I ,,
p = r sin 6 J y = p sin = r sin sin J
by means of which formulae an equation may be transformed
from one system to the other.
To determine an element of volume in terms of such polar
coordinates. Let E be the point whose coordinates are r, 6, $,
and H the point whose coordinates are r + dr, + d0, </> + c?$;
and which are thus determined; let the meridian-plane ZOE, in
which E is, be drawn; and let OE be increased by the infinite-
simal element EJ = rfr, and let the radius vector OE revolve in
the meridian-plane through an infinitesimal angle EOF d0 } so
that EF = r^0; then the area EFGJ == rdrdQ; now, let the me-
ridian-plane revolve about oz through an infinitesimal angle
NOQ = d((), whereby the rectangle EFGJ will come into the posi-
POLAR COORDINATES. 185
tion i D H K, and each one of the angular points will have passed
through a distance equal to E i or N Q, neglecting infinitesimals
of the highest orders ; but NQ = ONxe?$ = p^<|> = r sin dcf) ;
and therefore the volume of the elemental parallelepipedon of
which E and H are opposite angular points is r 2 sin 6 dr dQ dty ;
and therefore by integration between assigned limits
volume = r 2 sinedrd6d(p. (12)
The order in which the integrations are to be performed is
scarcely arbitrary ; for the form of the equations to surfaces iu
most cases renders it necessary to integrate first with respect to
r ; and as the infinitesimal expression represents an element of a
pyramid whose vertex is at o, and whose base is an element of the
bounding surface, so does the r-integration produce the volume
of the pyramid, complete or truncated according as the inferior
limit of r is or a given finite quantity : and every solid admits
of being resolved into such pyramids ; but the order in which the
6- and 0-integrations are performed is arbitrary ; and if the 0-
integration is performed next after that of r, it produces a sec-
torial or cuneiform slice of the solid, complete or truncated,
according to the limits of r ; and the subsequent (^-integration
will produce the aggregate of all such slices : or if the (^-inte-
gration be performed next after r it produces a complete or
truncated cuneiform conical slice, the aggregate of all which is
found by the subsequent ^-integration.
The particular values of the limits of course may be different
for every case, and are assigned by the geometrical conditions
of the problem. If the origin be in the interior of a closed
surface the volume contained within which is to be determined,
then, if the equation to the bounding surface be
this value, as the superior limit of the /--integration, it is con-
venient to express by r; whereby, if v be the volume,
v = / / f r r z sm6drd0d(l) (13)
JQ -A) ^o
(14)
'o
the labour of determining which definite integral between the
PRICE, VOL. n. B b
186 CUBATTJRE OF SOLIDS.
given limits will frequently be much diminished by the con-
siderations explained in Art. 82, Theorem VIII.
147.] Examples illustrating the preceding formulae.
Ex. 1. To determine the content of a sphere.
Let the radius of the sphere = a ;
pn /V [a
.-. volume = / / / r 2 sin0drd0d(f)
/0 JQ JQ
03 T27T fw
= ^ 8il
o Jo Jo
2 a 3 p*.
= -Q-/ ^>
o Jo
Ex. 2. To determine the general expression of the volume
included within a conical surface and a given base.
Let the origin be at the vertex of the cone, and let the axis
of z fall within the conical surface ; then, by Vol. I, Art. 301,
the equation to the surface is
;-/(;)'
which equation, in terms of polar coordinates given by equa-
tions (11) above, becomes
tan 9 sin <j> = /(tan cos <), (16)
and which let us suppose to be put in the form
tan0 = F(sin$); (17)
suppose the required volume to be contained between the conical
surface, and a plane perpendicular to the axis of z at a distance
c from it ; then the limits of the r-integration are c sec and ;
of the ^-integration, tan -1 F (sin</>), which I shall call , and 0;
and of the (^-integration, 2 TT and ; so that
/Zit /* /*
JO JO
c sec 6
r 2 sin dr d0 d<j)
g / % 2ir /*
= ^ / / tan (sec 0) 2 rf0 rf0
O Jo '0
(F (sin <^>)} 2 c?^>'; (18)
1 47-] POLAR COORDINATES. 187
but by (17) c F (sin <f>) = c tan 6 = the radius vector of the plane
base of the cone drawn from the point where it is pierced by
the axis of z ; therefore by equation (10), Art. 123,
c 2 /* 2jr
/ (F (sin </>) } 2 d<f) = area of base of cone ;
A JO
. . v = - x area of base of cone ;
o
and therefore the volume of a cone is one-third of its base mul-
tiplied into its altitude.
Ex. 3. The vertex of a right circular cone is at the centre of
a given sphere ; it is required to find the volume of the inter-
cepted space.
Let the axis of z be the axis of the cone ; and let its equa-
tionbe
or in polar coordinates tan 9 = - ,
a
and let b be the radius of the sphere,
/*27T Aan- 1 ^ /*&
v = / / / r 2 sin0drdOd<t>
JQ JQ JQ
tf ft, /-tan- 1
= fL / / * a
OJQ JQ
-.tan- 1 !
COS i
Jo
2irb 3
f
~\
/ '
B b 2
188 MULTIPLE INTEGRALS.
CHAPTER IX.
ON THE GENERAL PROPERTIES OF MULTIPLE INTEGRALS, AND
THEIR TRANSFORMATIONS.
148.] THE geometrical investigations of the last two Chapters
have led to the creation of double and triple integrals, and
herein to a large extension of the Integral Calculus. The course
which we have hereby adopted is that best suited to a didactic
treatise, being the one along which the mind of the inquirer is
led from the more simple to the more complex case : now how-
ever the subject requires discussion in its full extension; and
therefore I propose to generalize what has heretofore been said,
and to consider the theory of multiple integrals in their general
form. We shall also have to examine the process of transforming
them into their equivalents in terms of new variables, and the
conditions subject to which such transformations may be made ;
the expression of such integrals will hereby in many cases be-
come simplified ; and sometimes a judicious transformation will
enable us to perform an earlier integration, and thus to reduce
the order of the multiple integral.
The symbolization which I shall adopt is the following : let
there be n independent variables, x\, x^,...x n ; and let the inte-
gral be expressed by
. / u dx n dx n _i . , . dx^ dx\ ; (1 )
the integration being performed first with reference to x n , next
with reference to <r ;i _i, and so on, until lastly it be performed
with reference to x\, the principle of the arrangement of the
symbols being that explained in Art. 117, and where
u = v(x l ,Xz,...x n );
and if the integral be definite, the expression for a definite mul-
tiple integral will be
rx, rx 2 /-x
At, J^ J* n
(2)
I49-] TRANSFORMATION OF MULTIPLE INTEGRALS. 189
u being, as before, a function of all the variables; and where
X w x n are, or may be, functions of all the variables except x n ;
X w _i x n _! are, or may be, functions of all the variables except x n _-^
and x n ; and so on ; and lastly, Xi Xi are constants : and the in-
tegration with respect to any one variable is performed of course
on the supposition that all the other variables are constant.
Another arrangement of the symbols is the following, and the
order of integration is plainly indicated by the form in which
the variables are placed :
/ dxi I dxi ... u dx n : (3)
but as this arrangement does not so clearly express the geome-
trical conception in which the subject of multiple integrals has
been introduced I shall invariably employ the preceding form.
As of single integrals, so of all multiple integrals, it is a ne-
cessary condition that the infinitesimal element-function should
not change sign or become infinite for any value of the variables
between the limits. Also the order in which the integrations
are successively performed is usually indifferent ; but in the
case of a definite integral, if the order be changed, the several
limits will have to be changed also, except in some cases where
the limits are constant.
The course which I now propose to take is, first to investigate
some of the more simple cases of transformation of such inte-
grals, and chiefly those to which the geometrical researches of
the last two Chapters have led us; next I shall consider the
subject in its most general form ; and then investigate certain
new systems of coordinates by means of which multiple inte-
grals become simplified ; and lastly discuss some of the more
simple forms of definite multiple integrals which occur in the
problem of the quadrature of the surface of the ellipsoid.
SECTION 1. Transformation of Multiple Integrals.
149.] Examples of transformation of multiple integrals.
Ex. 1. To transform / / dy dx into its equivalent in terms of
r and Q, having given x = r cos 6, y = r sin 0.
190 TRANSFORMATION OF MULTIPLE INTEGRALS.
Differentiating we have
dx = dr cos 6 r sin dd )
r ; (4)
dy = dr sin -\- r cos Q dd J
but since x is constant during the y- integration, dy must be
calculated on the supposition that dx = ; hence we have
dy = dr sin 6 + r cos 6 dd
= dr cos r sin d0 ;
and therefore by elimination of dr,
dy = - - dd ;
cose
r
hence dy dx becomes - -d9dx;
COS0
but as the very existence of this last expression requires that 6
should not vary when x varies, dx must be replaced by dr cos 6
from (4) ; and therefore
jjdy dx = ffrdr dB; (5 )
If the limits of the given double integral are assigned, the re-
sulting double integral will also be definite, the limits being
determined from the equations connecting x, y, r and 0.
It will be seen on referring to Chapter VII, Articles 117 and
123, that equation (5) proves the equivalence of the expressions
for the infinitesimal element of the area referred to rectangular
and polar coordinates respectively.
Ex. 2. To transform / j e-(* 2 +.^) dy dx into its equivalent in
terms of polar coordinates.
By the last example dy dx = rdr dd ; and also x z + y 2 = r 2 ,
W + M dy dx = \e~* rdr dQ.
Hence if the limits of x and y are oo and 0, those of r must
be QO and 0, and of 0, = and ; and therefore
f
(00 / /* /*
/ e-^+Mdydx = / / e~ rZ rdrd0
.'0 .'0 ^0
77
4*
I49-] TRANSFORMATION OF MULTIPLE INTEGRALS. 191
Ex. 3. To transform F(X, y) dy dx into its equivalent in
terms of f and ry, having given a?=/i (, ij), y =/ 2 (f, 7;).
Hence calculating efe on the supposition that y does not vary
and that therefore dy = 0,
dx\ idy\ tdx\
and therefore, for the reason given in Ex. I, when dx = off 0,
and d y = ( tdr ]) (8)
and replacing ^ and y in terms off and rj, so that F (#,?/) = ^(f,^),
we have
If the order of the integrations in the original double integral
had been reversed, the final result (10) would have been effected
with a contrary sign; but as we are calculating an absolute
value, the sign is immaterial.
i I f I /7 z \ t ft \ *\ *
Ex. 4. To transform / Idacdyl 1+ \-r~] + (-T-) [ , having
given
The form of the expression implies that z is a function of x
and y ; hence by substitution it becomes a function of f and r) ;
>dz
(11)
dz\ _ /dz\ d idz\ drj
ff.T' \f/,f' (J,,V \ff,n' f/,,T
(dz\ (dz\ d (dz\ dri
\dyt " \d* dy \dri' dti
192 TRANSFORMATION OF MULTIPLE INTEGRALS. [149.
Now from (7)
and similarly
dx
drj
dy_
/dy
dx /dx\ idy\ idx\ ldy\
V7/J \*V ~~ \dff VrfJ -
(12)
and similar values are true for -5- . and - : substituting which
dy dy
in (11) we have
/dz
dz\ (dy\ (dz\ idy
~dV \dn> ~ \dn) \~dt
] (-] ( dx .\ (ty.}
dt) \dr}> ~~ \dJ \dt>
T) \dt> \dr]<
(dz\ idx\ (dz\ fdx\
\dJ \di) ~ \dt) \dJ
*dy' idx\ tdy\ /dx\ (dy\
and as dx dy is given by equation (9), we have
(13)
Ex.5. To express I dzdydx in terms of r, ^ <^>, having given
z = r cos -N
y = rsin^sin^ W, (15)
x = rsin0cos$ ^
the latter expressions being those of rectangular coordinates in
terms of polar coordinates ; see Art. 146 : to calculate dz, dx
and dy must be equal to zero,
dz = drcosd rsinOdO
= dr sin 6 sin < + r cos sin QdO + r sin cos < d$ > ; (16)
= dr sin 6 cos </> + r cos cos dd r sin sin <f>d<f>J
1 49.] TRANSFORMATION OF MULTIPLE INTEGRALS. 193
whence dz - - ; (17)
COS0
and therefore when dz 0, dr = ; and therefore to calculate
dy, dx and dr are equal to zero : whence we have
dy r cos 6 sin < dO + r sin cos $ e?0 -\
= r cos cos (pd0 r sin sin < rf< j '
r sin
.-. ay = - - #$,
COS0
and therefore d<f) = 0, when dy = Q; hence to calculate cfo?, e?r=0
(19)
= J U
r 2 sin 6 dr dO d$. (20)
Let this result be compared with Articles 142 and 146.
A shorter mode of effecting the same transformation is given
in Gregory's Examples on the Differential Calculus, Chap. Ill,
Sect. 2, Ex. 9 ; but the above process is better suited to the
illustration of transforming multiple integrals.
By a similar process may we transform into its equivalent in
terms of r, 6, and <p a triple integral of the form
1 (x, y, z} dz dy dx. (21 )
In the case of the bounding surface being one of revolution,
the triple integral (20) may be easily reduced to a single inte-
gral : for suppose the axis of revolution to be that of z, then
the equation to the surface is independent of <, and the integral
becomes r r
2 IT/ Jr 2 sinedrd6;
and if r sin = p, and r cos = z, then rdr dO = dp dz, and the
integral becomes r r r
2-rr / pdpdz = 77 p 2 dz,
which is plainly identical with equation (1), Art. 137.
Ex. 6. The equation to a surface being F (x, y, z) = c, and
u, v, w being the partial differential coefficients of F (x, y, z), it is
required to express in terms of polar coordinates as given by
equation (15) the double integral
dy dx,
where Q 2 = u a +v 2 + w 2 ; see Art. 134, equation (28).
PRICE, VOL. II. C C
194 TRANSFORMATION OF MULTIPLE INTEGRALS. [149.
By equations (11) Art. 146, we have
X ^^ 7* COS U j 30 ^^ p COS CD J /of*\
r > * C*^^/
p = r sin J y = P sm < J '
and by the method of Ex. 1, dy dx = pdp d(p.
By substitution, the equation to the surface becomes
p(p,<i, z) = c,
^ * * i j/j i\ji/j
v p/ d7 V a0'' a?
(dF\ idv\ sin <6
^ cos^-f^-j - Z 23)
ap' v aa>/ n
' " dy
according to the process of Ex. 1, Art. 93, Vol. 1.
fd~F\ (d?\ (d\ sin
Simdariy w = ( s ) = (^) cos - y __
and
and substituting (26) in (23) and (24) we have, squaring and
adding, , o rf al d 2 x _
: W + (del ^ + IjJ r 2 (sin^ ;
and therefore the double integral becomes
/^F\ 2 2.
"'
, .
/dv\ sm
cos 9 ( -r-
^rftf r
. , ,. . m ,.
r sin 6 d (r sm 6) d<b. (28)
And if the equation to the surface be given in the explicit form
r=f(0,(p), so that
then the integral becomes
2 2 (29)
150.] ON ELIMINATION. 195
If the surface be one of revolution round the axis of z, the
equation to it is independent of <, and therefore (29) becomes
and which, if the surface of the whole solid be required, admits
of being reduced to the single integral
2 TrJ{(rd6) z + c?r 2 }* r sin 6. (30)
This result is manifestly identical with that given in equation
(15), Art. 129.
150.] Before we consider the problem of transforming mul-
tiple integrals in the most general case, it is necessary to make
some observations on the forms which are assumed by results
arising from elimination of variables from linear equations.
Suppose that we have the following system of linear equa-
tions involving n variables :
=
a 2 < v l + ^2 -^2 + + **2 %n ,
(61)
a n x l + b n x 2 + ... +r n a? n = 0-
the result of the elimination of the variables is an expression of
which the right-hand member is zero, and the left-hand member
consists of a series of terms each of which is the product of n
factors, each factor having a different suffix, and the number
of the terms is 1 x2x3x ... x(n l)xw, or is equal to the
number of permutations of the suffixes taken n and n together
and the terms are formed from the coefficient of a?i x 2 . . . x n in
the product of the given equations by assuming the normal
term, such as a^ b z c 3 . . . r n , to be positive, and applying to every
other term a positive or negative sign according as it involves
an odd or an even number of quantities which are different from
those in the assumed normal term.
Thus, suppose that there are three equations ; then, assuming
i6 2 C3 to be the type-term, the result is
*
* For a proof of these results of elimination, I must refer the reader to
Peacock's Algebra, Vol. II, Chap. XLIV, Cambridge, 1845 ; and to a paper
by Mr. G. Boole in the Cambridge Mathematical Journal, Vol. IV, p. 20.
C C 2
196 TRANSFORMATION OF MULTIPLE INTEGRALS. [151.
As it is convenient to have a distinctive symbol for the result
in the most general form, and when the number of equations
is n, we shall express it by 2 . + i b- 2 c 3 . . . r n ; and therefore the
result of the elimination of the variables from (31) is
2.M 2 c 3 ...r n = 0; (32)
which result, in accordance with Mr. Boole's suggestion, I shall
call the "final derivative" of the set (31) and shall symbolize by EJ.
Suppose again that we have a set of simultaneous linear equa-
tions of the form
n = 0) 2
then by elimination, as is shewn in Peacock's Algebra,
fOA^
(34)
the numerator of the fraction differing from its denominator in
having w with its suffixes in place of the coefficient (with the
same suffixes) of the unknown quantity whose value it expresses.
And results similar to (34) are true of the other variables.
And in (33) if &> 2 = o> 3 = ... = &> = 0, then
151.] We proceed with our problem. Let the given multiple
integral be of the nth order, and be
s = ... F (xi, %%,... x n } dxi dx z . . . dx n) (36)
and let A = /i (f i, &, n)
^2 = /2 (^1, 2, ... n)
^n =/n(l, 2 ,...?)
whence, using an abbreviated notation,
151.] TRANSFORMATION OF MULTIPLE INTEGRALS. 197
where ai,a^,...a n ,...r n are the partial derived functions of Xi,% 2 ,..*
x n . Now, wh en x\ varies, dx- 2 = dx$ = . . . dx n = ; and therefore
w - l (39)
Ondi + b n df z + > + r n d n =
and therefore by (35)
Again, in calculating dx 2 , dx\ = dx^= ... =rfir n =0, and there-
fore by (41) ^ = 0; hence equations (38) become
And proceeding in a similar manner to calculate dx$,.
we shall ultimately find (q being the letter preceding r)
<? B rffs+ ... 4-r B df = J
whence, as before.
' .
(48)
n _i
**-i= 'r2*dt-i, (44)
2. r n
dx n = S.+ r n rff n , (45)
continuing the same systems of symbols, although of course
2.r n = r n .
Hence substituting (41), (43), . . . (44), (45),
dxidxz...dx n = '2. + a,ib' 2l ...r n .didi...d n ; (46)
and in F (xi, <z? 2 , . . . #), replacing x\, a? 2 , . . . x n by their values in
terms of 1, 2 , n> and symbolizing the function thereby ob-
tained by 3> ( x , 2 , ... n ), we have
//.../ F(^l, #2, #w) ^1 C^2 .- <^n
...^ n . (47)
From (46) therefore it follows that
The product of the differentials of the old variables is equal
198 TRANSFORMATION OF MULTIPLE INTEGRALS. [151.
to that of the differentials of the new variables multiplied by
the derivative of the set of equations formed by differentiating
the equations connecting the two sets of variables. The sign of
the final derivative is ambiguous, inasmuch as it depends on the
order in which dgi d 2 ... d n are introduced ; and this is of course
indifferent.
In illustration of the above rule let us consider the following
examples :
Ex. 1 . To transform into its equivalent dx dy, having given
x = r cos 0, y = r sin ;
.-. dx = drcosOrsinddO )
r ; (48)
dy = dr sin -j- r cos d6 j '
the derivative of which two equations is
r (cos 0) 2 + r (sin 0) 2 ;
.-. dxdy rdrdd.
Ex. 2. To transform into its equivalent dxdydz, having given
x=lr, y = mr, z=nr; l,m, and n being subject to the condition
By a process similar to those above, we have the following
results :
r 2 dr dm dn r 2 dr dn dl r 2 dr dl dm
dx dy dz = - - = - - = - .
/ m n
Ex. 3. To transform into its equivalent z dy dx, having given
se == a cos a, y = b sin a cos ft, z = c sin a sin ft.
dx = a sin a da,
dy = b cos a cos (3 da b sin a sin ft dft ;
therefore the final derivative is a b (sin a) 2 sin ft ;
.-. zdxdy = a^c(sina) 3 (sin ft) 2 da d ft ;
the integral of which expression between certain limits is the
volume of the ellipsoid.
From the given equations we have
and if for convenience we represent -(a 2 # 2 )* by Y, then
tt
TRANSFORMATION OP MULTIPLE INTEGRALS. 199
fa r-r f y& y2 -\ 4
volume of ellipsoid = 8/ / c-Il = ^\ dy dx
Jo JQ (, a 2 b 2 }
ro ro
= SJ^J^abc (sin a) 3 (sin /3) 2 da dp
4
3^
Ex. 4. Given x = i x -f b y' + c : z ~\
y - 2 ^ zy ^ 2 ~ r [ '
z = a$x +b 3 y +c$z J
where i 2 s are the nine direction-cosines connecting two
systems of rectangular coordinates ; it is required to prove that
dx dy dz = dx'dy'dz.
152.] * Suppose however that the n equations connecting
x\ x. 2 . . . x n 1 2 in are implicit, and of the forms fi 0, f z = 0,
.../ n -_0; then, making convenient and obvious substitutions
for the partial derived functions, we have
n =
,..-{-p n d n = O
and to calculate ?o?i, dx^ dx$= ...= dx n = 0, and we have
= ii-i
l = OzdXi I ft
(DUj
whence by elimination
a . ../); (51)
fi (52)
Substituting for <TI and for c?^ in the given differential expres-
sion, and eliminating 2 B n by means of the given equations,
* A full discussion of the theory of the transformation' of multiple integrals
will be found in a paper on the subject by M. Catalan, in the Memoires
couronnes par TAcademie de Bruxelles, Tome XIV, 1839, 1840.
200 TRANSFORMATION OF MULTIPLE INTEGRALS. [153-
the differential expression becomes a function of 1, x z , x$,..x n ;
and therefore in order to calculate dx%, dgi = da?3= ... = dx n = ;
and therefore equations (49) become
(53)
and therefore v .
* = - f^r 1 ^*- < 54 >
2. + 6ia 2 ...p n
And proceeding by processes similar to the above
2. + 0i# 2 .../3 w , ,_>,.
<te = ~ =^= -T- - c?^n ; (55)
2,.
and therefore by substitution
Which result may be expressed as follows : To transform any
multiple integral /// d#i ^2 ^ into its equivalent in
terms of 1 2 n, when equations are given connecting the
two sets of variables, differentiate the equations relatively to
#1 #2 %n, and to fi2"-n separately; let EI and E 2 be the final
derivatives obtained respectively by eliminating dx\dxi,..dx n ,
did 2 ...dg n , then
T*o
dx\ dx 2 . . . dx n di d 2 ... dg n . (57)
EI
153.] Examples illustrative of the above. *
Ex. 1. Let #1 = rcos0!
x z = r sin 61 cos 2
#3 = r sin 61 sin 2 cos 3 }> ; (58)
# = r sin 0! sin 2 . . . sin n _i J
whence, squaring and adding,
= r 2 ; (59)
* For much of the present Chapter I am indebted to papers of M. Catalan
inserted in Liouville's Journal, and to some papers by Jacobi in Crelle's
Journal.
1 5 3.] TRANSFORMATION OF MULTIPLE INTEGRALS. 201
which equation we shall find it convenient to use instead of the
last of group (58) ; differentiating with respect to x\, a? 2 , ...x ny
dxi + +...+ = ~]
+ dx> + ...+ = 0^ . (6Q)
of which the derivative is, 2a? n = E!.
Again, differentiating with respect to r, 61, ... n _i,
2rdr =
cos 6 1 dr r sin : dd : =
= !>, (61)
rsin0 1 sin0 2 ...sin0 w _ic$ w _ 1 = 0^
of which ( ) n - 1 2r w (sin0 1 ) M - 1 (sin0 2 ) n ~ 2 ...sin0 n _i = E 2 is the
final derivative, and therefore
If n = 3, we have the usual transformation from rectangular to
polar coordinates in three dimensions, and
dx\ dx 2 dx$ = + r z sin d\ dr dd\ dd 2 .
Ex. 2. Let the equations of transformation be
_ _ S~l f \
In differentiating these equations for x\, % 2) ...x n , we have
dxi = dx z = ...= dx n ; therefore E X = 1 : and differentiating
them with respect to 1 2 ... f w , we have
5 =0
-fiMfo =0 ^, (64)
the final derivative of which is ( ) M fi n ~ 1 f2 n ~ 2 ...fn-i = E a;
. . dx l dx 2 ...dx n = (- ) n &"- 1 f a n ~ 2 . . . f n-i ^i ^2 . . ^n- (65)
PRICE, VOL. II. D d
202 TRANSFORMATION OP MULTIPLE INTEGRALS. [154.
Hence we have the following transformation, when n = 2,
+l (l-ti) m (I-&rdtidfa (66)
154.] If the original integral be definite, the new one will be
definite also, because its limits will be fixed by means of the
limits of the original integral and the equations of relation.
Each problem will depend on its own circumstances, though the
principle is the same in all : to exhibit it, however, let us con-
sider the following case, which is of some importance :
Suppose x\x^...x n to be n independent variables of the ori-
ginal definite integral, and x'lXj, x' 3 x 2 ,...x' n x n to be the limits
of integration, and let
/x'j /*x' 2 r
= / / .../
^Xj */X 2 /X M
and suppose the integral to include all systems of values of
%i #2 x n which render negative the function f(x\ } #2, #) ; so
that x' M x n are functions of x n _\ . . . a?i ; x' B _i x n _! of x n _ 2 ...x^,x\' t
x' 2 x 2 of x\\ and x'lXx are constant. And suppose 1,^2
to be the new variables, connected with the old variables by
equations of relation, and let the new integral be
/*B'j rS' 2
u= / ...
./H Jn
suppose the equation which determines the limits to become
<i> (i> 2, n)> so that the integral includes all systems of values
f b 2? which render (f l5 2 , . . . ) negative ; then the limits
must be determined as follows :
Observing the order of the successive integrations as indi-
cated by the order in which the variables are arranged, H' M and
E,, are functions of n _i...f2>i an( i are determined by putting
<j) = 0, for thus will be included all values of , 4 which render <
negative. By this process the new element-function will involve
-i z,i '} an d the limits of integration must include all values
of the variables for which n is not impossible. To determine
this condition, let it be remembered that the roots of an equa-
1 54.] TRANSFORMATION OF MULTIPLE INTEGRALS. 203
tion pass from being real to being impossible, through being
equal ; the limits therefore of the new integral will be given by
the conditions which render equal two at least of the values of
n as found from = 0; that is, by the equations = 0, and
-7T- = 0. From these two equations therefore if we eliminate ,
n
we shall have a resulting equation of the form <t>i( n -i,---2,i) = Q,
from which E' M _i H,,_i are to be determined, because they are
the values of _! which satisfy X = 0, for thus shall we include
all values of n _i which render negative the expression 0i ; and
similarly by eliminating g n _i between the equations fa = 0, and
, = 0, will the limiting values of n _ 2 be determined ; and so
fn-i
on for all the other variables.
Hence we have the following method of determining the
limits : those for are determined by solving for the equa-
tion = 0; those for _! by solving with respect to n _i the
equation which results from the elimination of between $ 0,
and -jj- = 0; those for w _ 2 by solving with respect to n _a the
0C
equation which results from the elimination of and n _i be-
tween = 0, = 0, and . = ; and so for the others.
d* d n -i
If the equations = 0, 0i = 0, and the others deduced in a
similar manner, give only two values of g n , re _i, ... for the limits,
then the case is free from difficulty, and the new definite inte-
gral assumes the form in which it is written above : but if any
of these equations has more than two roots, and if there is
nothing in the conditions of the problem which excludes them,
we must resolve u into a series of integrals according to the
roots, and the limits of the several integrals will always be
given by the equations found as above. *
Thus, for instance, in calculating the volume of the ellipsoid,
we must include all values of the variables which render negative
Let the z-, y-, x- integrations be performed in the stated order :
* See a Memoir by M. Ostrogradsky on the Calculus of Variations in the
Memoirs of the Imperial Academy of St. Petersbourg, Vol. I, 1838, p. 46,
D d 2
204 CURVILINEAR COORDINATES. [155.
{ X^ t/^^* ( X^ 1/^1 Z
then the limits of z are cjl -- 2 ~~~ 72 f an( * ~~ c {* -- 2 ~ 72 } '
the limits of y found by eliminating z between / = 0, and
, Jf\ , y
l-j!-}=(), which latter condition gives = 0, are #|l --
9 JL
f X \ ^
and b \\ -- ^ f > an d tne limits of x found by eliminating
and y by means of/=0, - = 0, and - = 0, are #= +a,
and #= a.
SECTION 2.* Investigation of properties of general systems of
Curvilinear Coordinates, and illustrations of preceding results
by means of them.
155.] Suppose that we are discussing the properties of a
curved surface whose equation in terms of rectangular coor-
dinates is \ rk /c<v\
F (x, y } z) = ; (67)
and suppose that x, y, z are connected with two new variables
and 77 by means of the equations
x = /i (f , >?) "I
2/=/2&>?) k (68)
*/!(& 9)
whence by differentiation we have
drj -i
L, (69)
J
a\ y at, a 3 , bi, b 2 , b 3 being convenient symbols for the partial de-
* The matter of the early part of the following section is in a great measure
taken from Gauss' celebrated Memoir entitled " Disquisitiones generales circa
superficies curvas," which is contained in Vol. VI of the Memoirs of the Royal
Society of Sciences of Gottingen, 1828; it has been reprinted as an Appendix
to Monge's Application d* Analyse &c., edited by M. Liouville, Bachelier,
Paris, 1850. A French translation of it by M. A. has been published by
Bachelier, Paris, 1852. Also the student desirous of further information may
consult a profound paper of M. Ossian Bonnet on the General Theory of Sur-
faces, in the Journal de 1'Ecole Polytechnique, Cahier XXXII, Paris, 1848.
156.] GAUSS' SYSTEM. 205
rived-functions of x, y, and z \ so that the equation to the surface
becomes of the form = Q (7Q)
Now, as and 77 are entirely arbitrary, we may choose such
values for them as are most convenient to our purpose. Imagine
then two systems of curved lines to be described on the surface,
of which one is formed by the continuous variation of when 77 is
constant, and the other by the variation of r t when is constant :
thus suppose rj to be constant and f to vary ; then from the eli-
mination of by means of the three equations (68) there will
arise two equations in terms of x, y, z, and 77, which will deter-
mine the curve on the surface ; and the variation of 77 will give
a series of such curves : on the supposition also of being con-
stant and 77 varying another line will be traced on the surface,
and similarly may a series of such lines be drawn : and thus, if
77 is constant, the line given by the equation = Ci is a member
of the first system, the other members of which arise from the
variations of Ci ; and similarly as to the second system. Now,
as the systems are continuous, every point on the surface will
be at the intersection of two curves, one of which is a member
of the first, and the other is a member of the second system :
and the point is determined whenever such curves are given.
Suppose XQ yo ZQ to refer to a point on the surface, corresponding
to which the values of and 77 are and 770 ; then it is conve-
nient to consider the intersection of o and 770 to be the origin,
and any other point to be at the intersection of two lines cor-
responding to and 77; and and 77 may fitly be called the
curvilinear coordinates to that point.
156.] Conceive now two points on the surface which are
infinitesimally near to each other : the first to be at the inter-
section of and 77, and the second at the intersection of + dg
and 77 + dr], and let ds be the distance between them ; then, sub-
stituting as follows,
E =
P = ,,.
G =
V 2 = (0 2 3 03 02> 2 + (3 0i i *3) 2 + (i a a 0i) 2
v 2 = E G
whereby we have 2 = _
206 CURVILINEAR COORDINATES. [ J 57-
and therefore by means of (69)
ds z = dx 2 + dy 2 + dz 2
= Ed 2 + 2vddr] + Gdr?, (73)
where ds is the infinitesimal length-element of a curve on the
given surface ; so that if s is the length of a curve described on
the given surface between the limits 1 ^ and 7o
s = f (E dp + 2 F d dr] + o drf}?, (74)
Jo
1 and being the subscript letters of, and so conveniently used
as abbreviated expressions for, the limits of the integral.
157.] Suppose that we pass from the point (, 77) to the con-
tiguous point ( -f dg, 77) ; then, if da be the distance between
these points, it is manifest from (73) that
do- = E*df. (75)
Similarly, if dv be the distance between the points (, 77) and
tf>f + ^>' d</=e*dr,. (76)
Hence also by means of (73) it follows that if o> be the angle
contained between the two curves corresponding to and 77 at
their point of intersection,
F
cos w = - -. (77)
{EG}*
Or, observing from (69) that a^ a 2 a 3 , bi b 2 b 3 are severally pro-
portional to the direction-cosines of the two curves , rj at their
point of intersection, we have
cos co = (78)
'
which is identical with (77).
Suppose however ds to be the element of a curve on the surface,
and the extremities of ds to be at (f , ?j) and (f + d,r) + drf) ; then,
if x' y z are the current coordinates to the tangent of the curve
at the point (, 77) or (x, y, z), the equations to the tangent are
x'-x = y'-y = z'-z _
a\df+bidi] a2,d + b 2 dr) a^d^ + b^dr]^
and if 6 be the angle between ds and the line of the first system
at the point (, 77), then
158.] GAUSS' SYSTEM. 207
_
< 80 >
sm 6 = -f
ds
E
and if 0' be the angle contained between ds and the line of the
second system at the point (, 77)
(81)
These results will be found to be of use in the sequel.
*
. 158.] Also we may thus find the analytical value of the sur-
face-element which is contained between two curves of the first
system corresponding to and + d%, and two curves of the
second system rj, 77 -f dr\ ; using da, d<r', and o> in the same mean-
ing as in the last Article, the element of the surface abutting at
the point (, TJ) = dv dv sin o> ; but
sino) = {1 (coso)) 2 }*
(82)
{EG}*
surface-element = v di\ d%
{EG F 2 }*flfye? (83)
and replacing ^ a 2 3 , bi b 2 63 by their values, the element of the
surface is equal to
dz\ idy\ dz\ 2 ( idz\ idx\ idz\ idx\ \ 2
d) \ - / \J S
and is the same expression as that before found in equation (14.)
If however the surface on which the systems of curves are
208 CURVILINEAR COORDINATES. [159.
drawn be plane and be the plane xy, then z = 0, and the sur-
face-element is equal to
Also as the equations (68) are most general, the results which
are deduced from them are applicable to every other system of
coordinates. I proceed to exemplify this in a single case :
Let the equation to a surface be
* = f( x > y}>
dz /dz
tz\ /z\ ,
.'. dz = ( ) ax + -7- ) ay,
\dx> \d>
comparing which with (73), and replacing d and dr\ by dx and dy,
dz\ (dz\
and therefore by (83)
f i /i l dz \ z ^\ f l*-i ^
surface-element = jl+(j-j +\j~) j dxdy.
159.] Let us now consider this subject from another point of
view. Having given
it is required to transform into its equivalent in terms of f and
r] the double integral /. ,.
jJT!(x,y)dxdy.
From the data we at once deduce
and therefore by means of Art. 151, equation (46),
1 5 9.] GAUSS' SYSTEM. 209
and replacing in F (x, y), x and y by their equivalents in terms
of f and rj, the double integral becomes of the form
And if the integral be definite the limits must be replaced in
terms of their equivalents given by the equations connecting
the new and the old variables.
Now the left-hand member of (87) represents the infinitesimal
area-element referred to rectangular coordinates, and therefore
the right-hand member expresses the analogous element in terms
of curvilinear coordinates. Suppose then in fig. 43, p to be the
point of intersection of the lines corresponding to and 77, and
p 3 to be the point of intersection of the lines corresponding to
+ d> *1 + dr\, so that p p 2 , PI PS are two consecutive lines of the
first system, and PPI, P 2 P3 are two consecutive lines of the second
system : then, in terms of curvilinear coordinates, P is (, r/), PI is
( + <& vj), Pa is (f, i? + <fy), PS is ( + , *) + dri) ; and the rectan-
gular coordinates are, to
The area therefore of the triangle p PI P 2 , in terms of the coor-
dinates of its angular points which are given in (89), is
and therefore the area of the elemental (approximate) paral-
lelogram is .
(9o)
Hence by reason of the geometry of the figure we have
(dx\
PRICE , VOL. II. E 6
210 CURVILINEAR COORDINATES, [l6l.
160.] Again, suppose the relation between x, y, f, and 77 to
depend on the implicit equations
FI (x, y} = fi (, 77) ~i
*afoy) =/2 (&*?)-> '
then, putting these severally equal to p and #, we have from
above
161.] Suppose however the infinitesimal surface -element
ppiP 3 p 2 in fig. 43 to be on a curved surface, the coordinates to
p being x, y, z, and the curvilinear coordinates being , ?j ; and
let C?A represent the element; then, by Art. 133, equation (21),
c?A 2 = dy z dz 2 + dz 2 dx 2 + dx z dy z ', (96)
and replacing dydz, dzdx, dxdy by expressions analogous to
that given in equation (87), it is manifest that we shall obtain
equation (14) of the present Chapter.
It is manifest also from the figure that
pp 2 =
PP 3 2 = E# 8 + 2F<#rfl7 + Gd77 2 >, (97)
COS P 2 PPi =
{EG}*
area PFiP 3 p 2 = ppix?P 2 x
162.] M. Lame in his researches into the properties of heat
has made extensive use of a new system of curvilinear coordi-
nates, a mere outline of which it is desirable to give, although
for want of space we can do no more, and can therefore only
refer the reader to the original Memoirs in Liouville's Journal,
and in the Journal de 1'ficole Poly technique, Cahier XXIII :
1 62.] LAME'S SYSTEM. 211
the system however, besides being otherwise productive of useful
results, supplies large and important applications of the process
of transforming multiple integrals.
M. Lame * imagines three surfaces to cut each other orthogo-
nally at a given point, and these three surfaces he calls conju-
gate to each other : the equation of each surface is supposed to
contain a variable parameter, by the variation of which a series
or family of such surfaces is formed ; and thus every point in
space may be imagined to be at the intersection of three such
conjugate surfaces: and the arbitrary parameters which these
equations contain are, for a given point in which the three in-
tersect, called the curvilinear coordinates of that point. Now,
of such surfaces the following are salient properties :
(1) Any two surfaces cut the third surface at the point of
intersection along its lines of curvature : this is manifest from
Dupin's Theorem.
(2) In every such triple system of conjugate isothermal sur-
faces, of the six principal radii of curvature corresponding to
any point, the product of three taken in a certain order is equal
to the product of the other three.
And, to simplify the system of such coordinates, M. Lame
imagines the three conjugate surfaces to be of the second order,
and confocal. Thus any point in space is defined by being at
the point of intersection of three surfaces whose equations are
A 2 + A 2 ^ 2 + A 2 ^ 2 = 1
y z
wherein b is <c, \>ob, n>b<c, v<b<c: so that of the
three surfaces the first is an ellipsoid, and the second and third
are hyperboloids of one and two sheets respectively. The focal
distances of these principal sections are, it will be observed, the
same in all, and for this reason the surfaces are called confocal;
and the variable parameters A, p., v, which vary as we pass from
* See Liouville's Journal, Tome V, p. 313, Tome IX, p. 401, Tome XI,
p. 217 and p. 261.
E e 2,
212 CURVILINEAR COORDINATES.
one surface to another of the same family, are called the curvi-
linear or elliptical coordinates to the point which is at their
mutual intersection, and whose rectangular coordinates are xyz.
We may express xyzio. terms of A \iv in the following manner :
Take any one of the three equations (98), say the first, and
arrange the terms in powers of A 2 , then we have
= 0;
of which equation the roots are evidently A 2 , p?, v 2 : therefore
whence we have
bcx =
by (c 2 - b 2 )? = { (A 2 - b 2 ) (ju 2 - b 2 ) (b 2 -v 2 )}* j> , (99)
cz(c?-b 2 ^ = {(A 2 -c 2 ) (c 2 -v?)(c 2 -v 2 )}*J
the second members being always affected with the required
signs.
163.] Now the three surfaces whose equations are (98) always
intersect at right-angles : take the last two, and let the direc-
tion-angles of their normals at the common point be a 2 /3 2 y 2 >
then
cos a 2 cos a 3 + cos /3 2 cos /3 3 -J- cos y 2 cos y 3
a? *
1
v~ {jj,- u~) \v- b 2 ) (fj?
= 0,
by reason of equations (99) : and as a similar result will follow
whatever two of the equations (98) are taken, so we infer that
the three surfaces cut each other at right-angles.
164.] Imagine now a certain point on the ellipsoid, and the
two hyperboloids to cut the ellipsoid orthogonally at the point ;
also conceive A to vary, while //, and v are constant : thus we
pass along a normal line to the ellipsoid, and which is the in-
tersection of the two hyperboloids, to a point in another con-
secutive ellipsoid ; let the length of this line be d/j ; and let itg
164.] LAME'S SYSTEM. 213
projections on the three coordinate axes be d^x, d^y, d K z : then,
from equations (99), taking logarithmic differentials,
= -e?A
A
whence we have
C (A 2 -,, 2 ) (A 2_
It is worth remarking, that if p be the length of the per-
pendicular from the centre of the ellipsoid to the tangent plane
at (A, p, V ), 1 1 f (A 2 -V)(A 2 -i; 2 ) -i *
p ~ Xl(A 2 -c 2 )(A 2 -6 2 ) j ;
-
P
Similarly, if d^s, d v s symbolize similar elements of length
arising from the partial variation of the parameters of the
hyperboloids of one and two sheets respectively, then
d,s =
(101)
--
t*l/0 - ^ 777; . o Sv f '**'
(,(6 2 V 2 )(C 2 -I> 2 ) J
If then </5 represent the length-element as we pass from
(A, p., v) to (A + d\,
_^ 2) (A 2_ C 2 }
.i-
"' 1- ^
the integral of which will give the length of a curve in space,
when its equations are expressed in terms of elliptical coordinates.
Also as the three elements d^s, d^s, d v s are the three edges
of a rectangular parallelepipedon, abutting at the same point,
214 CURVILINEAR COORDINATES. [165.
so will the volume of the parallelepipedon be equal to the pro-
duct of the three elements, and be the volume-element. And
therefore, if v represents the volume contained within given
bounding surfaces,
v (103)
^
the limits being assigned by the equations of the bounding
surfaces.
Thus, if the equation to an ellipsoid be
+ + = 1 ' < 104 >
then the whole volume
-i> 2 ) dv dp d\
o b o { (A 2 - b z ) (A 2 - c z ) (^ - b 2 ) (c 2 -/x 2 ) (6 2 - ^) (c 2 -
but we know that the whole volume of the ellipsoid is equal to
3
'A re n> ( X 2 _ M 2) ( X 2 _ V 2) (^2 _ ,,2) ^ ^ ^
/7T
/O *^6 ^0
Jo J* Jo {(\ 2 -<
(105)
and therefore the value of the triple definite integral is hereby
determined.
M. Chasles has determined geometrically the value of a double
definite integral of the same form : see Liouville's Journal,
Vol. Ill, page 10.
165.] Also the systems of conjugate surfaces intersect each
other along their lines of curvature. We will prove this propo-
sition for the ellipsoid, and thence it may be easily inferred for
the other surfaces.
The differential equation of the lines of curvature of an
ellipsoid whose equation is (104) is by reason of equation (7)
Art. 346, Vol. I,
(C 2_2 ) * c2 |_ + 2^. =0 ;
dx dy dz
in which if dx, dy, dz are replaced by d^x, d^y, d^z, or by d v x t
d v y, d v z, the equation is satisfied : for
1 66.] LAME'S SYSTEM. 215:
x . , ny dp. j zp, dp,
substituting which in the above equation, it is rendered iden-
tical : hence the confocal hyperboloids intersect the ellipsoid
at any point in its lines of curvature : and of course a similar
result is true of the other two surfaces.
Imagine then a series of confocal hyperboloids formed by the
continuous variation of the parameters to be drawn, and to in-
tersect a given ellipsoid ; they will trace on the ellipsoid all its
lines of curvature, and the surface of the ellipsoid will by them
be divided into infinitesimal rectangular surface-elements. Now
the area of any one of these infinitesimal elements is equal to
d^s x d v s ; and therefore, if S = the surface of the ellipsoid,
S =
o C f b (p? V Z ) (h? P?)* (h? V 2 )^ dv dp, /-in^x
= o / / , . (106)
Jb Jo {(p? O 2 ) (C 2 p?) (0 2 V 2 ) (C 2 V 2 )} 5
166.] Jacobi has modified the expressions for x y z, given in
equations (99), by the introduction of two subsidiary angles <
and ty, which are connected with A p, v by the following equations :
v = bcos\}r
whence we have
y = (\ 2 * 2 ) sin ^ cos
Z = \-
and although in these expressions there is no ambiguity of sign,
for the signs will be given by the trigonometrical quantities, yet
they are not in general so convenient as the formulae (99).
216 EXAMPLES OP TRANSFORMATION OF L 1 ^.
SECTION 3. Miscellaneous illustrations of the preceding
principles.
167.] If the limits of the first integration of a multiple inte-
gral be functions of the variable in reference to which one or
more of the other integrations are to be performed, it is fre-
quently possible so to change the variables that the limits shall
be constant, and thereby the first integration may be performed :
let us take the case of a double integral ; and suppose it be
= f(x,y)dyd X , (107)
^o *Vo
where y\ and y are functions of x,
Let y = yo+(yi-yo)t, (108)
where t is a new variable ; therefore
dy = (yi-yo)dt; (109)
and observing that / = ! when y = y\, and t=Q when y=yo, we
/*, ri
= (yi- yo) / f{x, y Q + (yi-yo) t } dt dx,
Jx ^0
wherein the limits of the first integration are constant.
Thus, for example, if
dy dx,
let y (1 + a? 2 )* t ; so that t = 1 when y = (1 + # 2 )*, and t =
when y = ; then
=/T-
^o -A> i
= f
(1
dtdx
the latter integral of which can be determined, as will be shewn
by and by : and we shall ultimately have
TT* P
u = -7T
2 Jo
dt.
169.] MULTIPLE INTEGRALS. 217
168.] The following transformation is useful in the theory
of definite integrals : let there be given the double integral
let x = u uv I
\ I (HO)
y = uv J
dx = (\ v}duudv 1
dy = vdu + udv] } '
.-. dxdy = ududv; (112)
and the given integral becomes
(u uv, uv} ududv.
This substitution may be geometrically interpreted as follows :
In fig. 44 let P be the point whose coordinates are x and y :
through P draw PN parallel to, and PS making an angle of 45
with, the axis of x, and join N s ; then o s = u, and tan o s N = v : for
X = OS SM y = MP 1
= OS ON = ON L . (113)
= u uv = uv J
Hence, if the limits of integration of the double integral be
given, it is easy to assign the limits of the transformed integral.
By means of this substitution the integral / / x m y n f(x + y) dy dx
becomes changed into / u m+n+1 (lv) m v n f(u) dudv.
Or again, if the given integral be / //(- + v) dydx, first let
and then substituting for and 17 in terms of u and v, the inte-
/ / a bf(u) u du dv.
gral becomes
169.] We shall conclude the present subject with the con-
sideration of some double integrals which refer to the quadra-
ture of the surface of the ellipsoid : for hereby we shall be led
PRICE, VOL. II. F f
218
ON THE QUADRATURE OF THE ELLIPSOID.
[l6 9 .
to formulae which have been introduced into the problem by
Jacobi *, and which indicate certain interesting facts connected
with it.
Let the equation to the ellipsoid be
_ _l_ _L _
+ + '
(114)
a, b, c being in descending order of magnitude ; and let S repre-
sent the surface of an octant ; then, by equation (29), Art. 135,
rr
=
JJ
dy dx, (115)
the limits of integration being given by the inequality
a 2 6 2 =
To simplify (115), let
(116)
c 2
1-4= a 2
1 2 2 ~
a 2
A 2
*
l = r]
.. dx = adt;
dy = bdri
1 a 2 2 j3
* f2
1 P 2 77 !
1
(117)
whereby we have
s =
(118)
the inequality which gives the limits of integration becoming
P + rj 2 < 1. (119)
Now let us consider the variables , rj, ( to be rectangular
coordinates of a new system; so that the integral (118) mani-
festly expresses the volume of a cylinder, the element of which is
a prism of height C and of base d drj, , f\, { being related by the
equation
= f*-l,
(120)
* See Crelle's Journal, Vol. X, page 101, Berlin, 1833.
169.] ON THE QUADRATURE OF THE ELLIPSOID. 219
and the limits of the - and r\ -integrations being given by
(119) ; so that if v be the volume of the cylinder
r i
= I I
JO JQ
s = abv, v = I I dr]d. (121)
JO JQ
The surface whose equation is (120) is represented in fig. 45,
, if], f being the current coordinates to it : for regarding as a
variable parameter and greater than 1, it appears that every
plane parallel to that of f rj cuts the cylinder in an ellipse such
as RQ, of which the semi-axes are
and when f=oc = l, the ellipse becomes a point; and when
C = oo, the ellipse becomes a circle with radius 1, and of which
the projection on the plane 77 is BA: so that the surface is
asymptotic to a right circular cylinder of radius unity, and
whose axis is the axis of . Hence we may consider the element
of v to be a cylindrical slice of height , and standing on an
annular base bounded by two ellipses corresponding to and
C+^C- As this elliptical annulus is the infinitesimal increment
of the area of an ellipse whose semi-axes are given by (122), it
follows that the base of it corresponding to an octant of the
ellipsoid is equal to
and therefore
77 T* t 2 1
= *r at 2 -i) r (t 2 -i)^
4 L(r 2 -a 2 ^ (<T 2 -/3 2 )* J (f 2 -a 2 )^ ({- 2 -/3 2 )
and therefore the surface of the ellipsoid is equal to
So that by this clever substitution, due to M. Catalan, * the
* See Liouville's Journal, Vol. IV, page 323. The same method is ex-
tended to integrals of higher orders and more variables, the discussion how-
ever of which is beyond the scope of the present work.
F f 2
220 ON THE QUADRATURE OF THE ELLIPSOID. [170.
double integral in equation (115) is reduced to the single defi-
nite integral of equation (126).
170.] And (126) may be further reduced to elliptic integrals
by the following substitution : let
C = a cosec <f>, (127)
d = a cosec </> cot < d<$,
and the values of <f> corresponding to the limiting values of ,
viz. oo and 1, are and /u, if sin p, = a ; and therefore the surface
of the ellipsoid is equal to
a 2 -(sin</>) 2 r a 2 -(sin<J>) 2
but
C a 2 -(sin0) 2
/ r dd>
J (sin<) 2 {a 2 -OSsin0) 2 }*
' rc K-^sm^ ____ i-/* 2 \ ^ (129)
tfouW a 2 -3sin 2 ^J
; (130)
substituting which in (128) and reducing, we have,
surface of ellipsoid
(a 2 (/3sin0)
and evaluating the former part at the limits we have,
surface of ellipsoid = 2^ c^-abT ^^^^$1. ( i 32)
4 {a 2 -(/3sin</>) 2 }* J
171.] Lastly I propose to investigate certain other integrals
which involve properties of the surface of an ellipsoid, and to
exhibit substitutions by means of which the order of the mul-
tiple integral may be reduced.
Let the equation to the ellipsoid be
n/t 2 ?/ 2
+ = ! < 133 >
172.] ON THE QUADRATURE OF THE ELLIPSOID. 221
and let it be expressed in terms of subsidiary angles 6 and <f> as
follows :
x = a sin 6 cos $ -^
y = d sin 6 sin (f) L; (134)
z =. c cos Q J
whence by differentiation and by the equation (22), Art. 133, if
ds represent the infinitesimal surface -element,
ds =
= {b z c* (sin 0) 2 (cos <) 2 + C 2 2 (sin 0) 2 (sin </>) 2
+ a 2 6 2 (cos0) 2 } i sin0d0cfy. (135)
But if p = the length of the perpendicular from the centre of
the ellipsoid on the tangent plane,
2 2 z
' (136)
and therefore
and if s = the whole surface of the ellipsoid,
. (137)
172.] Again, let us introduce two new subsidiary angles i]
and \l/, such that sin rj cos \|r, sin 77 sin \^, cos 77 respectively may
be the direction-cosines of the normal at any point of the ellipsoid,
whereby we have (employing p as in the last Article)
3B '
sin 77 cos ^ = p -^
sin 77 sin ^ = p
cos T; = p 2
c _,
(138)
2 . (139)
From which expressions we obtain
222 ON THE QUADRATURE OF THE ELLIPSOID. [172.
sm^(dp\ sin 77 cos TJ cos ^
p ^ay p
sin TJ cos 17 sin \fs
jo* (^ p \d\lr> p
+ ^T- \ sin r] cosry (-^ ) [ dr? 2 c?\^ 2 ; (140)
whence e?s = -- j- smrjdrjdty; (141)
-
2 '
To simplify this, and with a view to the geometrical inter-
pretation of it, let
a 2 (sin 7]) 2 + c 2 (cos q) 2 = m 2 j
A 2 (sin r)) 2 + c 2 (cos r]) 2 = n 2 j
(144)
_
" '
Let n tan \jr = m tan o>, (146)
then, observing that the limits of ty, and therefore of o>, are TT
and ir, we have
f T 77 sin rj C?TJ r ff sin r? c??? ")
1 I + / -r- h (147)
(.Jo * Jo ww 3 J
s =
and therefore the surface is expressed in terms of two single
integrals. Let us consider the geometrical meaning of this
expression.
From (138) it appears that the relation between rj and the
coordinates of the point on the ellipsoid to which it belongs is
2 z 2 ~~%
(148)
which is the equation to an elliptical cone whose vertex is at
the centre of the ellipsoid ; and as 77 is the ^-direction-angle of
the normal of the ellipsoid, the axis of z is the axis of the cone,
I 73.] ON THE QUADRATURE OP THE ELLIPSOID. 223
and the ratio of the semi-axes of any plane elliptical section
of it perpendicular to its axis is that of a 2 : b 2 . Now the ^-in-
tegration, which has already been performed, between the limits
TT and IT gives an annulus on the surface of the ellipsoid, the
breadth of the annulus being due to the variation of 77. Imagine
therefore two cones, represented by equation (148), to be de-
scribed corresponding to rj and to 77 + drj ; the lines of intersec-
tion of these cones with the ellipsoid will be two curves, infi-
nitesimally near to each other, which contain between them the
band of the ellipsoidal surface expressed by
and the sum of all which bands between the limits IT and will
be the whole surface of the ellipsoid.
173.] In review of the processes of the two preceding Articles
let it be observed, that in equations (135) and (141) we have
a 2 b 2 c? sm.r]dr]d\l/
ds =
{a 2 (sin rj) 2 (cos \^) 2 + b 2 (sin ?j) 2 (sin \|r) 2 + c 2 (cos r;) 2
the former of which is irrational and the latter is rational ; so
that by means of the following substitutions we have been able
to transform an element-function involving irrational quantities
into an equivalent in terms of rational quantities only, viz. by
substituting . .
. . a sin ri cos \|r
smtfcosft = -
P
b sin rj sin \lr
}, (149)
P
c cosrj
~Y~
where
p 2 = a? (sin rj) 2 (cos ^) 2 + b 2 (sin rj) 2 (sin \|f) 2 + c 2 (cos rf) 2 ; (150)
or where, as in (136),
_!_ __ (sin B) 2 (cos ft) 2 (sin 0) 2 (sin ft) 2 (cosi
Art 2 "^ .9, i" 7 o '-"-- -'*" 1 "~ o
sin sin $ =
cos 6 =
224 ON THE QUADRATURE OF THE ELLIPSOID. [ J 73-
and therefore from (137) and (142)
p 3 sin dO d<f> =. abcsm.t]dr]d^r. (152)
Hence by the substitutions of (149), the double integral
u sin 6 dd dfy
II-
JJ 5
{b*c 2 (sin 0) 2 (cos 0) 2 + c 2 a 2 (sin 0) 2 (sin 4>) 2 + a*b z (cos 0) 2 }*
in which u is a rational function of sin 6 cos $, sin sin <j>, and
cos 6, may be transformed into the following, which involves
only rational quantities, viz. into
a 2 b 2 c 2 u sin 77 drj d\j/
!f a
a 2 (sin rj) 2 (cos \^) 2 + b 2 (sin rj) 2 (sin \^) 2 + c 2 (cos r/) 2 '
the limits of the new variables being easily obtained from those
of the former variables by means of equations (149).
Again, from (152) we have
nf rfrf
P 3 sin0d0d<j) = abet I s
Jo Jo
an integral which occurs in the determination of the volume of
an ellipsoid the equation of which is expressed in terms of polar
coordinates.
I 74.] VARIATION OF CONSTANTS IN DEFINITE INTEGRALS. 225
CHAPTER X.
ON DEFINITE INTEGRALS AS AFFECTED BY INFINITESIMAL
VARIATIONS OF CONSTANTS INVOLVED IN THE ELEMENT-
FUNCTION, AND IN THE LIMITS.
174.] WE return to the consideration of other properties of
single definite integrals, and especially of those which arise from
the infinitesimal variations of constants which are involved in
the element-function and in the limits of integration, and which
are for the time variable parameters : we shall hereby be led to
a wide extension of the Calculus, and to the solution of a class
of problems of the utmost importance, and which would other-
wise be beyond its range. And first let us investigate the effects
of an infinitesimal variation of a parameter involved in the ele-
ment-function but not in the limits, and which is constant so far
as the operation indicated by the sign of integration affects it.
Let the definite integral under inquiry be
*f(x, a) dx,
wherein a is a variable parameter, of which x, x n , x are inde-
pendent ; and let us for the sake of convenience symbolize the
definite integral by u : so that
/*<*
u = I f(x,a)dx; (1)
Jx
then u is a function of X 0) x n and a. Let a become a + da, and
let the change of u due to the change of a be d a u ; then
/**"
u + d a u = / f(x, a + da) dx,
(*x f*x
. . d a u = / f(x, a + da) dx f(x, a) dx
= / *{f(x,a + da)-f(x,a)}dx
= f X *d a f(x,a)dx; (2)
PRICE, VOL. II. G g
226 THE VARIATION OF CONSTANTS [175.
da J Xo da
or, /(*, a) dx = -l dx , (4 )
aaJ Xo J Xo aa
From (2) therefore it appears that the differential of a defi-
nite integral with respect to a variable parameter involved in
the element-function is the definite integral of the differential
of the element-function with respect to the variable parameter.
The two operations therefore of differentiation and of integra-
tion being performed in respect of different variables may be
interchanged without any alteration of the result.
The same result is also manifest from -the very form of a
definite integral when stated in the serial value ; the left-hand
member of (2) is the differential with respect to a of a series of
values of f(x } a) dx formed by the continuous growth of x from
XQ to x n -, and the right-hand member is the sum of the values
within the same limits of d a .f(x,a)dx ; and by first principles
these two sums are identical.
As the order of the operations is in the above case indifferent,
so will it still be indifferent whatever be the number of them ;
and therefore
)7 *- /* / \ 7 /K\
dx I -r-:/(#,a)aa?j (5)
and similarly if a be independent of x, y, . . . and of their limits
of integration,
~daJ J "f(x>y>-'- a )~' d y dae =J J *----j a f(x,y,~-a-)'-'dydx.
175.] The preceding method of differentiating a definite in-
tegral with reference to a parameter contained in the element-
function often gives the value of another definite integral; as
the following examples shew:
r x dx l , , x
Ex. 1. / -5 = = tan" 1 ;
. 'o a -\- x a a
differentiating with respect to a,
1 x x
= tau -1
a 2 a a (
dx I ,x x
= -zr- -, tan" 1 - -
a
1 75.] IN DEFINITE INTEGRALS. 227
^ O r dX 2 lW a - 6 \*4- X )
Ex.2. / - -, - = - -tan-M( - j-ltan-t;
Jo a + ocostf 2_2\i (\a + o' A }
differentiating with respect to a, and reducing,
r x dx _ b sintf 2a
Jo ~
a? b 2 a + b cos x s a z _ ml
dx
Jo ( + 6cos<r) 2
/* rtv> IT ,
Ex.3. /
Jo
2
and differentiating (n 1) times in succession with respect to a,
(2n-3)(2ro-5)...3.1 TT -^f- 1
/*
'"'Jo
r
Jo (1 + <r 2 ) n "" (2w 2)(2w 4)...4.2 2'
and if a = 1,
dx (2n 3)(2n 5)...3.1 w
/'OO
Ex. 4. / e~ ax dx = a- 1 ;
Jo
therefore differentiating n times with respect to a,
/*
Jo Xne "* = I-*. .
Ex. 5. Let / ~ = u :
Jo a + ox + ex*
differentiating r times with respect to a, and m times with
respect to b,
d r+m u
m >J (a+bx + cx*Y+^- '
Let r + m = w; .*. r = n m;
x m dx ( n d n u
C
Jn
JQ (a + bx + cx 2 ) n+l 1.2.3... n da n - m db m '
so that a definite integral is expressed as a partial derived-
function.
eg a
228 THE VARIATION OF CONSTANTS [176.
176.] The process of reasoning by which in Art. 174 is shewn
the legitimacy of differentiating a definite integral is of sufficient
width to include the reverse operation of integration with respect
to a variable parameter involved in the element-function; the
following however is an independent proof of the proposition :
/*
Let the definite integral be / f(x,a) dx, wherein x n , XQ and x
Jx
do not involve a, and let us suppose that it be required to inte-
grate this definite integral with respect to a between the limits
a and a n , that is, to determine
/" C x
/ / /(*,
JltQ JX
a) dx da,
provided that no value of a between a n and a is such as to
render the element-function discontinuous, or to make it change
sign; then, since
(<* /**
f(x,a)dx = I d a -e(x,a)dx,
. *o J*t
therefore if v(x,a) = I f(x,a)da,
JOQ
rx n /*O B fx n
d a l I f(x,a)dadx - I f(x,a)dx;
Jx J<LQ JZQ
whence, integrating with respect to a between the assigned
limits, we have
r x n /*<* [*. fXn
I I f(x,a)dadx = f(x,d)dxda;
JXQ JOQ JOQ J&Q
that is, the order of integration may be changed without an
alteration of the final result ; and similar propositions are true
of successive integrations and of multiple integrals.
177.] Examples in illustration of preceding Article.
T 1 1
Ex. 1. / x n ~ l dx = -.
.'o
/""
Operate on both sides of the equation with / dn, so that
Jm
r 1 r n f n dn
I I x n ~ l dndx = -;
^o Jta Jm n
f
n
, ,
dx = log .
log<r m
Ex
a
DEFINITE INTEGRALS. 229
1
/>
. 2. / e-
Jo
r r a C a da
. / / e- ax dadx = ,
*e "C
I
!x. 3. / e~ ax si
a 00
e-^si
_
X C
b
~ 2 i A2'
A da
smoxdadx
.
dx
r. .a~\ a =
= tan- 1 T
L *Ja=a
,
Let = 0, therefore
&p , TT
f 00 sin
^
/*
= /
Jo
= / e-^si
r Jo
g-* cos rx
Ex. 4. u
the limits of the last integration being such that u = when
r = 0, and therefore when, from the given value of u, a = b.
Ex. 5. To evaluate
JQ
Let the definite integral = u ; and let it be differentiated with
respect to b,
r e -a^ sm ^-. 00 ^ - w
= 5-5 x 5 / c- ** 8 cos bx dx
L 2a 2 J 2 2 J
230 THE VAKIATION OF CONSTANTS
du _ b
' : ~ '
.-. u ce 4a ;
c being the value of the given definite integral when b =. ; and
which will hereafter be shewn to be ~- .
2a
178.] Suppose however that not only the element-function, but
that also the limits of integration are functions of the parameter;
let us investigate the changes which the definite integral under-
goes by reason of the infinitesimal variation of this parameter.
Let the definite integral be of the form (1), and the limits of
integration be functions of a ; that is,
then the integral may be represented as follows,
r<t>(a-)
F(O) = / f(x,a)dx, (6)
/>(a) + cf.</>(a)
= / f(x, a + da) dx ; (7)
^ira+d.a
r<f>(a.)
I f(x,a)dx;
and since
ia+da
ib+db
a+da r- -i
f(x)dx = \f(x}
~u+db L -i
= f(a + d
= /() +/() da -f(b) -f(b) db
infinitesimals of the second order being neglected ; therefore
/*4>(<x)
rf F(a) =J {f(x,a + da)f(x,a)}dx+f{<t>(a},a}d.<t>(a)
-f{^(a),a}d^(a) (8)
(a),a} c?.0(a)-/{^(a),a}^(a) ; (9)
and this result is also manifest by general reasoning : the varia-
1 79.] IN DEFINITE INTEGRALS. 231
tion of the parameter in the element-function of course pro-
duces that change of the element-function which was investi-
gated in Art. 174, and this is the first term of the series (9);
the increment of the superior limit adds a term to the series,
of which the definite integral as expressed in (1) is the abridged
form and is the sum, and therefore if the superior limit grows
from <(a) to $(a) + e?.<(a), the additional term isf{(f)(a),a}d.(f)(a)
according to equation (1), Art. 81, and this is the second term
of (9) : and lastly, if the inferior limit is increased, say from
\}/(a) to \ls(a) + d.\lf(a), the first term of the series in equa-
tion (1), Art. 81, is taken away, and the series is diminished by
Of the general proposition contained in (9) the following are
cases : let -^
u = I f(x)djs
JXQ
fit/
dx n
du
= /(*), (10)
= -/(*o),
and similar theorems are of course true of multiple integrals.
179.] As a clear conception of the result (9) is important for
future investigations, a geometrical interpretation is subjoined ;
see fig. 46.
Let p PP n be the curve whose equation is y=f(x,a); let
r&
OM O = XQ, OM W = x n , so that the area M M n p n p = / f(x, a) dx.
Let the parameter a vary, and first let the element-function
alone contain a, and let the new position of the curve which is
due to the variation of a be Lp' Np' n , so that the area becomes
increased by the quadrilateral LP O P W N : therefore
LF P n N = / d a .f(x,a)dx:
now let the limits alone be functions of a, and such that by
the change of a, OM O becomes OM' O , and OM W becomes OM' W ;
and therefore the area is increased by p ;i M n M' n N' and dimin-
ished by POM O M'OI/, which are respectively represented by
f{(j>(a),a}d.<f)(a) and f{\ff(a), a} d.-^(a). But when all these
variations are simultaneous, so that the definite integral ex-
232 THE VARIATION OF CONSTANTS [l8o.
presses the area p' M' M' n p' n instead of p M M n p n , the two qua-
drilaterals LL', NN' are omitted, because they are infinitesimals
of a higher order ; being, in fact, quadrilaterals, each of whose
sides is an infinitesimal, and which are therefore infinitesimals
of the second order, and have for their analytical expressions
d a .f{^(a),a} xd.^(a) and d a ./{0(a),a} x d.<J>(a),
which are the terms omitted in the formation of equation (8).
180.] The preceding process at once suggests and resolves
the following corollary : Suppose that it be required to deter-
mine a, the parameter involved in the element- function of (1),
so that the definite integral should have a maximum or a mini-
mum value.
For the sake of simplicity let us at first assume that the
limits of integration are independent of a ; and let the definite
integral = F (a) ; then
(12)
which must be equal to 0, by reason of the theory of such sin-
gular values ; and the corresponding critical value of a can
easily be determined, if the integration can be performed : but
if the integration be impossible, we can only construct the curve
by points, or determine approximately the definite integral by
one or other of the methods of Chap. IV, Section 3, and thereby
find the required value of a.
And again suppose that a is a function of x, and therefore a
quantity varying with x through the extent of integration, and
which it is convenient to replace by y, so that it may accord
with the ordinary notation ; and suppose that it be required to
determine y, so that
may have a maximum or a minimum value.
By a process similar to the preceding we must have
3 Xr,
, y) dx = 0.
Now it may be that this problem is capable of resolution with-
l8o.] IN DEFINITE INTEGRALS. 233
out previous integration : for suppose that y is found in terms
of x by means of the equation
then each of the elements of the definite integral will have its
maximum or minimum value; and therefore the definite inte-
gral being the sum of all these separate singular Values will
have such a singular value itself. It is necessary however that
the value of y be such that /(a?, y} does not become infinite or
discontinuous or change sign between the limits, and that the
values of f(x, y} thus determined be either all maxima or all
minima, and not some maxima and others minima.
And we are hereby led to the examination of a yet more
general case, that viz. Avherein the element-function of the defi-
nite integral involves an, y (a function of x), and one or more of
the derived functions of y with respect to x ; and wherein it is
required to determine y in terms of <r, so that the definite in-
tegral may be a maximum or a minimum.
And similarly we are led to the yet more general case of a
multiple integral, the element-function of which involves many
variables, independent or not as the case may be, and their suc-
cessively derived functions with respect to one of them, say, x ;
and wherein it is required to express one in terms of others, so
that the definite multiple integral may have a singular value.
These and other like problems are those of the Calculus of
Variations, which I proceed to inquire into in the next and
following Chapters.
PRICE, VOL. ii. H h
234 CALCULUS OF VARIATIONS. [l8l.
CHAPTER XL
EXPOSITION OF THE PRINCIPLES OF THE CALCULUS OF
VARIATIONS.
181.*] THE subjects of investigation in the preceding parts
of our treatise have been functions whose forms are known and
determinate ; such as those symbolized by cos, tan- 1 , log, log" 1 ,
and other such like : and the inquiry has been for the most
part confined to the properties of such functions, which arise
from the continuous and infinitesimal variation of their subject-
variables; and we have had no occasion to consider the func-
tions themselves undergoing continuous change as to form:
certain invariable relations have been shewn to exist between
certain functions; for by the process of derivation we pass
from one function to another ; but these are nevertheless deter-
minate, and the relation arises from a continuous growth of the
subject- variable, and not from a continuous and infinitesimal
change of the function as to form : this distinction is important ;
* The authors and titles of the principal works on this branch of infinitesi-
mal calculus are the following, and from them much assistance has been
derived :
Euler, Methodus inveniendi lineas curvas maximi minimive proprietate
gaudentes, Lausanne, 1744.
Lagrange, Lecons sur le Calcul des Fonctions, Paris, 1808.
Lagrange, Theorie des Fonctions Analytiques, 3 me edition, par J. A. Serret,
Paris, 1847.
Poisson, Memoires de 1'Institut de France, Tome XII, Paris, 1833.
Jacobi, Journal fur Mathematik, Crelle, Band XVII, Berlin, 1837.
Ostrogradsky, Memoires de 1'Academie de St. Petersbourg, Tome I, St. Pe-
tersbourg, 1838.
Ibid. Tome III.
Delaunay, Journal de 1'Ecole Polytechnique, XXIX Cahier, Paris, 1843.
Delaunay, Journal de Mathematiques, par Liouville, Tome VI, Paris, 1841.
Sarrus, Memoires presentes par divers savants a 1' Academic des Sciences,
Tome X, Paris, 1848.
Strauch, Theorie und Anwendung des Variations-calcul, Zurich, 1849.
Jellett, Calculus of Variations, Dublin, 1850.
Schellbach, Probleme der Variationsrechnung, Crelle, Band XLI, p. 293,
1851.
1 8 2.] ORIGIN OP THE CALCULUS. 235
for there is no conceivable reason why functions should not be
continuously variable as to form, as well as numbers be as to
magnitude. Thus for instance suppose the subject of investiga-
tion to be y = sin x; the value of y may manifestly be changed
either by a change in the subject-variable x, or by a change of the
functional symbol into any other, as tan" 1 ; changes due to the
former cause are considered in the Differential Calculus; but those
arising from a continuous change in the form of the function
require another mode of investigation ; and whereas heretofore
we have passed per saltum from one function to another, the new
calculus requires a continuous passage: a wide extension then
is opened before us, one the subject-matter of which is not
number but functions : and as a functional symbol expresses the
law of combination of its subject-variables, we shall have to
consider laws, and not subjects of laws. Functions then, as
they are the subject of this new calculus, are free from all
concrete or applied signification, and express laws ; and the
proper end and object of such a calculus of functions is to inves-
tigate their origin and their principles, their growth and extent,
their laws of combination, and to deduce from these, properties
with which they are pregnant. As differential calculus investi-
gates properties of continuous number, so does the new calculus
properties of continuous functions ; and as there is an integral
calculus of number, so will there be also an inverse calculus of
functions.
182.] Apart however from these general considerations, let
us view the calculus in the light of an easy problem of that
class, the attempt to solve which gave rise to it. Suppose that
it be required to determine the form of the function connecting
x and y which expresses the shortest distance between two
given points : if the function were given, the problem would be
one of rectification and would be solved by the integral calcu-
lus : also a posteriori we know that the required function is that
which expresses a straight line : but the direct solution of the
problem requires a different process ; viz. the assumption of a
general functional symbol undetermined as to form, and the
expression for the distance between the two points in terms of
it : so that if an infinitesimal variation of the distance due to an
infinitesimal variation of the form of the function be calculated,
the required form will be determined by equating to zero that
H h 2,
236 CALCULUS OF VARIATIONS. [182.
variation : provided that the form so determined is such as to
make the first variation change its sign ; or what is equivalent,
such as to make the second variation either positive or negative
for all values of the determined function within the given points :
for such an operation it is necessary (1) to calculate the infini-
tesimal change of the distance due to the infinitesimal change of
the form of the function, (2) to be able to determine the form of
the function by equating to zero the variation of the distance ;
in other words, we must be able to differentiate functions as to
form, and to determine functions by means of given conditions ;
also if these conditions give many results, we must be able to
discriminate according as one or another is applied. Such a
process then requires a knowledge of functions as accurate and
complete as that of number required in the differential calculus.
It will be observed that, as the two points which are to terminate
the line are given, the only variable quantity of the problem is
the form of the function.
Suppose however that the problem is, to determine the form
of the function which expresses the shortest distance between
two given curves in space ; let the distance be expressed by
means of a general undetermined function, as in the former
case, and in terms of the current coordinates of the two curves
which it is to meet ; then it becomes dependent on the form of
the function, and on the coordinates of these two curves : and as
these quantities are independent of each other, they may be
considered as independent variables, and their variations may be
taken separately : that arising from a change in the form of the
function may be estimated as in the former case, and thence-
may be deduced the form that gives the least distance : and
those which arise from the coordinates of the points on the
given curves at which the required curve is to meet them, must
be calculated according to the rules of the differential calculus,
and by equating them to zero we shall be able to determine the
points of meeting. In the solution of this problem therefore
two kinds of variations will be required, one arising from a
change in the form of the function, and the other from the dif-
ferentiation of the coordinates of the given curves.
183.] The infinitesimal variations therefore of the calculus of
functions and of the differential calculus are essentially distinct
1 84.] ORIGIN OF THE CALCULUS. 237
in kind : in the former they result from a change of form of an
undetermined function ; in the latter from a change of the sub-
ject-variables of a determinate function : and to use language
borrowed from the geometry of curves, a variation of the former
.kind leads from a point on a curve to a point on another curve
infinitesimally near to it ; a variation of the latter from a point
on a curve to a point on the same curve infinitesimally near to
it. It is convenient therefore to have different names for quan-
tities so different, and to express them by different symbols : in
the former calculus they are called variations) in the latter
differentials : hence arises the name " calculus of variations,"
and so henceforth we shall employ the term "variation" in a
technical sense, to indicate the particular infinitesimal change of
this calculus : also we use d to express differential, and b to
express variation : d therefore indicates a passage from one
system of variables to another, both of which satisfy a given
determinate function ; 8 indicates a passage from a system which
satisfies one function to a system satisfying a function infini-
tesimally different from the former one : thus a variation as
applied to a function may be defined as the infinitesimal change
of the value of the function due to its change of form,
184. ~] The symbols in relation to their subjects stand as follow:
let u be an undetermined function ; then bu is the change of
form of M; and let a certain operation symbolized by F be per-
formed on u, (it might be differentiation or integration) and let
v = FW;
then 5v = b.ju (1)
and 8v is the change in V( = F.W) due to a change in form of u.
As in the differential calculus there are partial and total
differentials of functions of many variables, according as one or
all of the variables change value ; so if the function, whose
variation is to be calculated, involves many undetermined and
independent functions, it is susceptible of different variations
according as one or more or all of these undetermined functions
vary, and therefore in the present calculus there will be partial
and total variations ; and by the principle of such infinitesimal
changes, the total variation is equal to the sum of the several
partial variations.
Thus let u\ u z ...u n be n undetermined functions, and let F
238 CALCULUS OP VARIATIONS. [185.
symbolize an operation performed on a certain combination of
them ; and let
V = P(Mi,2,...tt w ), (2)
then o v = I 1 oMi + ( 1 OH* + ... + I ) ou n : (3)
> OUi ' \ 8^2 u n
using brackets to symbolize partial variations. But (and this
remark is important) so long as the relation between F and Ui
remains the same, the ratio of the infinitesimal changes of F
and Ui must be independent of the particular species of them,
that is, whether the changes be of magnitude or of form ; and
therefore ,
()-.
and similarly for the others ; and therefore
185.] Thus far as to the general principles of the Calculus
of Variations : we proceed to investigate methods by which it
may be applied to the solution of problems which are of the
greatest importance in the present state of mathematical science,
and which the Differential Calculus fails to solve.
Of functions in their integral and determinate forms our
knowledge is too scanty for the attainment of the present ob-
ject ; but there are^ certain general expressions for infinitesimal
elements, independent of the functions of which they are ele-
ments, and therefore the same for all, provided that the func-
tions satisfy the law of continuity within the range for which
they are considered; thus ds = {dx 2 + dy 2 }^ is the distance
between two points (x, y) (x + dx, y + dy) on a plane curve,
whatever be the form of the function y = f(x), which is the
equation to the curve. Thus also {dy 2 dz 2 4- dz 2 dx 2 + dx 2 dy 2 }^
is the surface-element, whatever be the form of F (x, y,z)-=c
which is the equation to the surface : similarly dx dy dz is the
volume-element referred to rectangular coordinates, and is in-
dependent of the particular form of the bounding surface.
Now these and similar general expressions for infinitesimal
elements are made the subjects of investigation ; and we calcu-
late their variations according to processes which will be de-
veloped hereafter : and if an integral function be the subject of
185.] FORM OF THE PROBLEM. 239
inquiry, it is considered as the integral or sum of elements ;
and to this sum we apply the conditions, so far as they are
applicable, for determining the unknown function. By this arti-
fice therefore we avoid the difficulty of making to vary the func-
tion in its general form. Thus, for instance, in the problem of
finding the shortest line between two given points, (x n , y n ) and
(XQ, y ) ; instead of assuming the distance to be F(<r n , y n , X Q , yo),
where F represents some indeterminate function, and then de-
termining F by equating to zero the change of the distance due
to the variation of F, we assume ds = {da? + dy 2 }^ to be an ele-
ment of the distance, so that the distance
*/*
= /
^o
and we make the latter sum the subject of investigation.
And in the most general case ; suppose that we have to in-
vestigate the form of the relation y=f(x}, where /is the symbol
of some unknown function, so that a given condition should be
satisfied, when that condition is the sum or integral of a series
of elements, each of which is a given function of x, dx, d 2 x, . . . d n x,
y, dy, d z y, . . . d n y, neither x nor y being equicrescent ; then, if the
element = F (x, dx, d^x,... d n x, y, dy, d 2 y, ...d n y), where F repre-
sents a given function, and if x\ y\, XQ yo are the given limiting
values of x and y, the unknown function
= / F(#, dx, d*x, . . . d n x, y, dy, d*y, . . . d n y), (6)
Jo
and the relation which exists between y and x, that is, the form
of/, is determined by means of conditions to which (6) is subject.
And a similar method is applicable if the element of the unknown
function involves more variables and their differentials.
186.] Now the principle of investigation explained in the
preceding article is of the greatest importance : the calculus of
variations in fact consists in the full development of it ; and
therefore I do not hesitate formally to enunciate it : suppose
that we have a quantity depending on a certain unknown function,
and that the form of the unknown function is to be determined by
* Instead of expressing the limits of integration at length, we have merely
put their subscript letters ; we shall find it hereafter convenient in other cases
to employ a similar notation.
240 CALCULUS OF VARIATIONS. [l86.
making the quantity fulfil certain given conditions ; in the general
case our knowledge of functions and of their laivs is insufficient
for the determination of the unknown function, and especially
when the conditions require an infinitesimal variation of it : but
as the form of many infinitesimal elements is known and the same,
whatever be the unknown continuous function, we may consider
the quantity which depends on the function to be the sum of
certain elements between given limits, and may make the quantity
in its latter form the subject of inquiry.
187.] When the problem has been put into the above form,
the following is the most convenient method of effecting and of
symbolizing the necessary operations : the unknown function is
made to assume a new form by an infinitesimal change of the
variables and their differentials which are involved in the given
element-function, the infinitesimal variations being functions of
the variables to which they are applied ; and as hereby the
element-function will have changed value, so will also the sum
of all these; and as these infinitesimal changes are not made
subject to the conditions of the original given function, they
may be, and generally will be, inconsistent with it, and thus a
new law will be introduced which will be expressed by a new
functional symbol. Or to employ the language of geometry:
suppose a certain curve to be expressed by the undetermined
function ; and suppose each point of the curve to be shifted,
and thereby each of the length-elements and each of the suc-
cessive differentials to change value ; the curve in its new
position will generally have taken a new form, and so will re-
quire a new function to express it. Thus suppose the curve
under consideration to be a curve of double curvature, and let
the position and form of it be changed ; then if b& by bz are the
variations (technically so called) of the coordinates, these being
functions of x, y, z, the point (#, y, z} becomes (x -f x, y + by t
z + bz); observe then the change; the point on the old curve
infinitesimally near to (x,y,z) is (x + dz,y + dy,z + dz), whereas
ae + bx, y + by, z + bz refer to the same point as x, y, z, but to the
point in a new position, and on a new curve, and when the
form of the function has varied. Similarly also b.dx, b.dy, b.dz,
b.d^x,...b.d"x express variations which the several successive
differentials undergo and which are due to the change of the
I 88.] ITS SYMBOLS AND THEIR LAWS. 241
form of the functions. It is necessary however to consider the
relation between variations and differentials with greater pre-
cision. It is to be observed of two such operations, that they
are subject to the commutative law, because they are of the same
nature though of different species ; so that
= d.b.dx =
Similarly, if y = f(x)
b.dy = l.df(x) = rf.
(8)
and similarly for other variables. And as d signifies an opera-
tion subject to the index law, the results of the operations per-
formed on x and on/(#) being true for positive integral values
of n will also be true for negative integral values ; that is, as
they are true for differentiation, so will they also be true for
integration. Thus
5. \dx = Ib.dx
J J (9)
and similarly also for successive integrations.
Similar results are also true for successive variations, so that
we have generally
b m d n f(x) d n b m f(x);
and also still more generally
m + n + r+... m+w + r+...
'
dx m dy n dz r ... ~~ dx m dy n dz r ... '
188.] And as these results, especially (9), are of importance
in the sequel, let us consider them in reference to a plane
curve ; see fig. 47.
Let p PQPi be a plane curve whose equation is y =/(#),
OM = XQ~\ OM = X~} MN = dx~\ O MI = X\~\
= \)
M O P O = yo Wcsjr GQ = dy y MI^I = y\
Q being a point on the curve infinitesimally near to P.
PRICE, VOL. II. I i
242 CALCULUS OF VARIATIONS. [189.
Now suppose each point of the curve to have shifted, so that
the points indicated by t*he letters in the figure assume the
points indicated by the letters accented, and suppose hereby
the form of the function to have changed, so that
MM = x NN = x x = x .dx'}
.dS '
P'L =tyJ KQ' = b(y + dy) = by + b.dy
Also as P', Q' are points on the new curve mfinitesimally near,
since o M' = x -f bx, therefore M'N' = d(x + bar) = dx + d. bx. Also
. . M N + N N' = M M' -f M'N'
.-. dx + bx + b.dx = bx + dx + d.bx
.-. b.dx = d.bx.
By a similar process we may prove that
b.dy = d.by,
and by repetition of a similar process that
b.d n x = d n .bx
b.d n y = d n .by.
It will be observed that we have made the limiting points of
the curve, viz. P O and p b to change position, so that there are
variations of tfo,yo>#i,yi' now in the most general case this may
be so, and the change of value of an integral of which these
are the limits must be calculated by the methods of the last
Chapter : if they are fixed, as in the first problem stated in
Art. 182, they of course have no variations : and if they are
constrained to be on certain curves, as in the second problem of
Art. 182, their variations are not arbitrary, but must be in
agreement with the equations to those curves.
189.] Problems within the range of this calculus may involve
either one single infinitesimal element, as, e. g., the volume-ele-
ment referred to rectangular coordinates, or the integral of such
elements between given limits : the former problem may be
solved by means of the principles already explained, and with-
out the intervention of any other formula : the latter require
longer processes, and it is only by the judicious employment of
integration, for which we are indebted to Lagrange, that we
can finally obtain practicable results.
First then let us consider a function of two variables x and y,
which are connected with each other by an unknown functional
i8 9 .]
CALCULUS OP VAKIATIONS.
243
symbol, which is to be determined ; and suppose that the ele-
ment-function is
F (x, dx, d*x, . . . d n x, y, dy, d 2 y, . . . d m y},
where F expresses a known function; and let u represent the
sum of these element-functions between the limits x\ y\ and
XQ y , so that
u = I F (x, dx, d z x, . . . d n x, y, dy, d z y, . . . d m y) ; (10)
t/Q
it is required to calculate the variation of u, the relation be-
tween x and y being an unknown function.
Let the variation be of the most general kind that is possible;
so that not only x, y, but also dx, d z x, . . . d n x, dy, d 2 y } . . . d m y re-
ceive variations ; and let
ft = F (x, dx, d*x, . . . d n x, y, dy, d z y, . . . d m y) ; (11)
and thus u = / li, (12)
Jo
.-. bu = 8. / il
/o
(13)
-/
Jo
then since & is a function of x, dx, d z x, . . . d n x } y, dy, d z y, . . . d m y,
by virtue of equation (5) we have,
dx
dy I ' \d.dy d.d z y> \d.d m y
but to acquire a more convenient notation, let
= x
x l
dy
= Y l
therefore
d.d n x
>>
(15)
...+Y w 8.rfy. (16)
112
244 CALCULUS OF VARIATIONS.
And similarly
[189.
(17)
= /
Jo
+ Y8y+Y!6?.8y + Y 2 c? 2 .8y + ...4- v m d m .by} ; (18)
the order of the symbols of operation having been changed in
accordance with the commutative law established in Art. 187.
But
/*l r -ii ri
I Xj d. bx =. Xi bx /. d\i
Ja I n 'n
o .o
| + /
Jo Jo
/
Jo
-)"/ rf"x
and similar results are of course true for the integration of the
Y'S ; therefore, substituting in (18)
5. / H =
bar
ri
/ (x c?xi
Jo
(20)
which expression it will be observed consists of two parts : one
190.] CALCULUS OF VARIATIONS.
of which depends on the variables at the limits and their varia-
tions ; and the other involves a sign of integration, and being
therefore dependent on the form of the function connecting x
and y cannot be determined unless that function be known :
but by means of which in many cases, as we shall see hereafter,
the unknown form may be found. It is also to be noticed that
the former part vanishes if the limits are fixed ; and if they are
constrained to fulfil certain conditions, relations will exist be-
tween their variations with which the former part of (20) must
consist.
190.] The variation of a definite integral whose element-
function involves two variables and their differentials up to
those of given orders, has thus been found in all its generality;
and hereby some advantage will be gained in future problems,
because we shall be able to preserve symmetry. In many cases
however it is convenient to make x equicrescent, so that
d 2 x = d?x = ... = d n x = ;
and therefore
fl = F (a?, dx t y, dy, d*y, . . . d m y), (21)
.-. x 2 =x 3 =...=x ?l = 0; (22)
./ & =
ri
+ / (x dxi} 8#
Jo
-d^ + d*^-... (-)rfY m } 8y. (23)
191.] We may further observe that it is in many cases un-
necessary to subject x to variation, because the form of the
function may be changed by making y alone to vary, provided
that the variation of y is a function of x and y : this is also
geometrically evident ; each point of the curve may move in a
direction parallel to the axis of y ; thus in fig. 47, P may- be
shifted to B ; in which case if all the points do not move
through equal spaces, but through spaces which are functions
246 CALCULUS OF VARIATIONS. [192.
of the coordinates of the point in its original position, the form
of the equation to the curve will change, although the point has
the same abscissa in both its positions. If however the extreme
points P and P! are constrained to move in given curves, at the
limits generally x and y must both vary, and consistently with
the equations to the limiting curves. Generally however it is
allowable to equate bx to zero in the above formula (23).
192.] Suppose however that in the element-function x is
equicrescent, and that the quantity whose variation is to be
calculated is n
u - I vdx, (24)
*A)
/ dy d 2 y d n y\
where v = r (,,, J,^,...^) ; (25)
F representing a known function, and the relation between a?
and y being undetermined.
To give to v the most general variation that is possible, let
//?/ CM 77
x,y,-jr-, -r-2, vary, and for convenience of notation let us
substitute as follows :
(27)
. 8/ \dx = I b.
* Jo Jo
= / {v8.
Jo
.bx + b
+ /
= I {vd.
Jo
^
CALCULUS OF VARIATIONS. 247
Again, let
V< n '
dy'~ ~ "~
\ (v\ , i v \
1 v I _ I V I _ I V
r \dy''~ ' \dy"t~ "
-. b. v<&?= v8a? +/
JQ L JQ JQ
. . . + Y< W > (dx by < n > 5a? dy ()
= v8a?l + / {Y% y'8.r) +Y'%' y" 8a?) + ...
L J J
x.
Let byy'bx = a,
b.dy dy
dx dx*
d.by dy d.bx d /dy\
dx dx dx dx \dx>
dx dx dx
da
dx
Similarly by"y'"x <
. b. vdx = \vd#\ +/ {(Y + <oV + ft)"Y // + ...+o)< n >Y< n >}^. (30)
^o L Jo Jo
Now,
-i *I'
a>dx
/i P -]i /*I^Y'
a>'\'dx = COY' / -j-
L Jo ^0 <2?
r 1 r dv"ni /*i // 2 v"
/ CO"Y"^ = U/Y"- co ^-1 + / ^-
Jo L dx Jo */o ^
/ CO (re| Y( n) ^ = r w (-l)Y()_ co (n -
248 CALCULUS OF VARIATIONS. [193.
substituting which values in (30) we have
r 1 r ( d\" d 2 y
b. vbx = \vbx+\y' r- +
Jo ( ax
rf Y <>
_L
(
3 Y (-!)_
(
]i
o
which expression*, it will be observed, consists of two parts;
the former depends on the values of the variables at the limits
of integration ; the latter involves an integration, and which
cannot generally be performed unless the function connecting
x and y is known.
Let it also be observed that the derived-functions of the Y'S are
calculated on the supposition that y, y , y", . . . are implicit func-
tions of x, so that these derived-functions do not vanish even
though the variable x does not explicitly enter into v.
193.] Let us notice certain properties of (31). Suppose
o> = 0, in which case
byy'bx =
*y. d y..
bx dx'
and therefore the ratio of the variations and of the differentials
of y and x is the same : and we have
b. vdx = \dx
JQ L Jo
that is, if we make the coordinates x y of a point on a curve to
vary, so that the ratio of the variations and of the differentials
of the coordinates is the same, we do not leave the curve,
* In the Memoir on this Calculus by M. Poisson, which was read to the
French Academy in 1831, and is printed in Vol. XII of the Memoirs of the
Institute, equation (31) is deduced from first principles.
CALCULUS OF VARIATIONS. 249
but pass to a consecutive point of it, and the definite integral is
increased by the value of its element-function corresponding
to the superior limit, and diminished by that corresponding to
the inferior limit (see Art. 178).
Also the geometrical meaning of &> dx deserves notice. Let
the variations of the coordinates of any point on the plane curve
under consideration be bx and by } and let the projections of the
space through which the point (x, y) has moved be estimated
along the tangent and normal of the original curve at the given
point, and let these projections be r and v; then
dx dy
ds ds
dy ** < 32 >
by = r-f- + v-r
ds ds
(33)
= by dx dy bx
= vds; (34)
ds
dx
d i ds\
, _ d i ds
d.r \ dun
fjn / ffo \
, \ Uf I Wo "
d.r n \ d.v>
(35)
substituting which values in (31), it will be seen that every
term in the part at the limits involves only v (the normal dis-
placement) except the first ; and the part of that involving r is
[dx~\ i r ~i ^
VT -r- , which is equivalent to v8# , if the variation is made
on the supposition that v = 0. Also the part under the sign
of integration involves v only ; the reason of which is, the varia-
tion in the form of a curve due to the shifting of its several
PRICE, VOL. II. K k
250 CALCULUS OF VARIATIONS.
points and elements arises from the infinitesimal normal dis-
placement only ; the effect of the tangential displacement being
to shift a point to another consecutive point on the curve.
If v involves a? y Q y' Q y" ... ,x\y\ y'\ y"% . . . , either one or more
of them, and if these are capable of variation, independently of
each other or subject to given relations, their variations must
be calculated, and hereby the former part of (31) will contain
such terms as
194.] In reference also to the general expression (20) it is
worth remarking, that if 5# and 8/ are replaced by dx and dy,
that is, if the shifting of the point takes place along the curve
only, and if there be no normal displacement, then the total
ri
variation of / SI is that which takes place at the limits ; thus
'O
in this case
8./ 11= \Xidx-\-Yidy\
r -a
+ x 2 era ax 2 dx + v 2 d 2 y d\ 2 dy
L Jo
-1- 1 ^ n d n x dx n d n - l x+ ... ( ) n ~ l d n - l -x. n dx
ni
F
*^o
Ff
Vo ^ Y Yl
and the last two terms, after integration by parts, become
C l
I
Jo
+
o ^o
4- dfydx + dYzdy X 2 d 2 x + Y 2 d*y +
L Jo L Jo Jo
and so on. Hence
1 95.] CALCULUS OF VARIATIONS. 251
= /
Jo
= Pda
JQ
- H'
L -Jo
by reason of equation (17) ; so that the total variation is re-
duced to the difference between the values of the infinitesimal
element-function at the first and the last limits.
195.] We proceed now to investigate the variation of a defi-
nite integral whose element-function involves three independent
variables and their successive differentials ; and to consider the
variations in their greatest generality let us suppose all the
variables and differentials to receive variations. Let
u =
where
i2 = F (x, dx, d 2 x, . . . d n x, y, dy, d z y } . . . d m y, z, dz, d 2 z, . . . d k z).
Let tis first substitute as follows :
I v I I v ( I 7
1 / A \ 1 / " \ 7 / ~~" "
dx I v dy i v dz >
Xl =:Yl :=Zl
\ / \ / \
Td"xl ~~ Xn \Td^y) = Ym \d^z) =
then, as in Art. 189,
811
= / 8
-'o
= / {
Jo
er}; (36)
and reducing these terms by partial integration, according to
the method of Art. 189, we have finally
K k 2
252 CALCULUS OF VARIATIONS.
o.
-i-z^bz
{x dxi
Jo
f {v-dv 1
Jo
z-dz 1 + d 2 z 2 -...(-) k d k z k }bz. (37)
When il involves more than three independent variables, the
expression for the variation of / A 18 of course similar.
Jo
If in (37) bx, by, bz are replaced by dx, dy, dz, so that the vari-
ables are changed by passing from one to another successive
system of values within the range of the function, then by a
process similar to that of Art. 194 it may be shewn that the
only variation which the function undergoes is that which takes
place at the limits, and that there is none due to any change of
form of the function.
196.] Suppose however that an equation of relation is given
between the variables and their differentials which are involved
in il ; and, to fix our thoughts, let us take the case of three
variables x,y,z; and suppose the equation to be
L = f{x, dx, d z x, ...y, dy, d-y, ...z, dz, d' 2 z, ...} =0. (38)
If L involves only x, y, z, z may be expressed in terms of x
and y, and thence dz, d 2 z, ... may be found, and substituted
196.] CALCULUS OF VARIATIONS. 253
in 12, so that ,12 will become a function of only two variables,
x and y : but as L involves the differentials of the variables,
such an elimination is generally impossible, and we are obliged
to have recourse to the following process. Take the variation
of L, viz.
- = 6l = ' < 39 >
and employing a convenient and abbreviated notation,
8L = gbx + id.bx + 2 d 2 bx + ...
+ rjby + r^d.by + rj 2 d*by + ... (40)
+ &* + frd.bz + &d*bz + ... = Oj
Now since the equation L = must be satisfied for all values
of x, y, z which are admissible into the problem, therefore the
variation of x,y,z must be subject to the condition SL = 0, that
is, to equation (40); but since
8v = \bx + -x.id.bx + x 2 e? 2 8# + ... 1
+ YSy + ^d.by + Y 2 c?% + ... >, (41)
+ zbz + Zid.bz + z z d 2 bz + ... J
it is plain that we may add to it the right-hand member of (40)
multiplied by an indeterminate quantity A. without destroying
the truth of the expressions, so that
Observing now the process by which (37) was deduced from
(36), a result similar to (37) will be deduced from (42), wherein
instead of x will be x + Af, instead of x b Xx + A^ . . ., instead of Y,
Y + ATJ, ..., and so on for the others; and therefore the equation
will still involve three independent variables, viz. A, and two of
the three quantities x, y, z.
The variation also will be found in a similar manner if the
254 CALCULUS OF VARIATIONS. [197.
original element-function involves more variables, and if these
are related to each other by many equations of condition.
197.] Suppose however that the element-function involves
three variables x, y, z ; that x is equicrescent, and that y and z
are two unknown functions of x, and independent of each other,
and that the quantity whose variation is to be calculated is
u = I vdx, (43)'
A)
/ dy d 2 y d m y dz d 2 z d n z'\
where v = *( x ,y,-, ,..., z,--, ,... ), (44)
F being a known function.
To give v the most general variation, suppose that not only
#, y, z } but that also the derived functions of y and z vary : then,
adopting the following substitutions,
dy , d 2 y d m y
_ *L it/ _ _ gf' _ 2. .(mi
dx ~ y ' dx* " " dx m ~ y
dz , d z z ,, d n z
. _ z _ z _ z^ n >
dx~ ' dx* ~ '"' dot" ~
v = F (x, y, y', y", . . . y<*>, z, z, z" y . . . *<>) ; (45)
and,
dyj- '\dy'>
5v =
+ z bz + z' bz' +z"bz"+ ... + z' w > 82f<" ; (46)
and following a process precisely similar to that of Art. 192,
and extending it to z, and putting
by y'bx = co bz z'bx = &>i
by y" bar o> bz' z'bx o/i
we have the following result :
1 97.] CALCULUS OF VARIATIONS. 255
d ^'" , d*-i Y <> -)
4- nr ...( V"- 1 = -p- f&>
^ 2 da? 1 *- 1 J
_^+ m _ 2 ^-^ Y ^i .
^ da?*-* J
dx
(-)
(ra) wj^- 1 )
-lo
an expression consisting of two parts; of which one involves
the values of the variables and their variations at the limits
only ; and the other involves a process of integration, and which
cannot be performed unless the relations between x and y and z
are given.
Let us however examine it from a geometrical point of view ;
and let us consider the general displacement of a point to be
due, (1) to two displacements perpendicular to each other in the
normal plane, and (2) to one along the tangent line ; now by a
process exactly analogous to that of Art. 193 it may be shewn
that the quantities under the signs of integration involve the
256 CALCULUS OF VARIATIONS. [198.
normal displacements only ; and that v bx\ is the only term
L -"o
wherein the tangential displacement appears.
If v explicitly involves the values of the variables at the
limits, viz. # yo Z Q} x\ y\ z\, and if these vary, then to the former
part of (47) terms must be added which arise from the variations
of these limits ; and these will be of the form
* + ^ *+
And if v contains any number of undetermined functions, the
variation of/ vdx will be calculated in a similar manner, and
^o
will consist of a series of terms and quantities similar to those
of equation (47).
198.] In the last Article y and z are considered to be inde-
pendent of each other; suppose that a relation is given con-
necting them and their derived-functions and x, and of the form
multiplying which by an indeterminate multiplier A, and adding
it to 8v, which is given in equation (46), we have
(50)
comparing which expression with that of (46), and noticing the
process by which (47) is deduced from (46), it is palpable that
(50) will lead to a result of the form (47), and with quantities
such that in the place of Y will be Y + A. (-r- j , in the place of Y',
y
(-7-,), ... in the place of z, z + A(-T-
' + A (-7-,), ... in the place of z, z + A(-T-J, in the place of z',
1 99-] CALCULUS OF VARIATIONS. 257
z' + A. ( -j-, ) , . . . and so on : and thus the variation will be reduced
\dz '
to the form of a definite integral, whose element-function in-
volves x and two unknown and independent functions of x.
199.] Certain processes in the sequel will require the calcu-
lation of the variation of a variation, that is, of the second varia-
tion of a definite integral. As the principles and the method
are the same as those explained and applied in the preceding
Articles, we will consider only one simple instance : viz. it is
required to find b 2 u, having given
=/'
v'o
where v = F (a?, y, y', y" , y'", ... y<>). (51)
bu = I b.vdx
= I {dxbv + vb.dx}, (52)
Jn
(53)
=j
;(55)
but {^r-}, (~r)> (TTV) are functions of a?, y, y', y"...,
/c?v\ / d 2 v \ /d^v\^ / d 2 v \ .
8 -( j-j = w // J 8lig + \"^~2/ ^ + \/7 V / ^ +
substituting which values in (55) we have,
+
PRICE, VOL. II. L 1
258 VARIATION OF A MULTIPLE DIFFERENTIAL. [2OO.
and therefore
+ [\tf.dx. (57)
Jo
By similar processes may 8 3 w, 8%, ... be calculated.
200.] Before we enter on an inquiry into the variations of
double and multiple integrals, it is necessary to investigate the
variation of a product of differentials of the form dx dx 2 dx s ... ,
x^x^Xz, being n variables independent of each other.
Let i, %, 3, ... be the values of #1, x%,Xz, ... in their varied
states, that is, after the displacements have taken place : so that
2 =
where &TI, bx 2 , ... are functions of x\ t x z ,Xz, ... : then, our object
being to calculate d i} d 2 , d z , ... , we have
so that, by the process of Art. 151, and neglecting infinitesimals
of the higher orders, we have
C d.bXi d.b#2
= -s 1 + 7 -- h 3 -- h
ax\ ax z
ax\
* Many of the brackets which are indicative of partial differentiation are
omitted in this and the following Articles, that the heaviness of the formulae
may be relieved.
201.] VARIATION OF A DOUBLE INTEGRAL. 259
r
1
-)
... v
j
,
(, dxi dx%
Hence if x and y are the coordinates to an element of a plane
area, the varied value of dx dy is
d.Sx d.b
and therefore the variation of dx dy, or
Similarly, for the variation of the volume-element referred to
rectangular coordinates, we have
d.bx d.b d.bz .
(59)
201.] Instead of investigating the variation of a multiple in-
tegral in its most general form, I shall consider, for the sake of
simplicity, the case of a double integral only : for the principle
on which the inquiry is founded is the same in all cases; and
the number of terms in the result increases so rapidly with each
new integral sign, that by taking any higher order the formulae
are so complicated as to require new symbols and new modes of
abbreviation, and no useful result is arrived at.
And I shall consider only a simple case of a double integral :
that, viz. in which the element-function involves a?, y, z (z being
an undetermined function of x and y) and the partial derived
functions of z of the first and second orders ; and in which also
the limits of integration are given by an inequality in accordance
with the principles explained in Art. 154 : for the process of deter-
mining the complete variation of a double integral in its most
general form is so long, that it would far exceed the limits of
the space which can be given to it ; and it is the less necessary
to enter on the investigation because (1) the simple case will be
sufficient for all the examples to which the process will be
applied; and (2) the student who desires information will find
the difficulties elucidated in M. Poisson's Memoir cited above (see
Art. 181), and in that of M. Ostrogradsky in the St. Petersbourg
Memoirs for 1838, and in M. Delaunay's Memoir in Journal
de 1'ficole Polytechnique for 1843, 29th Cahier: the Memoir
however by M. Sarrus, which is mentioned in the foot-note of
Art. 181, is the only treatise, which I have met with, wherein the
values of the terms at the limits are fully developed.
L 1 2
260 CALCULUS OF VARIATIONS. [2OI.
Let the definite double integral be
u=T 1 f \dydx, (60)
c/xo A
and suppose v to be a function of
tdz\ fdz\ (d*z\ i d z z \
x >y> z > \)* \d)> /' \dxdl '
dy)> fe' \dxdy
or, substituting as follows,
idz
\~
dz\
t "
the upper and lower accents referring to partial derivation with
regard to x and y respectively,
v = F (a?, y, z, /, z t) z", z' t) zj (61)
and let the limits of integration be given by the inequality
4> (x, y) < 0.
Now u may vary in consequence of (1) a variation of the
limits due to a variation in the form of $ (x, y} ; (2) a variation
of the element-function : these we shall consider in order.
Suppose that only two values of y are given by <f> (x, y} = 0,
viz. YI and Y O ; and that their variations are YI and 8 Y O ; and let
Vi and v be the values of v, when y = Y X and y = Y O : then, ob-
serving the order of the integrations in (60), the variation which
the definite integral undergoes at the ^-integration is
f
I
*'*
(62)
And to obtain the variations of these quantities which arise
from the variations of the limits X! and x , we must successively
substitute in them Xi and XQ for x, and 8x1 and 8xo for dx; but
since Xi and XQ are the values of x determined by the equation
which results from the elimination of y between = 0, and
-j- = 0, and therefore, when the values of y are equal, it follows
dy
that YI = Y , both when x = x x and when x = x , and therefore
vdy
f
Jvn
vanishes, for x = KI and x = XQ, and therefore the terms arising
from the variations of the limits Xj and x vanish : hence the
whole variation of the double integral arising from the variation
of the limits is that expressed in (62).
202.] VARIATION OF A DOUBLE INTEGRAL. 261
We may also, by the way, observe, that in the case of a mul-
tiple definite integral of any number of variables,, if the limits are
given by means of an inequality, as in Art. 154, the variation of
the integral due to the variation of the limits is that which arises
from the variation of the limits at the first integration only.
202.] Next let us consider the variation of the element-
function.
nVj
v dy dx (63)
bu = / / b.v dy dx
Jx JY O
n Y l
{dydxbv + vb.dydx} (64)
substituting for b.dydx from equation (58); now integrating
by parts the last terms, and remembering that the order of
integration is indifferent provided that attention is paid to the
value of the Limits given by the inequality <p (x, y) < :
f x ir ~I Y I /* Y ir ~l Xl T Xl f Yl ( /dv\ /dv\ )
\vby \ dx+ fv8^ +/ / ]8v-(5- )8a?-(5-)8y^y<fo?. (66)
^xo L JY O /Y O L Jx JXQ Jy Q ( X ^ 7 X y 7 )
Substituting x, Y, z, z', z y , ... for the partial differential coeffi-
cients of v with respect to a?, y, z, z', z j} ..., we have
Sv = xbx + vby + z8z + z'bz+z^ + z"bz" + z j 'bz; + z^z jt (67)
dv = \dx + vdy + zdz + z'dz +z ( dz / + z"dz" + z'dz / ' + z /i dz / , (68)
and .-. &}te= {x + zz' + z'z" + z z' + z" z'" + z'z," + zz'} bx
\ttX'
Substituting these, the last part of (66), which is affected with
the double signs of integration, becomes
nY]
{z (bz z'bx z,by) + z (bz z" bx z'by )
+ z, (bz t z' t bxz^by)
+ z"(bz"-z'"bx-z,"by)
+ z ji (bz t jzjbxz, / by)} dydx. (69)
Let bzz'bxZjby = w,
262 CALCULUS OP VARIATIONS. [2O2.
, , f
and let = o> , =. o> , -^ = co , , , , -j-
</# c?y ' dx* dxdy ' dy 2
and for the sake of simplicity consider bx to be a function of x
only, and by to be a function of y only * : then
^ dz d 2 z ^ d 2 z .
bz' z"bx-z by = 8.-, -- __8a?
2
-__8 .
c?^? to 6?^? da? 2 " dxdy ^
d d (dz \ d (dz \
~dx dx\dx ' dx-dy '
dx
= O)'.
Similarly, bz / z'bxz // by=Q> / ;
and so for the others : and therefore the last part of (66) becomes
/* x i /* Y I
/ / {z&) + z / G)' + z / &) / + z"(d"4-z / '&)/ + z // ci) // } dydx; (70)
% ^Y O
and integrating by parts,
n x i r v i r ~i x i r x i T v i </z'
z'vdxdy= \ z '>\ dy / -ju>dydx, (71)
^Y O L Jxo Jx,, -Ar <W?
( x l /* Y i /* x i p T Y J /*X] /*YI ^g
/ i^dydxl z^ dx / ^co dydx, (72)
- - ^Y -'x L J Y A A y
T Y I r x ' /* YI F n Xi C^^rdz" ~i xi C Xi f yi d 2 z ff
/ / z" at" dxdy = / z'V c?y / -760 %+/ / -j~T^dydx } (73
Jy Jxo -'YO L J x () ^Y O L # J x -'x A **
/* x j p -jYj f v irdz' n x ' r xj T Yl <? 2 z '
.' i <* t i dydx= z>' <fe-J h^ ^ + / / d^bi dydx > (7A
' X L -|Y O -'Y O L y Jx -'xo A- ^y
Or if we commence by integrating with respect to x, bearing in
mind the alteration in the values of the limits,
( x l /* Y i /*YJ /*x,
/ z'<a / 'dydx= / z^' (a 'dxdy
- JLO ^Y O ^Y O A
/* Y T / 1 X1 ^ /* x T rfz / "l^j T x > /* Y I c? 2 z'
= ["J.Tj [*-J + s^ wrf ^ ;
* This condition is shewn by M. Poisson not to affect the subsequent
values of &>', ,, . , . . See Mem. de 1'Institut. Vol. XII. p. 290.
203 ] GEOMETRICAL INTERPRETATION. 263
and for the sake of symmetry taking half the sum of (74) and
(75)
n Yl / I
__ . i -"'** r =2
+ -^dydx. (76)
Jx A datdy
And similarly let the other term be integrated ; so that finally
substituting in (66) we have
1 dz' dz \ 1 ! Y I
I ,
d 2 z"
203.] Now this expression consists of two different classes of
terms, viz. of single and of double definite integrals : the latter
class, which is contained in the last line of (77), does not admit
of reduction ; because its element-function involves <a, which is an
arbitrary quantity, depending on bz, bw, by, and therefore on the
form of the function, connecting these three variables, and which
is undetermined. The former class consists of a series of single
definite integrals ; of which some involve 8,2?, by and o>, and
manifestly do not admit of reduction : but others, which involve
to' and co 7 might at first sight appear capable of further integra-
tion by parts : but they are not so, because the values of a/ and
co y which enter into them are not the partial derived functions
of a function of y and x, but are particular values of these de-
rived functions obtained by substituting for x and y, xi, x , YI, Y O ,
as the case may be ; and therefore it is only after such substi-
tutions have been made that the expression can be further
integrated.
204.] One or two points of the variation of the double defi-
nite integral given in equation (77), as interpreted geometrically,
deserve notice; the integral / / \dydac represents either a
Jx Jv
volume contained between given surfaces, or the area of a
curved surface (see Art. 134 and Art. 142), the boundaries of
CALCULUS OF VARIATIONS. [204.
which in both cases are cylindrical surfaces perpendicular to
the plane of xy, and which are determined by the limiting equa-
tion <f) (x, y) = 0. With a view to simplification, let the limits
of the x- and the ^-integrations undergo no variation ; that is,
let there be no variation of the equation < (x,y) 0. And, as the
interpretation is similar in both cases, to fix our thoughts, let us
suppose the double integral to represent a volume ; then, as z is
an undetermined function of x and y, and as v contains r, and
its derived-functions, the variation of the double integral causes
a variation in the unknown function, and thereby in the volume,
and therefore produces a change in the form of the surface
bounding the volume ; the variation, that is, involves a passage
from one to another and consecutive surface. Observe now the
meaning of o> ; for the sake of symmetry, let us suppose the
equation to the surface when determined by the conditions of
the problem, whatever they are, to be
*fo&*) = c; (78)
then replacing / and z t by their values, (see Art. 50, Vol. I,)
= z z x z
dz
and let us suppose the total displacement of any point on the
surface to be the effect of three combined displacements, two
in the tangent plane and at right angles to each other along
the lines of curvature, and the third in the direction of the
normal : let this latter displacement be represented by v ; let
\dx> \dy' \dz'
() (*!) (*1)
then x y Z are the direction -cosines of the
dy> y \dz
normal to the surface, and therefore ^ -^
is the projection on the normal of the three partial variations
2,05.] AN INFINITE NUMBER OF VARIABLES. 265
along the coordinate axes, and therefore is equal to i>; and
thus, if y is the ^-direction angle of the normal,
o> = t>secy; (81)
therefore the last part of (77), that namely which alone involves
the double sign of integration, depends on the normal displace-
ment only.
Suppose that co = 0; then from (81), v 0, and there is no
normal displacement : whatever displacement therefore any
point on the surface undergoes, it is in the tangent plane only;
and thus it involves a passage from one point to another conse-
cutive point on the surface, and does not require any change of
form of the surface : hence the expression for bu in equation
(77) becomes
/ x ir T Y I T Y ir ~i x i
vcty das + vS# dy; (82)
-0 *- -" Y 'YO L Jx
And these terms involve the variables at the limits, and there-
fore express the variation of u due to the variations of the limits
in the tangent plane of the required surface along the lines
wherein it meets the limiting curves.
Also if co = 0,
bz = sfbat + zjy, (83)
but if the equation to a surface be z = /(#, y ),
dz = z 1 dx -f z t dy ; (84)
and as z' and z t are variables, from (83) and (84) we infer that
? = ^ = ^. (85)
dx dy dz '
that is, the variations and the differentials of the coordinates of
the point under inquiry are proportional to each other ; the
new position therefore of the displaced point is on the original
surface, and therefore the displacement has been wholly in the
tangent plane.
205.] It is good to consider a difficult subject, such as that
under discussion, from another point of view. We have con-
ceived the quantity involving the unknown function to be resolved
into its elements, and the definite integral of these elements
to be the finite quantity which is the subject of inquiry; and
the limits have been taken to be values whose symbols have
subscripts 1 and 0. Now imagine the definite integral to re-
PRICE, VOL. ii. M m
266 CALCULUS OF VARIATIONS. [205.
present some property of a plane curve, and between the values
x\ and # ; this restriction is convenient to fix our thoughts ;
and let the quantity X\XQ be resolved into n elements, and
1 2 3"- n -i be the values of x corresponding to the points of
division, and the corresponding values of y be y\ yz.-.y n -\ ' then,
as the definite integral is the sum of a series of quantities, of
each of which the element-function is a type ; so if we replace
the definite integral by its equivalent series given in equation
(12), Art. 6, it will be a function of Xoiz---n-itfn, that is of
n + 1 variables ; and when the elements are infinitesimal, of an
infinite number of variables : this then is a distinguishing mark
of the Calculus of Variations ; its immediate subjects of inquiry
are functions of an infinite number of variables generally inde-
pendent of each other ; but as these functions consist of a series
of terms, all of which are of the same form, the differential, or
variation of the sum of them, is equal to the sum of the differen-
tials or variations of the separate terms : hence the cause of
b and / being subject to the commutative law. The principles
of the calculus of variations therefore are only different from
those of the differential calculus, because its subject is a func-
tion of an infinite number instead of a finite number of varia-
bles.
It will also be observed, that if for the definite integral the
equivalent series of terms involving intermediate variables be
substituted, the number of variables that enter into each term
will depend on the order of the highest differential which enters
into the element-function ; thus if the element-function involves
d 2 y, three consecutive values of y will enter into each term;
and so for other forms of the element-function.
I will not however enter on further inquiry into this method
of the calculus of variations, because the process is much larger
than, and ultimately leads to the same results as, the preceding;
but because the principles of the calculus become hereby re-
solved into their most simple elements, nay more, because the
processes of perhaps the most transcendental analysis hereby
become capable of geometrical interpretation and construction,
I shall take an opportunity, in the next Chapter, of solving a
simple problem by this method ; and the mode of application
will thereby be evident to the student.
206.] PROBLEM OF MAXIMA AND MINIMA. 267
CHAPTER XII.
APPLICATION OF THE CALCULUS OF VARIATIONS TO PROBLEMS
OF MAXIMA AND MINIMA.
SECTION 1. Determination of the critical values of a definite
integral whose element-function involves variables and their
differentials.
206.] WE proceed to apply the principles of the last Chapter
to a large class of problems of maxima and minima involving
unknown functions.
At this part of our treatise it is superfluous to repeat the
conditions and the criteria for determining maxima and minima
values of known functions, and which depend on particular
values of the subject-variables of these functions, for the whole
question has been fully discussed in Chapter VII of Vol. I, and
the reader is supposed to be familiar with it. Suppose however
that the problem is to determine the form of a curve or curved
surface between certain limits, so that a property of it, such as
its length or the area enclosed by it, may have a maximum or
minimum value ; the principles of Vol. I are plainly insufficient,
because the form of the function is unknown ; and we have
recourse to the following mode of solution: let the property,
whose value is critical, be resolved into its elements, the element
being a known function of the variables and their differentials,
and this being independent of the relation between the varia-
bles ; then the sum of all these, or, in other words, their definite
integral, is the quantity whose critical value is to be found, and
by which means the form of the function is to be determined.
The definite integral therefore is the subject of inquiry, and is
such as those whose variations have been calculated in the pre-
ceding Chapter.
207.] Let u represent the definite integral, of which the
critical value is to be determined; and first suppose that the
M m a *
268 CALCULUS OF VARIATIONS. [207.
variables and their differentials of which it is a function are
independent of each other ; that is, that there is no equation of
relation amongst them : a maximum or minimum of such a
kind is termed absolute : then, by the theory of maxima and
minima, it is plain, if u has a critical value, that bu = and
changes its sign ; and that the change of sign may be deter-
mined by the sign of b 2 u ; so that if bu = 0, u has a maximum
or minimum value according as b 2 u is negative or positive ; the
solution of the problem therefore requires the calculation of bu,
and of b 2 u ; and by the condition bu = 0, the form of the func-
tional symbol connecting the variables is to be found.
Suppose then, as in Art. 189,
= fq,
^o
(1)
where 12 = F (x, dx, d z x, . . . d n x, y, dy } d 2 y, . . . d m y), (2)
and F is the symbol of a known function. On referring to the
value of bu given in equation (20) of the last Chapter, it will be
observed that it consists of two parts ; one of which is inte-
grated, and depends on the values of the variables, their dif-
ferentials, and their variations at the limits ; the other is under
signs of integration, and cannot be further reduced, because
bx and by are unknown functions of x and y, and because the
other factors in the element-functions involve the undetermined
function and its differentials. What conditions therefore are
requisite that bu = 0? For convenience of reference let
+ x n d n - l bx = a; (3)
and let the analogous quantity involving by and its differentials
= /3 : also let
-...(-)d'%, = H, (4)
...(-) m d" ! Y m = H; (5)
so that we have
bu = a + i\\ P(Eaff+H$y). (6)
208.] MAXIMA AND MINIMA 269
Now, as bx and by are arbitrary functions of x and y, bu
cannot vanish unless a + /3 = ; whence we have
= 0, (7)
0; (8)
and also, 5 = 0, (9)
H = 0; (10)
and these are the conditions which are primarily necessary to u
having a maximum or a minimum value.
208.] Although it is desirable, both for symmetry and for
the discussion of an expression in its most general form, to re-
tain all the terms in bu thus far, and although in many of our
subsequent examples we shall retain them throughout, yet it is
necessary somewhat to abridge them, that we may point out
some general properties of the above equations.
First, let the difference between (7), (8) and (9), (10) be ob-
served : (7) and (8) involve limiting values of bx, by, and of their
differentials ; whereas H = 0, and H = 0, being differential ex-
pressions, will after integration give general relations between
x and y, and therein the required functional connection; and
the same function will be deduced both from H = and from
H = 0, provided that (and this is a necessary condition) the same
limiting values are taken in the integrals of both equations:
for the form of the function involved in them will depend on
the form of function of XI, and from fl they are derived by a
similar process; and therefore the same functional form will
appear in the final result of each.
Again, let us suppose that there is no variation of x, save at
the limits ; and that therefore the shifting of any point from a
curve to the next consecutive curve is due to a variation of y
only ; then bx = (except at the limits), dbx = d 2 bx = . . . = :
so that (6) becomes
(11)
270 CALCULUS OF VARIATIONS. [209.
209.] Suppose that O, which involves d m y, is not linear with
respect to d m y, then Y m is a function of d m y, and therefore d m \ m
involves d 2m y : the equation H = involves therefore d 2m y ; and
as, during the process of integrating H = 0, an arbitrary con-
stant is manifestly introduced at each successive integration,
so does the complete integral involve 2m arbitrary constants;
thus, if T is the complete integral, it involves c\, c 2 , c 3 , ... c 2m ,
that is, 2m unknown constants : and these must be determined
by means of the former parts of equation (11), which are func-
tions of the limits.
Now if the limits are not restricted by any given conditions,
the former parts of (11) will contain 2(m + l) arbitrary quan-
tities, viz.
# , fyo, %o, d 2 by 0> ...d m - l by 0) &? b by l} <% b d 2 by 1} ... d m -^y l} (12)
of which therefore the coefficients must be separately equated
to zero : hereby we shall have 2(m + 1) different and independent
equations to determine 2m arbitrary constants, and which are
manifestly more than sufficient : and this was to be expected :
for if there is no restriction on the limits or their variations, the
definite integral might be of any magnitude, and would not
therefore have either a maximum or a minimum value.
Suppose however that equations are given connecting the
variables at the limits, that is, that equations are given between
XQ and y 0) and between x\ and yi : then, if T = is the integral
of H = 0, there will be given
dT\ (d 2 T\ ( d m - l r\ (dT\ (d m ~ l r\
) > \w>o''''\W^'o' '\Jy'i''''\'tir^'i' (
which added to the 2m + 2 expressions of (12) give us 4m + 2
different quantities whereby to determine 2m constants c\, c 2 , ...
c zm , and the 2m + 2 quantities
#o, yo, dy , d 2 y Q , . . . d m y , #1, yi, dy 1} d 2 y l} . . . d m yi. (14)
When ii is linear with respect to d m y, H = will be a dif-
ferential equation of the order 2m 1, and therefore its com-
plete integral will contain only 2m I arbitrary constants ; and
the number of equations relative to the limits of the general
integral being the same as before, the problem is manifestly
indeterminate.
210.] MAXIMA AND MINIMA. 271
210.] We proceed to consider certain cases wherein the dif-
ferential equation H = assumes particular values, and hereby
admits of integration.
(1) Suppose that 12 does not contain y; then H becomes
-dYi + d a Y 2 -...(-)'- 1 dY lll = 0, (15)
which admits of integration without any determination of rela-
tion between y and x.
(2) Suppose that 12 does not contain the first k terms of
y, dy, d 2 y, ... , then H = becomes
(_)*-l^Ya-)M* + hr* +1 -...(-)-l^Y TO = 0, (16)
and which admits of being integrated k times in succession.
(3) Suppose that 12 does not contain x, then, according to
equation (17), Art. 189, and from H = 0,
^12 = Y dy + YI d.dy + Y 2 d.d 2 y + Y 3 d.d 3 y + ...
= y-d.Yi + e^.Ya-cF.Ya + ...
therefore, eliminating Y,
= Y! d.dy + dy C?.YI + Y 2 d.d z y dy d 2 .? 2 + Y 3 d.d s y + dy d 3 Y B + ...
and which admits of immediate integration, viz.
12 = d + YI dy + Y 2 d 2 y dy d.\ 2 + Y 3 d 3 y d 2 y c?.Y 3 + dy d 2 Y 3 + ... (17)
211.] Now suppose that the variables x and y which are in-
volved in the element-function 12 are not independent of each
other, but are restricted to certain values expressed by the
equation, integral or differential as the case may be,
L = 0; (18)
then, as explained in Art. 196, we have
8L = 0, (19)
and a relation is given which the variations of the variables and
their differentials must satisfy; multiplying therefore 8L by an
indeterminate constant multiplier X, and adding to bu, we have
8{w + AL} = 0; (20)
and we may operate on W + AL in a manner precisely the same
as that by which we have determined the necessary conditions
/i
12. Such are called relative
_
272 CALCULUS OF VAKIATIONS. [a 1 2.
maxima and minima, and the method of determining them is
hereby reduced to that of finding absolute maxima and minima.
It is also manifest that if the problem be the determination
of the maximum or minimum value of u, when the variables
and their differentials are subject to conditions expressed by a
series of equations, which may be in the form of definite
integrals or other, viz.,
LI = /i, Lg = /2- L * = 4> (21)
then it is sufficient to determine the absolute critical value of
u + \ l li + \ 2 l 2 + +**!*; (22)
where A! X 2 . . . A* are undetermined constants, but which will Jbe
determined by means of the necessary equations arising from
equating to zero the variation of (22), and from the equations
(21).
212.] The above principles are also applicable to the deter-
mination of the critical values of u where (see Art. 195)
u =
and 12 = F (#, dx, d*x, . . . d n x, y, dy,... d m y, z, dz,... d k z) ; (23)
and employing substitutions similar to those of Art. 207,
bu =
f (Z
Jo
+ (Zbx + Hby + Vbz); (24)
and as bu = 0, and bx, by, bz are arbitrary functions of x, y, z,
we must have _ x
L-f/3 + y = (25)
-lo
E = 0, H = 0, * = 0: (26)
of which (25) is a series of equations at the limits, and (26)
when integrated will give the general functional relation be-
tween the variables : it is also to be observed that the same
function will be given by any two of the three equations (26),
for as H, H, ^ are all deduced by a similar process from fi, the
functional form of 12 will be (at least implicitly) contained in
each; and therefore all the integral equations which may be
arrived at from them, provided that they be taken between the
same limits, will have the same functional form : of this result
many examples will occur in the sequel.
2 1 3.] MAXIMA AND MINIMA. 273
213.] To determine the shortest line joining, (1) two given
points, (2) two given curves in the same plane.
(1) Let (XQIJQ) (x\yi) be the coordinates of the given points,
then f l j /OT\
u = / ds (27)
A}
.-. bu =Pb.ds. (28)
But ds z = dx? + dy 2 ,
.. dsb.ds = dxb.dx + dy b.dy,
.-. b.ds = d.bx + d.by, (29)
ds ds
Jo (ds ds
(30)
' ds
= 0, if
. dx ,dy
and c?.-^- = d.~- =
a* a*
dx dy
.-. ~j- = a -f = /3
cfe cfe
a? = a* + a y = (3s + b
xa yb
(32)
which is the equation to a straight line, and which is therefore
the shortest line ; a, /3, a, 6, being four arbitrary constants in-
troduced in integration; and which may be determined as
follows : since the limits are fixed &r = 0, by = 0, &a?j = 0,
8yi = 0; and therefore equation (31) is satisfied without any
relation between the constants of the straight line and the
limits : but as the line is to pass through the two points, x and
y must satisfy simultaneously (# ^o) and x\, y{) ; therefore (32)
becomes
XXQ _ yy
X\XQ ~ yiyo'
which is the equation to a straight line passing through the two
given points.
PRICE. VOL. II. N n
274 CALCULUS OF VARIATIONS. [2.14.
(2). The process of determining the unknown function is the
same in both parts of the problem ; but in the second part the
constants a, /3, a, b must be found as follows : from (31) we have
Let the equations to the limiting curves be
FO (#0, yo) = 0, FI (^ yO = ; (34)
then as XQ, tyo are the variations of x and y as we pass from
one point on the limiting curve to another consecutive one, they
are subject to the relation
(fl V \ ( fill \
-r-j (~r) refer to the required straight line at the
limit, it follows from (33) that the straight line cuts the curve
at right-angles; and a similar result is also true at the other
limit : hence we have
F (M) =
MM) = 0.
By means of which four equations we can determine , a and b,
and thereby definitely fix the line whose equation is (32).
214.] To determine the form of the longest or shortest line
which can be drawn from one curve to another curve in space.
Let the equations to the curves be
y\ = f\ (#
ri
then u = I ds
Jo
=A
Jo
rf.v
2 1 5.] MAXIMA AND MINIMA. 275
, dy , dz
efe
and as bu = 0, we have
(37)
, dx
' ds
4 A
ds
=
dz
' ~j~
as
dx
ds
a
dy_
ds
= *
dz
ds
y
' a =.
a.v
y b
= p
z c =
y.v
xa
y b
z c
(38)
P y
Hence a straight line whose equations are (38) is the longest or
n dx dy dz
shortest line joining two curves in space ; and , -~, -7- are
cis cts as
the direction-cosines of the line, and 8^0 8yo teo, ^\ Syi 8^1 are
the variations of the limits, and therefore, along the limiting
curves; and thus it follows from (37) that the line cuts both
curves at right-angles; and from these conditions combined
with the equations to the limiting curves the unknown con-
stants of (38) may be determined.
215.] It is required to determine the maximum value of
/ p,ds, ds being an element of a plane curve and p being a
Jo
function of the coordinates x and y.
bu
f 1
= I fj.ds
JQ
=
JQ
N n 2
276 CALCULUS OF VARIATIONS. [2 1 5.
also V = te+|F. ' (39)
Therefore integrating by parts
dx ^ .. rf y*..T
and since 8w =
(40)
^J =
(41)
~) =
Therefore from (41)
,c?d? /<4,\ , dx
P-d-j- \^r-) ds -r
ds \dx> ds
dx ( /d^i\ /dp.\ )
= (-l-)ds -=- <(-/- }dx+ (-f)dy >
ds (\dx> \dy) y )
dx dy
ds d> ds
similarly
whence, squaring and adding, and substituting from equation
(19), Art. 236, Vol. I, we have
H _ /dfj.\ dy /dp.\ dx
p ' \dx' ds \dyl ds '
where p is the radius of curvature at the point (x, y~) ; and
therefore , i c ,j , j /^^^^^
1 _ l\(dp\dy _(dfi\d*\ ( ^.
p~ H\\dx> ds \dy>dsV
This equation gives a geometrical property of the curve ; and
2 1 6.] MAXIMA AND MINIMA. 277
we cannot proceed further with the integration unless the form
of p. is given.
If the limiting values of x and y are given, the equations (40) are
satisfied without any relation between #0^0? x \y\> (~j~) > be-
cause bx = tyo = S.TI = tyi = : if the limits of integration are
on two given plane curves, then (40) shew that the required
curve cuts both the limiting curves at right-angles.
216.] Suppose that ds in the last problem is an element of a
curve in space, and that p. is a function of x, y, z, then the equa-
tions of limits and of the indefinite terms become
(44,
dsl J
From (43) we infer, that if the curve is to be drawn between
given limiting curves, it cuts both these curves at right-angles.
Also from (44)
7 dx /du.\ 7 7 dx
ds
dz id\i\ dz
pd. -j- = i-j- }ds dfji-j-;
ds \dz> ^ ds
therefore, squaring and adding and substituting by means of
equation (23), Art. 325, Vol. I,
278 CALCULUS OF VARIATIONS.
and this equation does not admit of further reduction unless
the form of /u, be given. It is worth observing that the line is
straight if ,"^'\ ,, ,
djj.\ id^\ /dg\
\dx> \dyl W
dx dy dz
217.] To determine the form of a plane curve which passing
through two points (xiy\), (#o*A>) generates by its revolution
about the axis of a? a surface whose area is a minimum :
f 1
= 27T/ yds,
Jo
dy
(47)
- *|-a (48)
fi?(2?
Integrating (48) y~ c, (49)
ftp
and therefore the projection of y on the normal of the curve is
constant.
Substituting in (47)
ds-cd.^- = 0,
dx
dx
c
2 1 7.] MAXIMA AND MINIMA. 279
dy^\ s \ x a
dx 2 ' ) " c '
= e
j .r a a; a
.. aw
. . 2~~ = e c e c
ax
xa xa
y-b = g{e~ + e~~}, (50)
which is the equation to the catenary, a, b, and c being arbi-
trary constants and to be determined. For the sake of sym-
metry, let us suppose the limiting values of y to be equal, and
let the axis of y bisect the line joining the extreme points of
the curve; then yo=yi, #0= &i, and therefore from (50) =0;
whence we have
y-b =
dii
Also since ~ = Q, when x 0. the curve cuts the axis of y at
dx
right-angles, and, as appears from (49), at a distance c from the
origin ; c being an arbitrary constant which we have no means
of determining; and therefore from (51) b = 0. Hence the final
equation becomes
~~' (52)
If the curve is to be drawn between two given curves, then
equations (46) shew that it cuts both at right-angles.
This example is plainly a case of Art. 215, where [J. = y, and
therefore (-j-l = 0, (.-*-) = 1, and therefore from (42)
\dx> \dy'
11 /J "Y 1
JL 1 1 .f
p ~ y ds'
ds
''' p = - y Tx'
that is, the radius of curvature is equal, and in opposite direc-
tion, to the normal ; which is a known property of the catenary.
280 CALCULUS OP VARIATIONS. [218.
218.] Of all plane curves which can be drawn between two
given points, to find that which contains between the curve, its
evolute, and the radii of curvature at its extremities the least
area.
Let p be the radius of curvature, and ds be the arc of the
curve ; then it is manifest that
u = o
( x \yi}-> (xoyo) being the coordinates of the limiting points.
.NOW,
bu = i i * {dsbp + p b.ds}. (53)
Jo
1 d 2 x dy d 2 y dx
- -j-o
p as 4
bp _ dy d 2 bx dx d 2 by + d 2 x dby d 2 y dbx _ 3(d 2 xdy d 2 ydx)dbs
~T 2 ~ ~ ds? ~~ds*~~
(54)
pds ' 9
p 2
.'. dsbp + p b.ds = 5s (dy d ^ x ~ dx d *ty + dPx dby d 2 y dbx}
-y- db,v + -j~ dby > , (55)
b.ds having been replaced by its value given in equation (29).
Hence integrating by parts
= = {dsbp + p b.ds}
n,p 2 dx C) d 2 x . dy) ^ ( 7 p 2 dy .d 2 y . dx) ^ "l 1
d --^-+P ^z-^P^rtty- ] d --^-f+P -/J + 4 PT7-c 8<z>
ds 2 ds 2 r ds } " ( ds 2 ds 2 r ds ) J
therefore if a and 6 be arbitrary constants,
l
(57)
9 , dy ,
p 2 -4p-f + rf.^ = a
2 ds 2
, d 2 y . dx
'
2 1 8.] MAXIMA AND MINIMA. 281
' P
d 2 s . (dx d*x + dy d 2 y) ds 2 - 2 ds* d 2 s
-r- + 2pdp + p 2 - ^"33 - = adx + bdy
as as
(58)
Also from (57) by subtraction
and, after some obvious reductions, we have
dx
(59)
Either (58) or (59) is a geometrical definition of the curve.
From (58) it appears that the square of the radius of curvature
is a linear function of the coordinates; and as the radius of
curvature is an absolute quantity and independent both of the
origin and of the particular system of coordinate axes, we may,
without thereby affecting the generality of the problem, choose
our system of reference such that a = 0, c = ; whereby
p 2 = dy, (60)
From (60) it follows that the curve lies wholly on the positive
side of the axis of x, and that the curvature is the same at all
points equally distant from that axis; also from (60) and (61)
fl T*
we infer that -- = 0, when y = : the curve therefore meets
ds
the axis of x at right-angles. And since
b dx
%Ts =
dx __ Zydy
The equation to a cycloid of which the starting point is the
PRICE, VOL. II. O O
282 CALCULUS OP VARIATIONS. [219.
origin, ^ is the radius of the generating circle, and the constants
o
are such that the origin is on the curve, and the axis of x is
the base of the cycloid.
219.] To find the relation between x and y, so that
n
(x 2 -f y^Y ds may be a rninimum.
/*i
u = I (xP + y^ds-,
-'0
and changing to polar coordinates
/
.
fi
u = I r n
Jo
ds
o
but ds 2 = dr z +r*d0*,
dr , r*dd , rdd*
.-. d.Ss = - r d.br+
r . . r
ds ds ds
= / $r n -j-d.br + r n+2 - r -d.W+ (r n+l -^-
Jo ( ds ds ds
ds ds
= a
Pi/ ^.d6*
I \(r n+l ^-
Jo ( \ efo
n
o, (63)
_. ,
ds ds
r n + 2ffa
and dj^- = Q; (64)
a*
r n+z d0
j
ds
.-. r n+l = asec(n + l)6; (65)
and as this equation satisfies, and might have been deduced
from (63), it is the complete function which gives the critical
value of u.
If n = 0, (65) is the equation to a straight line, and the result
is in accordance with that of Art. 213.
220.] MAXIMA AND MINIMA. 283
220.] We proceed now to the solution of problems of relative
maxima and minima; those namely wherein the variables are
not independent of each other but are connected by some given
relation, which may be integral or differential, or in the form of
a definite integral. These problems are often called isoperi-
metrical, because the given condition when interpreted geome-
trically is frequently equivalent to the length of the curve being
given between certain fixed points or limiting lines.
And although the method of introducing indeterminate mul-
tipliers, indicated in Art. 211, is most convenient for explaining
the course to be adopted in the general case, yet as in the fol-
lowing problems only one condition or relation will be given, it
is better to use a process which results from the theory of inde-
terminate multipliers as explained in Art. 142, Vol. I, and which
consists in equating to a constant quantity the ratio of the
several coefficients of the variations of the variables in both
the definite and unintegrated parts of the given equations : see
Art. 143, Vol. I.
To determine the form of a plane curve which being of given
length revolves about a given line (the axis of #) and generates
a solid whose volume is a maximum or a minimum.
f
Jo
= TT
JQ
(66)
= c = a given length ; (67)
/i
, bu = TT \ (2ydxby + y* d.bx)
Jo
= TT [y z bx\ TT / (2y % 8a? 2y dx by). (68)
L JQ JQ
Also taking the variation of (67),
ri
5c = = / S.ck
A>
whence equating to a constant A. the ratio of the coefficients of
8a? and by in the unintegrated parts of (68) and (69), we have
002
284 CALCULUS OF VARIATIONS. [221.
Zydy_ -Hydx --- ... (70)
d.% d. d ^-
ds ds
the last term of the equality being deduced from the first two
by means of equation (19), Art. 236, Vol. I ;
that is, the radius of curvature varies inversely as the ordinate.
//2?
Also 2y dy = \d.-j-,
whence we have
dx dy ds
expressions which do not admit of further integration, but are
the equations of the elastic curve, the mechanical form of which
will be the subject of investigation hereafter.
If the limiting points of the curve be given, then 8<r = 8^0 = 0,
and 8^1 = 8^1 = 0, and therefore in (68) and (69) the terms at
the limits disappear : but if the line is to be drawn between two
given curves, the arbitrary constants will be determined by
means of the equations to those curves at the limits.
221.] To determine the form of the closed plane curve which
is of given length and encloses the greatest area.
f 1 1 f 1
Since the length = / ds, and the area = - r z dd,
Jo &JQ
; (73)
.-. bu = = i /* {2r dd br + r 2 d.W}
<&Jo
= I fr 2 80] '+ f {r d6 6r - r dr 80} ; (74)
*> L Jo JQ
r 1
8c = = / b.ds
Jo
221.] MAXIMA AND MINIMA. 285
*
ds
_ + rf.,0; (75)
ds ds J J ( v ds ds I ds y
whence, equating to a constant A. the ratio of the coefficients of
the variations of the variables in the unintegrated parts of (74)
and of (75), we have
=+x = + (76)
dr ' dffi
fj __ /v* _
tt-. ^ / j M% 7
ds ds ds
the last term of the equality following from the first two terms
by means of equation (48), Art. 249, Vol. I. Hence it follows
that the radius of curvature is constant, and therefore that the
required curve is a circle.
To find the integral equation to the curve, let us suppose the
origin to be on the curve ; then from (76)
, , ,d9
r dr = Xd.r 2 -r ;
ds
~
ds
ds =
dr
= d0
i * fk
.-. r = 2\cos0; (77)
the equation to a circle whose radius = A, and whose centre is
on the prime radius vector.
Also from the above equation
ds = 2\d6,
whence, as c is the length of the curve, we have by integration
c = 4-rrA,
.-. r = cos0;
and therefore the radius of the circle is expressed in terms of
known quantities.
286 CALCULUS OP VARIATIONS. [222.
222.] Of all isoperimetrical curves joining two given points, to
find that, the product of whose length-element and the square of
its distance from the line joining the two points is a maximum.
Let the line joining the two points be taken as the axis of x,
and let the origin be taken at the middle point of this line, and
let 2 a be the distance between the two points ; then x, y, z being
the coordinates of any point on the curve corresponding to the
commencement of the element,
u =
following a process similar to that of the last Articles, we have
dx , dy dz
d.-j- d.-f- d.
ds ds ds 1
dx 9 dy dz A
ds ds ds
from the second and third of which terms we have
*3r-V*%;\ = 0; (79)
but by the particular system of reference which we have chosen,
when z = 0, y = ; therefore c' :
and as y z +z 2 A cannot vanish for all points of the curve, the
above equation can be satisfied only by
zdyydz = 0,
.'. y -=k; (81)
Z
and therefore the curve lies wholly in one plane passing through
the axis of x : let this plane be that of xy : then z = 0, and
from (78) we have , ,
X.-y == d.y 2 -^-,
ds ds
223.]
MAXIMA AND MINIMA.
287
.-. (X-J)- = *i (82)
which is the differential equation to the required curve, and
does not admit of further integration.
223.] To find the line of constant curvature whose length is
a maximum or a minimum.
In this example I propose to follow the general method for
resolving problems of relative maxima and minima; and for the
purpose of shortening the process and formulae, let s be supposed
to be equicrescent.
Let k = the constant radius of absolute curvature -, so that
and therefore
,oox
- ; (84)
then, applying the symbols of Arts. 195 and 212, we have
=
d.dx ds
_ dx 2A dx
ds k ds 2
(87)
and therefore substituting in H = 0, (see Art. 207,) and as the
values of H and * will be similar, we have
dy
_
ds*
_
_
~
(88)
288
CALCULUS OP VARIATIONS.
[223.
and by integration
dx
Ts
dy_
ds
dz
ds
_^-dx_
2A dy
ds*
2A dz
d 2 z
(89)
a, /3, y being constants introduced in the integration : to deter-
mine them, let it be observed that the definite part of bu given
in equation (37), Art. 195, becomes in this case
Kdx 2A dx
5-T- < ar
(dy 2\
i -/- - -T
(~ds k
2A dz
~k
- J_ _ A d.
j
ds 2
d 2 z
ds*
~)
\
j
+
fa d 2 y
d.bz] ; (90)
Jo
ds* ds*
and which must vanish by virtue of the reasoning in Art. 207 ;
and as no relation is given at the limits between 8x, by, bz, ...
the coefficients of these quantities must separately be equal to
fdx 2A dx
ds k ds 2
, ,
~ L
=
(91)
.ds k ds 2
and as these are particular values of the first equation of (89),
it must be consistent with them ; therefore a = ; for a similar
reason ft = 0, y = : whereby, and differentiating, bearing in
mind that * is equicrescent,
2A dx
T~ds 2
2\ dy
dx
ds
dy
ds
dz
ds
ds*
=
k ds 2
2\ dz
Tils 2
ds*
ds*
= u ;>;
(92)
and employing the symbols of Art. 325, Vol. I, equation (6),
223-1
MAXIMA AND MINIMA.
289
multiplying the above equations successively by x, Y, z, and
observing that X<fe + T rf y + Z fo = 0,
there results
d*z = 0; (93)
and therefore, by reason of equation (40), Art. 330, Vol. I, the
radius of torsion is infinite ; and therefore all points of the re-
quired curve lie in one plane.
Again, from (92), since A is an arbitrary constant, and ds is
also constant, we may replace \ by A' ds : and also, replacing
^'
1 2-7- by h, and k\' by h', we have
K
dx
ds
ds
, dz
h -7-
ds
ds 3
_
ds 3
=
(94)
whence by integration
hx
d' 2 x
ds*
also, because s is equicrescent,
h (x dx + y dy + z dz) =
(95)
(96)
which is the equation to a sphere : and therefore, combining
(93) and (96), it follows that the curve is a plane section of a
sphere, and therefore is a circle.
It may also thus be proved that the curve is plane : from the
last two equations of (95) we have
h(zd 2 yyd 2 z) = c 2 d z y <
.'. h(zdyydz) =
.'. also h(xdz zdx) =
h(ydxxdy} =
PRICE, VOL. ii. p p
290 CALCULUS OP VAKIATIONS. [224.
multiplying these severally by dx, dy, dz, we have
kidx + kidy + k^dz = 0;
.-. 1 x + k 2 y + k 3 z = k;
the equation to a plane : and therefore the curve required is a
plane section of a sphere.
224.] In Art. 205 it has been stated that the calculus of
variations may be considered as a particular form of differential
calculus, wherein the number of subject-variables of any func-
tion is infinite : I propose to illustrate this mode of viewing the
calculus by the following simple example : Between two given
points to draw a curve of given length, so that the area contained
between it, the ordinates to the two points, and the axis of x,
may be a maximum or a minimum *.
Let the coordinates to the two points be x Q y Q , x n y n \ and
suppose the distance x n x Q on the axis of x to be divided into
n parts, and let the abscissae corresponding to the points of
division be x\,x^, ... x n -\, and let the corresponding ordinates
be 1/1, y<L,...y n -\> and also for convenience of notation let
yiyo=&yo, yzyi = &yi, , %\ X O =^X Q ,. # 2 x l = ^x l ,...-,
and suppose the several points, to which these coordinates refer,
to be joined by straight lines, of which let the lengths be As >
A*i, ... As n _! ; and let the sum of these lengths be equal to the
given length c ; then, if A = the required area,
... +2(x n -Xn-i)(y n +y n -i), (97)
c A
^n-l) 2 }. (98)
Let u = the required critical function ; then, A being an unde-
termined constant,
U ss A -J- A C }
wherein u is a function of (n 1) independent variables, viz.
y\ } yzi yn-i) an( l therefore, taking the partial differentials of
* For other examples of maxima and minima solved by this process, see
Schellbach, Variationsrechnung, Crelle, Band XLI, p. 293, 1851.
224.]
MAXIMA AND MINIMA.
291
u with respect to them, and equating them to zero, we have the
following series of equations :
=0
=0
As m
=
J>;
(99)
and because x m+z x m = x m +i + Ao? TO +i # m = 2A# w -f A 2 # m ,
the above equations may be expressed as follow :
=
=
=
=
(100)
all of which are manifestly of the same form, and therefore any
one, say the (m+l)th, is the type of all: now suppose the
number of the points of division of x n XQ to become infinite,
then, taking x, y, s to be the general types of their corresponding
particular values which (100) contain, we have, as the type of all,
as
= 0;
(101)
in which, neglecting d 2 x, because it is added to dx, we have
dy
dx = \d.
ds'
(102)
whence by integration, a and b being arbitrary constants, we
f|Q VP
-*) a = A 2 . (103)
292 CALCULUS OF VARIATIONS. [225.
And to determine a and b and A, we have
(*>*+(*-*> =*1. (104 )
(x n -a? + (y n -b)* = A 2 J
/n
rf*
.
A A A
.-. (#<> a)(y n b) (ac n a)(yob) = A 2 sin-; (105)
A
and from (104) and (105), a, b, and A may be determined.
225.] In the last Article #1, #3, ... x n -\ have been considered
constant, while y^, y 2 , ... y n -\ have been considered variable;
but we might manifestly have divided y n yo into n parts, and
y\> Vz, y n -\ being the ordinates corresponding to the n 1
points of division, have considered these to be constant ; and in
this case the x's would have been the variables; and hence a
process similar to that employed above would have led us to
the equation ,
dy = \d.?f; (106)
as
and also the problem might have been treated more generally ;
the afs and the y's might both have been considered variable ;
and in this case, as the coordinates would be independent of each
other, we should have two simultaneous groups of equations
similar to (100), and from them, by a passage to infinitesimal
subdivision, should obtain two simultaneous equations, viz.
dx \d.-r-
ds
dx
7 . j UJ,
ay = A a. -=-
as
whence, integrating, squaring and adding,
If the student will carefully examine the process by which this
example has been solved, he will perceive that the method
which has been employed in the previous cases, and which was
explained in all its generality in the last Chapter, is precisely
226.] GEODESIC LINES. 293
the same, though it lies concealed under signs of integration
and variation, and is thereby likely to escape his notice. Also
it will be good for him to solve other problems by this simple
method.
SECTION 2. On Geodesic Lines.
226.] The following example, in its primary form, is only a
simple illustration of the calculus of variations, and the dif-
ferential expressions which characterise the curve are found
without difficulty ; yet, as the lines possess important properties
in the theory of Geodesy, and thus especially in relation to the
ellipsoid of three unequal axes, whence their name of Geodesic
Lines has been derived, it is desirable to consider them at some
length, and in reference to Gauss' and Lame's systems of curvi-
linear coordinates.
Geodesic lines are the longest or the shortest lines which can
be drawn on a curved surface between two given points or
between two given curved lines.
Let the equation to the surface be
Tf(x,y,z} = c; (109)
and to abbreviate the results, let us, as in Art. 346, Vol. I, use
the following symbols :
: W, (110)
(111)
d 2 F\ id z v\ /fl? 2 F\ ~]
j? 2 > ( 112 )
/ 2 F \ _ , / d 2 \ _ , I d*F \ _ ,
\dnd2>~ U> \d^r!~ V} \d^bj)- W '\
dxdy >
Let s = the length of the line on the surface drawn between
two points on the two given lines,
/
Jo
= ds
. dy , dz
.-
ds y ds
294 CALCULUS OF VARIATIONS.
but bx, by, bz are subject to the relation
vbx + vby+wbz = 0, (114)
and therefore, as this must consist with the part of (113) which
is under the sign of integration, we have
. doc , dy . dz
a. -j- -r " ~j~
as as as ,-, -, ,-^
u v w
and which are the differential equations to geodesic lines on a
given surface : the complete integrals of them have never yet
been found, but many properties may be deduced both in the
general case and in the particular case of the ellipsoid.
, dx , dy . dz ,, ,.
227.1 Since a.-y-, d.^-, d.-r- are proportional to the direc-
J ds ds ds
tion cosines of the principal normal, or of the direction of the
radius of absolute curvature of a curve in space, and since
u, v, w are proportional to the direction-cosines of the normal
to the surface at the point (x, y, z) ; from (115) we infer that
the radius of absolute curvature of a geodesic line drawn on a
surface is coincident in direction with the normal to the surface ;
or, in other and equivalent words, that the osculating plane of
a geodesic line is a normal plane to the surface.
If the geodesic line is drawn from one given point to another
given point on the surface, then, as there are no variations at
these limits, the definite part of (113) vanishes; but if the geo-
desic line is drawn from one given curve to another given curve,
then, since
r^te+^y + 8*T=o, ( ii6)
\_ds ds * ' ds J
and as , rj- are the direction-cosines of the tangent to
ds ds' ds
the geodesic line, and bx, by, bz are proportional to the direc-
tion-cosines of the tangent to the limiting curve at the limit, it
appears that the geodesic line cuts both the limiting curves at
right-angles : this is also manifest by general reasoning.
228.] The equations to a geodesic line on a surface may be
put under the following form :
Since the osculating plane of the geodesic line contains the
normal to the surface, we have
22,9.] GEODESIC LINES. 295
u (dy d 2 z dz d 2 y) + v (dz d*x - dx d 2 z) + w(dx d 2 y dy d z x) = 0, (117)
or (vd 2 zwd 2 y)dx + (wd 2 x-ud 2 z)dy + (vd 2 yvd z x)dz = Q, (118)
also vdx + vdy + wdz 0; (119)
whence we have the equality
dx _
u (u d^x + v d 2 y + w d 2 z) ~
Q 2 (dx d*x + dy d 2 y + dz d 2 z)
dv dx + dvdy + dw dz
Q 2 (dv d^x + dv d 2 y + dvf d 2 z) (udu + vd\ + wdw) (\sd z x + vd 2 y + wd 2 z)
and since udx + vd + vrdz
', (120)
(du dx + dv dy + dw dz) ;
and hereby (120) becomes
dv d z x + dv d 2 y + dw d 2 z vdv + vdv + wdw dx d*x -f dy d 2 y + dz d 2 z
- - - 1- ^ 2 Q (121)
dv dx + d\ dy + dw dz Q 2 dx 2 + dy 2 -}- dz 2
dv dPx 4- dv d 2 y + dw d 2 z do, d.ds
dv dx + dvdy + dw dz Q ds
/dv d 2 x + dv d 2 y + dw d 2 z ds
dv dx + dv dy + dw dz Q
229.] Again, let p be the radius of absolute curvature of a
geodesic line, then, by equation (23), Art. 325, Vol. I,
ds 2 i , da?\ 2 i , dy\ 2 / , dz\ 2
- = (d.^-) + (d.-f) + (d.-j-) ,
2 > \ '
- . .
ds> \ ds' S ds)
therefore, from (115),
dx dy dz
U, 7~ i*. f~ U. j ,
ds ds ds ds
u v w pQ
by means of either of which equations the length of the radius
of absolute curvature at any point may be determined.
Also let p be the radius of curvature of the normal section
of the surface which contains ds ; then, from (123),
7 dx , dy . dz
, vd. -j- +vd.~ + wd.- r -
ds_ _ ds _ ds ds
pQ ~ Q 2
296 CALCULUS OF VARIATIONS. [230.
ds u d 2 a? + v d z y + w d 2 z
reason f ( 12 )' Art - 347 > Vo1 - L
p Q (IS
.'. P = P; (124)
that is, the radius of absolute curvature of a geodesic line is
equal to the radius of curvature of the normal section of the
surface, which at their common point touches the line.
This result may also be inferred from the property stated in
Art. 227; viz., the osculating plane of a geodesic line is a normal
plane to the surface.
230.] Hereby also may the radius of torsion of a geodesic
line be determined.
Let X, \t., v be the direction-cosines of the binormal of a geo-
desic line : so that employing the same symbols as in Chap. XVII,
Vol. I, we have
A = *, M = I, , = |j (125)
and if R be the symbol for the radius of torsion,
*. (126)
n
Now as the binormal is perpendicular to the tangent and to
the principal normal, we have
dy dz
+ '*-' < 128 >
therefore, differentiating (127),
d\ dx + dp dy + dv dz = 0, (129)
also, vdx + vdy + wdz = 0, (130)
d\ d^i. dv ds
.'. = =; = ; (lol)
U V W RQ
from either of which equations may the length of R be found.
231.] For the sake of illustrating the preceding formulae, let
us investigate the following properties of geodesic lines on an
ellipsoid, which are due to M. Joachimsthal. See Crelle's Journal,
Vol. XXVI, 1843.
GEODESIC LINES. 297
Let the equation to the ellipsoid be
~ + ~ + . , - = 1, (132)
a 2 b 2 c 2
so that u = -4, v = -Jf , w = -J ; (133)
/Z ^ /*
and hence equations (115) become
dx dy dz
' ds __ 'rfj _ ' ds
x y z ' '
a 2 6 2 c 2
each of which is equal to the following equalities,
1 do? dx \ dy dy \ dz , dz
a 2 ds ' ds b 2 ds ' ds c 2 ds ' ds
= , (135)
x ctx y fty z dz
4 ds A 4 ds c 4 ds
and to
x dx y dy z , dz
//2 /jo h 2 /TO /2 //o
g gg _ as _ c " g . (136)
/Y>2 '3/2 /^ 2
muz
a 4 A 4 T 4
now from the equation to the ellipsoid we have
x dx y dy z dz
_ |_JL_*L_j = o, (137)
whence by differentiation
x dx y dy z dz _ (1 dx 2 1 dy 2 1 dz 2 )
a 2 ' ds b 2 ' ds ' c 2 ' ds \a 2 ds b 2 ds c 2 ds }
so that from (135) and (136) we have
1 dx , dx 1 dy . dy 1 dz , dz
// I _ /-I _ I f(
a 2 ds ' ds b' 2 ds ' ds c 2 ds ' ds
x dx y dy z dz
I// /w2 "I //>/- //y2
f ' . / X f '^f/ X te ^
^-di + ^-di + ^-di (139)
X
PRICE, VOL. II. Q q
298 CALCULUS OF VARIATIONS. [>3 2 -
1 da? 1 dy 2 I dz* ,,,
= 1? ~d? + F ~d& + c 2 ds* '
so that from (139) we have
dv du
.. uv = a constant; (142)
whereby the equation to a geodesic line becomes
x 2 y 2 z 2 \ i 1 dx 2 I dy 2 1 ^ 2 \
i+TT + -* (-2 ;T2+T2 A + -T^) =
a 4 o 4 c* y v 2 cfo 2 2 a* 2 c 2 cfc 2/
and this combined with the equation to the ellipsoid will be the
equation to the geodesic line on the ellipsoid.
It is evidently a differential equation, and also involves one
arbitrary constant : there is no known method of deducing the
integral equation, but (143) admits of the following geometrical
interpretation :
Let p be the perpendicular from the centre of the ellipsoid on
the plane which touches the ellipsoid at any point (x, y, z) of a
geodesic line, and let d be the central radius vector of the
ellipsoid parallel to the tangent line of the geodesic at the same
point, then 1 1
W =L n) V = Ja)
p 2 d*
.-. pd = a constant k 2 (say)*. (144)
232.] Again, since each of the expressions in (134) is by
ds *D ds
reason of (123) equal to , that is, to - ; and also since each
pQ p
ftj f] Q
is equal to - by reason of (139) and (141)
u
pds vds d 2
p u ' p
o 2 yfc 4
= $i*i =-? < 145 >
that is, along the same geodesic line the radius of absolute cur-
vature varies inversely as the cube of the perpendicular drawn
from the centre of the ellipsoid on the tangent plane at the point.
* An elegant proof of this theorem, founded on the geometrical infinitesimal
method, is given by Professor Charles Graves of Dublin, in Crelle's Journal,
Vol. XLII, 1851.
233-1 GEODESIC LINES. 299
It would exceed the limits of our present inquiry to proceed
to other similar and equally curious properties of the geodesic
lines on the ellipsoid ; but the reader desirous of further in-
formation will find an ample supply in a most masterly paper
on surfaces, and especially on geodesic lines traced on surfaces,
by M. Ossian Bonnet in the Journal de FEcole Polytechnique,
Cahier XXXII, Paris, 1848, and chiefly in the supplement to
the memoir.
233.] I propose now to investigate other properties of geo-
desic lines on surfaces in general, and shall first consider two
theorems of geodesic parallel lines and geodesic circles, because
these are, as is manifest, pregnant with many important results
for the elucidation of which however I must refer the reader to
the memoir of M. Bonnet which has just been mentioned.
Let there be a curved line, see fig. 48, PQR on a given surfacCj
and through p let there be drawn a normal plane ; and of its
intersection with the surface, let PP' = bs be the first infinitesimal
element : let the coordinates to p be x, y, z, and to Q, which is
a point infinitesimally near to p and also on the given curve,
x-\-dx, y + dy, z + dz; and to P' let the coordinates be , rj, ;
then we have , .
Suppose that from other points on PQR lines are drawn on
the surface similar to and of equal length with bs ; then for
points infinitesimally near to p and p', we have by differentiation
(x ) (dx d) 4 (y 77) (dy drj) + (z f) (dz d() = bs d.bs
= 0,
because bs is of constant length: but since PP' is perpendicular
= 0,
= 0; (146)
and therefore the several and successive elements of P'Q'R' are
perpendicular to the lines PP', QQ', ... ; and from the points
p', Q', R', . . . let lines of infinitesimal and equal lengths be drawn
perpendicular to the successive elements P'Q', Q'R', ..., and let
their extremities be at the points p", Q", B", . . . : through which
let a curve be drawn which shall have to P'Q'R'... the same
relation that P'Q'R'... has to PQR: then it is plain that each of
the elements p'p", Q'Q", ... is perpendicular to the length-ele-
ment of the curve P"Q"R"... : and similarly may any number of
Q q 2
300 CALCULUS OF VARIATIONS. I>34-
such curves be found; and the lines PP'P"..., QQ'Q"... are geo-
desic because the plane containing every two consecutive ele-
ments such as PP' and P'P", which is the osculating plane to the
curve, is a normal plane to the surface at P' : and as we can
predicate of the sum of such infinitesimal lengths, that which is
true of each of them, we infer that if on a surface geodesic lines
of a constant length are drawn from the points of a given line
perpendicular to the given line, the curve-locus of the extremities
of them will be perpendicular to them. Two lines such as
PQR, P'Q'R' are called geodesic parallel lines; and of course any
number of such may be drawn.
Also suppose as a particular case that the curve PQR is col-
lected into a point: then all the lines PP', QQ', ... start from a
point, and as the equation (146) equally holds good, we conclude
that if from a point on a given surface a series of geodesic lines
is drawn in all directions, the curve-locus of points on them
which are at equal distances from the given point is such as to
be perpendicular to all the lines ; and as the analogy is exact
between this property and that of a circle which cuts orthogo-
nally all its radii the curve-locus may aptly be called a geodesic
circle. Thus, in fig. 49, let o be the point whence the geo-
desic lines start, and of them let OP, OQ, OR,... be infinitesimal
equal lengths; then, by reason of equation (146), the curve PQR
cuts all these orthogonally. Similarly, if PP', QQ', RR' are equal
infinitesimal lengths, the curve P'Q'R' cuts all orthogonally, and
so on for infinitesimal lengths, until ultimately the locus of all
points on the geodesies drawn from o which are at a finite distance
from o may be shewn to cut all these geodesies orthogonally.
And if from a point o on a surface two geodesic lines or
and OT' be drawn to two points T and T' infinitesimally near to
each other in a line on the curved surface, then if T T' = d<r, and
T T'O = 0, ,
OT OT OUT COS 0.
234.] Let us now apply directly the calculus of variations to
lines drawn on a surface, and referred to Gauss' system of
curvilinear coordinates, which were the subject of consideration
in Section 2, Chapter IX, of the present volume.
By equation (74), Art. 156,
dri*}*; (147)
235-] GEODESIC LINES. 301
and taking the variation of s, we have
f/ d d-n\ . ( d dri\ ^ I 1
8 = 0= \ U - + v -!)& +(v -. + Q-!. )br,
LA ds dsl \ ds ds' J
da '
_ ~-bE - ^-^8F ^80^ j (148)
but since E, F and G are functions of and 77,
and similar values are true for SF and 5a, so that substituting
in (148), and equating to zero the coefficients of 8 and 8r/ which
are under the sign of integration, and bearing in mind equa-
tions (80) and (81), Art. 157, we have
from which we have
and from (80) and (81), Art. 157, we have
E d F
cot 9 = - j-
V di\ V
. G drj F
COt 6 = -r4 +
V dq V
by means of which equations 9 and 6' may be eliminated from
(150), and thereby a differential equation found which will be
that to the geodesic lines on the surface.
235.] The formulae above are much simplified if and 77 are
so arranged that the angle at which the lines of one system in-
tersect those of the other system is 90 ; in which case, see
equation (77), Art. 157, cos o> = 0, and therefore F = : and the
equations to the geodesic lines become
302 GAUSS' SYSTEM. [235 .
(152)
i E \
cote =
\ G /
the equations in terms of tf become identical with these.
This also admits of further simplification : let us suppose the
systems of lines and 77 to be geodesic : then if 77 = a constant,
6 = 0, and therefore it follows from the first of equations (152)
that = 0, and therefore that E either is a constant or is inde-
#77
pendent of 77, so that
2(G)*d=-(^
(153)
cot = ( I ~
by means of which, 6 may be eliminated, and the resulting dif-
ferential equation will be that to the geodesic lines.
And to take the simplest case of all : let a series of geodesic
lines of equal length originate at the point o ; and let it be re-
ferred to a system of geodesic polar coordinates analogous to
that of plane polar coordinates : but to avoid the inconvenience
of new symbols, let be the geodesic radial-distance of any
point from o, and 77 the angle between the first elements of
and of an originating prime radius which abut at o ; then, by
virtue of Art. 233, the condition of orthogonality is satisfied, and
by reason of equation (75), Art. 157, since d<r = d, E = 1 : hence
(153) become
(154)
cot e = ^- ^
G* dr l
to simplify which let G* = m,
dO = ( rrjdri; (155)
v '
and eliminating d, we have
d
which is the differential equation to the geodesic lines, but does
not generally admit of integration.
236.] GEODESIC LINES. 303
m is generally a function of both and r], and m dr], by reason
of equation (76), Art. 157, is the element of a line of the second
system ; but if all the lines of the first system originate at a
common point o, r\ manifestly = 0, when = ; and as above
taking ?/ to be the angle between the first elements of the ori-
ginating geodesic, and of any other geodesic corresponding to f >
the element of a line of the second system may be considered
as the arc of an infinitesimal circle when is infinitesimal, and
therefore is equal to gdrj : therefore for an infinitesimal value of
, dm
, = m, and = 1.
A further inquiry into the subject of geodesic lines from this
point of view is beyond the scope of our work; but it has
important applications in the determination of curvature of
surfaces, according to the principles of the system invented by
Gauss, and explained in his memoir, " Disquisitiones generales
circa superficies curvas ;" and for these I must refer the reader to
that work. There is also much information on the same subject
in the notes appended by M. Liouville to his edition of Monge's
Analyse appliquee, &c., Paris 1852.
236.] It will be observed that the general equation to geo-
desic lines, whether in the form of equation (122) or in that
of (156), is a differential equation of the second order, and of
such a form as not to admit of integration : in the case of the
ellipsoid, however, the first integral has been determined, and
has led to that which is known as Joachimsthars theorem ; see
Art. 231 : M. Jacobi also has discovered a first integral by
means of his elliptical coordinates, the forms of which are given
in Art. 166, and has expressed it as the sum of two Abelian in-
tegrals*.
But as the problem of geodesic lines on an ellipsoid is of
great importance in questions of geodesy, I propose to consider
it by the aid of the method of elliptical coordinates, such as is
developed in Art. 162 166, and hereby to prove some of their
salient properties, and which are chiefly due to Mr. Michael
Roberts of Dublin.
* See Liouville's Journal, Tome VI, p. 268, and Crelle's Journal, Band
XIX, p. 309.
304 LAME'S ELLIPTICAL COOKDINATES. L 2 37-
Let the equation to the ellipsoid under consideration be
r 2 W 2 ~2
X? + 1J=& + W=^ 2 = l > (157)
and let a point of the geodesic line be at the intersection of
the ellipsoid by the two orthogonal and confocal hyperboloids
whose equations are
/>- i/2 *>2
j i
(158)
it* z* ' '
y i j
the relative magnitudes of A, /tx, v, b, c being those explained in
Art. 162 : then, as we are considering points on the same ellip-
soid, A. is constant, and ^ and v are the current elliptical coor-
dinates to the geodesic line ; and in reference to which it is to
be observed that systems of lines of curvature are formed on
the ellipsoid by either /j, or v varying, while the other is constant.
237.] Let ds be a length-element of a geodesic line, and let
d^s, d v s be the projections of ds on two lines of curvature, which
meet at the point where ds commences : and let i be the angle
contained between ds and d v s, so that ^ i is the angle between
ds and d^s : whence also
d v s = dscosi
} . (159)
= dssiai J
Also for convenience let
rA2_,,2w,, 2 _^) _ ^ 60)
so that by reason of equation (101), Art. 164,
J- f '
C? V 5 = g*dv j
.'. ds 2 = pdp? + q dv 2 . (162)
Let u represent the length of the geodesic line : therefore
u = I ds;
JQ
.-. 8 = / b.ds = 0;
.'o
and from (162)
237.] GEODESIC LINES. 305
du. dv . du? . dv 2
b.ds = - b.dJ. + -b.
.. .
ds ds 2ds 2ds
du?
r i ( du. 7 dv 7
= = lp-rd.bu.+ q d.
/o ds ds
ds f * ds
Let r-
~\ l /*U4i 2 * dt? . da^ , dv ^ }
+ / J^-8j9 + q -d.p- ^-d.q-f-Svl. (163)
J JQ (2ds * 2ds a ^ ds ds )
(6 2 -i; 2 ) (c 2 -i; 2 )
so that m and n are functions severally of u, and i> only : whence
p mdj 2 ^), q = ndj? v z ), (164)
.-. bp = (pt-v
bq = (]u 2 v 2
and substituting in
-
TT
(2ds dp
t s^T ?rj- Tr
(2ds ' dv 2ds 2ds * ds
and equating to zero the coefficients of 8/x and of bv under the
signs of integration, we have the following results :
2 ,, dm du? _ dv 2 _ du.
- d * = ' (167)
but d.p = d.m(u?-v 2 )
ds ds
', (168)
substituting which in (167) we have
/x, u? v 2 du,dm
,
2mv--dv
'ds^ 2 ds ^ ds "ds
du?)
( dv 2
= m -{nr-
( ds
or, as it may be expressed, replacing m and n by their values
from (164),
PRICE, VOL. n. R r
306 LAME'S ELLIPTICAL COORDINATES. C 2 37-
' v z )ds
K i
whence by integration
//,,2
(169)
Ci being the constant of integration.
From the coefficient of bv in (166), by a parallel process, we
obtain ,72
n(p?-v*)*^ = c 2 -v*, (170)
and, to determine Ci and c z , add (169) and (170), whence
md^ + ndv* ,,
- -iTa -- (|" 2 -z> 2 ) 2 = Cj + ca + y 2 -* 2 ; (171)
C*>
d^i
but wz f?u 2 + n di? = -5 - 5 ;
= 0;
and therefore replacing c 2 by an arbitrary constant /3 2 , we have
whence by division ^ ^-^
= J
and extracting the square root, and replacing m and n by their
values, we have
in which equation the variables are separated ; and therefore
(theoretically) the equation can be integrated, and when inte-
grated is that to a geodesic line on the surface of the ellipsoid.
It will be observed that the final equation will involve two arbi-
trary constants, viz. /3 and that introduced in the last integra-
tion ; and these will have to be determined in terms of the co-
ordinates to the points at which the geodesic line commences
and terminates.
239-] GEODESIC LINES. 307
238.] Hence we may determine the length of the geodesic
line. For , a , , ,
ds 2 = p dy? + q dv 2
ds =
dv;
032-1,2)*
_ dp + I V n t dv. (176)
239.] Again, equation (174) may be put in the form
2 _ mv 2 dfj? + n^j? dv 2
p dp? + q dtp
v 2 dp.8* + p? d^
ds 2
= v 2 (sinz) 2 + ^ 2 (cosi) 2 , (177)
which is manifestly a differential equation, but being expressive
of a condition to which all geodesic lines on the surface of the
ellipsoid are subject, may be considered a definition of them.
Let us investigate these properties more at length.
(a) Suppose a series of geodesic lines to originate at a point
(MI> v \}> an d to touch the line of curvature (/xj) ; then at that
point i = 0, and therefore /3 2 = pi 2 , and we have
rf = n 2 (cos i) 2 + v 2 (sin i) 2 ; (178)
therefore for all geodesic lines touching the line of curvature
(p, = n/,i), ft 2 has the same value.
(/3) Suppose a geodesic line to originate at an umbilic ; then,
see Art. 356, Vol. I, joi 2 = v 2 = b 2 , and we have fi=b, whatever
direction the geodesic line takes.
(y) From (178) it appears that if two geodesic lines which
R r 2
LAMES ELLIPTICAL COORDINATES. L 2 39-
touch the same line of curvature, that is, for which ^ is con-
stant, pass through the same point (/x, v), they make equal angles
with the lines of curvature which pass through that point ; for
P.J 2 = p 2 (cos i) 2 + y 2 (sin i) 2 = ^ 2 (cos i') 2 + v 2 (sin i') 2 ;
.-. i = i'.
(8) If from two umbilics situated on opposite sides of the
least axis of the ellipsoid, geodesic lines are drawn to any point
on a line of curvature, of which the equation is p = a constant,
the sum of the lines is constant along the whole line of curvature.
For suppose r and r 2 to be the lengths of two such umbilical
radii drawn to a point (p., v) : then
dr\ = d v s cos , dr 2 = d v s cos i,
.'. r*i + 7*2 = a constant.
By reason also of the theorem of geodesic lines at the end of
Art. 233, viz., ox' OT = da cos 0, it may be proved in the same
way as the analogous theorem in plane geometry, that the
geodesic radii vectores make equal angles with the curve of
curvature.
It is scarcely necessary to call the reader's attention to the
obvious analogy which exists between foci of a conic with refer-
ence to the curve and the umbilics of an ellipsoid with reference
to the lines of curvature : the preceding theorems are only two
out of many which indicate the resemblance.
(e) Let two geodesic lines touching two lines of curvatures
which are determined by ^ = ^1 and /x = f/ 2 , intersect at right-
angles in (/*, v), then
Hi 2 = /x 2 (cos i) 2 + v 2 (sin i) 2 ,
fj^ 2 = p 2 (cos i') 2 + v 2 (sin i') 2
= fj? (sin i) 2 -j- v 2 (cos i) 2 ,
that is, the locus of the point of intersection of two such geo-
desic lines on the surface is such that p 2 + v 2 is constant.
And since by Article 162
= a constant ;
and we infer that the curve-locus of the point of intersection of
240.] GEODESIC LINES. 309
two such orthogonal geodesic lines is a sphero-conic, that is,
the intersection of a concentric sphere with the ellipsoid. And
this theorem is parallel to the well known one in plane geo-
metry, viz., Tangents to two confocal conies intersecting at
right angles intersect on a concentric circle.
For other properties of geodesic lines I must refer the stu-
dent to
(1) Two memoirs by M.Chasles in Liouville's Journal, Vol. XI,
p. 5 and p. 105.
(2) A memoir by Mr. M. Roberts in Liouville's Journal, Vol.
XIII, p. 1.
(3) A paper by Mr. H. J. S. Smith, Fellow of Balliol College,
Oxford, printed by the Ashmolean Society in 1852.
(4) The memoir of M. Ossian Bonnet, referred to in Art. 232.
240.] It will be observed that we have now found two first
integrals of the differential expressions for geodesic lines on the
surface of an ellipsoid : that, viz. in Art. 231, which is known
geometrically as JoachimsthaFs theorem, and that of equation
(174) ; and perhaps it might hence be inferred that the same
differential expression would be found in both, and that it
might be eliminated, and that thus the equation to a geodesic
on an ellipsoid might be expressed in the integral form : this
however is not the case : the two results at which we have
arrived, although by different methods, are identical ; and their
identity may thus be shewn :
Take the central plane section of the ellipsoid (which of
course is an ellipse) parallel to the tangent plane at the point
(ju, v] or (x, y, z) : then, as the direction-cosines of the normal
to this plane are proportional to
A 2> A 2 -6 2 ' A 2 -c 2 '
the principal semi-axes of this ellipse are the values of r deter-
mined by the quadratic equation
A 2 (r 2 -A 2 ) T (A 2 -6 2 ) (r 2 -A 2 + 5 2 ) T (A 2 -c 2 ) (r 2 -
Now from equation (98), Art. 162, it is manifest that A 2 , p,
and v 2 are the values of r 2 in the equation
310 LAME'S ELLIPTICAL COORDINATES. [240.
a? 2 y 2
-- 1_ i -- 1
also we have
x*
A 2 + JJ
therefore by subtraction
a? 2 y 2
= '
(A 2 -A 2 ) (r 2 -* 2 ) (A 2 -c 2 )(r 2 -c 2 )
the roots of which quadratic are // 2 and v 2 : therefore the roots
of (179) are manifestly A 2 /x 2 and A 2 v 2 : these therefore are
the squares of the semi-axes of the central elliptical section of
the ellipsoid*; and their directions are evidently parallel to those
of the lines of the curvature at the point (/*, v).
Hence if d is the central radius vector of this elliptic section,
and inclined at an angle i to the semi-axis whose length is
( COS ) 2
(\ 2 -i; 2 )f, 2 2
also if p be the perpendicular from the centre of the ellipsoid
on the tangent plane at (p,, v), by Art. 164,
J_ 1 ( A .2_ At 2 )(A 2_ y 2^
p 2 ~ = 7? (A 2 -^) (A 2 - 2 ) 5
since, then,
p 2 = v 2 (sin i) 2 + fjL 2 (cos i) 2 ,
. . A 2 - /3 2 = ( A 2 - y 2 ) (sin i) 2 + (A 2 - M 2 ) (cos i) 2
ini) 2 (cos) 2
A 2 (A 2 -6 2 )(A 2 -c 2 )
and therefore p d = a constant : and this is Joachimsthal's
theorem.
* A proof of these properties of the principal axes of the section will be
found in Gregory's Solid Geometry, Chap. VI, Cambridge, 1845.
241.] MAXIMA AND MINIMA. 311
SECTION 3. Investigation of the critical values of a definite inte-
gral^ whose element-function involves derived-functions.
241 .~\ In all the above problems of maxima and minima, the
differentials contained in the element-function have been taken
in their most general forms ; no supposition has been made as
to one or more being equicrescent, and they have not been put
in the forms of derived-functions ; and the solutions, it will be
observed, have been deduced from first principles in every in-
stance, and without the intervention of any general formulae :
the results arrived at are left in their symmetrical forms, and
hereby have we been able to infer geometrical properties, which
are frequently the only available definitions of the function
which satisfies the maximum or minimum property that is re-
quired : and for elegance and symmetry nothing else can be
desired : but we have not investigated any critical function
whose element contains differentials above the second order;
the simplest cases only have been considered, and a slight in-
spection of the general results of Art. 207 will shew that the
complexity of the formula rapidly increases if higher differen-
tials enter into the calculation : in this latter case then, it is
desirable to simplify the formulse as far as is possible, ere they
become the subjects of inquiry ; and as such a simplification is
obtained by making one of the variables equicrescent, and by
using derived-functions instead of differentials, although it is
with the loss of symmetry, it is necessary to consider the con-
ditions under which a definite integral, whose element-function
involves derived-functions of different orders, may have a critical
value. And there is also another reason why the subject must
be investigated from this point of view : it is only when the
element-function is of this form that criteria for discriminating
maxima and minima have been constructed. We proceed then
at once to the investigation.
Let the definite integral, whose maximum or minimum is to
be determined, be ri
u = / v dx ;
JQ
where v = f(x, y, y' , y" , . . . y<>) ; (180)
using the notation of derived functions : then, for convenience
of reference, let
312 CALCULUS OF VARIATIONS.
(
Y "-
V
= H; (182 >
so that equation (31), Art. 192, becomes
8w = fa] + / H w dx ; (183)
L Jo v/o
and as u is a maximum or minimum, bu = ; and to satisfy
this condition it is manifest that
(184)
H =
of which expressions the former depends on the values of certain
variables and their derived functions at the limits; the latter
by integration gives the general functional relation, and thereby
the form whence the required critical value may be found.
d^ij d n Y( n )
Now since v contains y( n ) or -7-^, H, which contains ,
dx n dx n
d^ n ii
will generally contain -j-sr- , and therefore will be a differential
(LOG
equation of the (2rc)th order : the solution of this equation will
therefore contain 2n arbitrary constants ; and the determination
of these depends on the values which a = assumes .at its limit-
ing values ; the process however of finding which, being similar
to that explained in Art. 209, it is unnecessary to repeat, but it
is desirable to investigate one or two cases in which the equation
H = assumes particular forms, and thereby admits of imme-
diate integration.
242.] First suppose v not to contain the first m of the quan-
tities y, y', y" y ,.., then the equation H = becomes
ffm^im) /7m + l-y(m+l)
(185)
243-] PROBLEMS OF CRITICAL VALUES. 313
which admits of m successive integrations, and therefore, by
reason of Chapter V, becomes
dan '" o i 2 m _ 1 - 1 . (186)
Secondly, suppose v not to contain x : then
dv = Y dy + Y' dy' + Y" dy" + , . . + Y< n > dy ("> ;
</Y' dV ; dY<>
also = Y -- - -- H r-5 -- ( ) 7 - ;
dx dx z dx n
.-. dv =
which is a differential equation of an order not higher than
2n 1 ; and therefore whenever v does not contain a?, the equa-
tion H = always admits of being integrated at least once.
Thirdly, let v =f(y'), then, by (187),
but as v and Y' contain y only, this may be put into the form
y = F (c)c2? + Ci; (188)
and thence we infer that a linear function, as (188), is such that
the variation of any function of ~ deduced from it vanishes.
ff T*
Lastly, if v= f(y, d ),
v = c + Y'y'. (189)
243.] We proceed to the solution of one or two problems by
the method just investigated. Let
r
dx. (190)
Jo y
Here v=y n ^ T} and therefore equation (187) is applicable.
fdv\
Y =
PRICE, VOL. II. S S
314 CALCULUS OP VAKIATIONS. [243.
_
'-
Therefore (187) becomes
_. c y n y" y n y"
y y y' y y
.-. c = ny n ~ l y e ;
whence by integration yn _ cx + ^
Again, for a second example, let us take that of determining
the shortest line between two given points.
(191)
here also, as v does not involve x, (187) is applicable; and
y'=(~) = ^-;
so that (187) becomes
~~ dx ~ c
(1-c 2 )*
.-. y = - c
which is the equation to a straight line ; and therefore gives the
same result as that arrived at in Art. 213.
Again, for a third example, let us consider a case where the
element-function is of the form investigated in Art. 197, viz.
where __ /., , /, , , . ,
v / v*.> y y ) y ,..*Z)Z,z ,z ,...),
and suppose that
' 1 (192)
.'o
then, by reasoning similar to that which has been frequently
243.] PROBLEMS OP CRITICAL VALUES. 315
employed, both the terms under the integral signs in equation
(47) Art. 197 must vanish : and therefore, as v involves only
y' and z', we have
Y = = r . Z
and therefore, since
dx_
dx
y' z'
**- = ^ =
dx dx
(1 + i
dy 6?Z
dx dx
y = CI^ + GI, z =
which are manifestly the equations to a straight line in space.
A straight line therefore is the shortest distance between two
given points.
And, for another example, let us investigate the following
problem of relative critical value :
To find the plane curve of given length enclosing the greatest
area.
Let A be a constant multiplier, then, since
ny
dy dx,
n
- I ydx,
length = / {l + y' 2 }^dx,
y'rf}dx; (193)
= [
Jo
so that v =
and because v does not involve #, (187) is applicable, and
.
\dy'>
whence (y Ci) 2 + (# c 2 ) 2 = A 2 , (194)
which is the equation to a circle, whose radius is equal to A;
S S 2
316 CALCULUS OF VARIATIONS. [244.
and A may be expressed in terms of the known length of the
curve by a process similar to that of Art. 221.
A comparison of the two methods by which problems have
been solved plainly shews that, although the former immedi-
ately involves first principles and from them is directly deduced,
yet, as the results assume complicated forms when all the dif-
ferentials are retained, it is convenient to make one of the vari-
ables equicrescent, and to express the element-function in terms
of derived-functions, and then to apply the process of these
latter articles.
SECTION 4. The discriminating conditions of Maxima and
Minima.
244.] The process which has been developed in the preceding
articles of this chapter, and which has been applied to the solu-
tion of problems involving maxima and minima of definite in-
tegrals, although necessary, is yet insufficient for the object
proposed, because no discriminating conditions of maxima and
minima have been investigated. For the existence indeed of
such critical values it is necessary that the first variation should
vanish ; but at the same time such vanishing is consistent with
the definite integral being either a maximum or a minimum or
a constant, and with being none of these : the truth of this
statement is evident from the ordinary theory of maxima and
minima. For a critical value it is necessary that the first varia-
tion of the definite integral should not only vanish, but also
change its sign : and I know of no process immediately applica-
ble by which to determine whether a function deduced from
the differential equation H = 0, (see Art. 241,) and involving 2n
arbitrary constants, will or will not cause the required change
of sign of bu. In accordance then with the theory explained in
Art. 130, Vol. I, we are obliged to have recourse to the second
variation of the definite integral, with the object of determining
its sign, and hereby to obtain the discriminating condition ; so
that when bu = 0, and if 8 2 w does not vanish, and does not become
infinite or discontinuous, and does not change its sign within the
limits of integration, u is a maximum or minimum according as
6 2 w is negative or positive. We proceed to the further develope-
ment of these conditions.
245-] JACOBl'S DISCRIMINATING CONDITION. 317
But, to narrow the investigation as far as possible, I will take
the case which has last been considered ; that, namely, in which
the infinitesimal element-function involves a?, y and the derived
functions of y, and in which also x is not only equicrescent but
undergoes no variation ; that is, bx is not one of the subjects of
calculation, but the variation is due to a variation of y only : or,
geometrically viewed, the displacement of the point on the curve
is in a direction parallel to the axis of y only : for it is to this case
that Jacobi, the discoverer of the criteria, has confined himself.
And first let the object of the research be clearly understood.
If the infinitesimal element-function contains a derived func-
tion of the nih order, the differential equation H = will gene-
rally be of the 2 wth order, and therefore the value of y deduced
from it is of the form
y = f(x, cj, c z , ... c 2n ), (195)
and contains 2n arbitrary constants which have been introduced
in the process of integration : and therefore, if u be the given
definite integral, it is plain that, after the substitution of y by
means of the above equation, u will depend partly on the form
of the function /, and partly on the arbitrary constants. It may
seem then that the critical value of u will depend on both these
quantities : as to the constants, however, it has been shewn that
all their values may be determined by means of the given limit-
ing values of the variables and of the derived functions ; and
hence, that as these are determinate constants, the value of the
definite integral cannot be made critical by any change of them :
and even more than this, did u depend on such quantities it
would become an integral (not differential) function of many
variables, and would have its critical value determined by the
ordinary rules of the differential calculus.
245.] It is then the other question which we have to discuss;
namely, whether the form of the function deduced from the
equation H = is such as to render the definite integral a maxi-
mum or minimum. For this purpose we must calculate b 2 u,
and determine its sign, subject to the conditions that
(1) b 2 u has the same sign for all values of the variables and
their derived functions between the limits.
(2) 8 2 w does not become infinite for any values between the
limits.
318 CALCULUS OF VARIATIONS.
(3) b 2 u does not vanish : for if so, we must, in accordance
with the theory of maxima and minima, proceed to the investi-
gation of b 3 u and 8%, and so on ; a work beyond our present
purpose.
Let the definite integral, which is the subject of inquiry, be
u = I v dx, (196)
o
where v = f(x, y, y', y" , . . . y <>) ; (197)
then, by equation (183), since o> = by, because bx= 0,
bu = fol + / Hbydx-, (198)
L JQ Jo
and as H = 0,
b 2 u = f 8H by dx, (199)
^0
the sign of which is to be determined.
d dv d 2 / dv d n dv
-" ( ->. (200)
; (2 o 2)
d
(_)n.nj( - -Jby + (7 -)by + ... +
now observing that , k
it appears that in the above expression for 8H, there are terms
of the form
245-] JACOBl'S DISCRIMINATING CONDITION. 319
wherein the order of the derived of ly is the same as the index
of -j-, which affects the whole of the subordinate subject ; and
it appears also that the other terms may be grouped in pairs of
rf g
__
d*
the form rf rf .
\*r _ _[p __ M . /9n^\
*')'
< 206 >
and the connecting sign in (205) is + or , according as s r
is even or odd.
Now first I shall shew that all the terms of which such a
series as (205) is composed may be put in the form (204) ; and
therefore that 8H admits of being expressed in a series of terms,
the type of each one of which will be
d k
where A ft is a determinable function of x.
By the theorem proved in the foot-note*, if P and Q are two
functions of x, whose derived functions are symbolized by
p' p" p() ' Q" QW
jr,jr,...r j W j v , "^ )
* Let p and Q be two functions of x, of which let the derived be repre-
sented by P', P", . . pW, a', a", . . a (n) : then
d do. dp
j .PQ = P-J + Q-T-.
ax ax ax
Let AI be the symbol of derivation as applied to p only,
A 2 ------- ...... a only,
A------------- any function of * ;
therefore from the above equation
A.PQ =
and, omitting the subjects,
A = A 2 + A!,
A = A A
. . .
1*2 1
A 2 n .Pa
1.2
n(n-l)
dx n dx n 1 dx n ~ l 1.2
320 CALCULUS OF VARIATIONS. [245.
dx n dx n 1 dx"- 1 1.2 dx"- 2
..(_ r -i^- p( "" lQ (_) P w Q . ( 007)
fi fjn
which theorem we shall apply to the subordinate subjects of
differentiation in (205) when some convenient substitutions have
first been made.
Let by = u, so that employing the ordinary notation of de-
rived-functions, we have , k .
dx k
also suppose that * = a + p, r = <r p } whence
(208)
s r = Vp J '
that is, we suppose first that sris an even number, and there-
fore that the pair of values in (205) is connected with a positive
sign ; then, by reason of (207),
d'.by _
dx*
c
_! _ * i_ "'" / / \p Q<p)^((r). ^209)
fiwP ft f>& 1 1 9 rl wfl 2
vjdj r j. lit " i t fy um r
~dx~ r \ C dx' $"
--- -- -
dsr~> dx? 1 dx- 1 1.2
p
. . 210
~L2 dx*~ z dx"-<>
Also by Leibnitz's theorem, Art. 53, Vol. I,
d* + (>
d' j d r &y\
dx' \ dx r } "
d" dp ( d
~ ~
.
+ ~T2 dx^ + '" + ~dx~* ;( }
245-] JACOBI'S DISCRIMINATING CONDITION. 321
adding which to (210), we have
d".cu w p (d*- l .c'u ( * } cKcV"- 1 '^
dx" ~ I ( dx"- 1 dx" \
fr> *"-'>
"" ( '
. ., d , . .
but '- 1
= --- = - - - - -=-.
7 - 1 cfo? " da*^ 1 [ . TO
^01 t
= -5 - - 1 c'M (<r) C 'M (<r) C' V'- 1
da?'- 1 (
Also again applying the theorems of Leibnitz, and that given
in (207),
^r-
dx
^-.c'V'- 1 ' -Sc'V'- 11 +2c"V- 1) -I- c
Mr
" w (o 1) //o 2 r "",.(o 2)
_ ? i - + fJlfJ; - ; (214)
- 1 -
and by a similar process, the other terms of (213) may be
transformed into equivalents of the required form ; so that
ultimately,
that is, 8H consists of a series of terms, in each of which the
order of the derived-function of u or by is equal to the order of
the index of -=- , which affects A A u ik) ; and in which AO, A I} . . . A are
functions of x.
A process similar to that pursued above is also applicable if
r s is an odd number.
It is manifest from (203) that A n , which is the coefficient of
/ c?^v \
w <n) , is ( ) M ( ^ )2 j ; but the other coefficients, viz., A O , A! ... are
of a form so complicated that it is useless to calculate them in
the general case.
PRICE, VOL. II. T t
322 CALCULUS OP VARIATION'S. [246.
Heuce we have from (199)
2 " n ( n
(216)
246.] Now we proceed to shew, that when 8H is expressed
in the form of the right-hand member of (215), bHudx is an
exact differential, and that therefore the second variation of the
required definite integral admits of integration : hereby we shall
be led to a reduction of the result to such a form as will imme-
diately indicate the sign.
<5H is manifestly a differential expression containing u and
its derived-functions up to the 2wth order inclusive.
Let z be a value of u satisfying the equation 8H = 0; that is,
suppose z to be such a value as when substituted for u renders
8H = : so that we have
z " d n .AnZ^
" + - = - < 2I7)
In 5H let uz be substituted for u, and let the result be multi-
plied by z, and be subsequently represented by u for convenience
of notation : so that
' 2 " nn
(218 >
Then the following investigations will prove
(1) that ndx is an exact differential, whatever be the value of u:
(2) that udx will have the same form as u, except that n will
be diminished by unity : in other words, that we shall have
___. __
and which consists therefore of a series of terms, in each of
which the order of the derived of u is the same as that of -=-
dx
which affects the whole subordinate subject.
Multiplying (217) by uz, and subtracting it from u as given
in (218), we have
^ . _ _
= z -j -- f- z - ^r -\- ...-+ z
- -- ...
dx dx 2 dx n
247-] JACOBl'S DISCRIMINATING CONDITION. 323
which series consists of pairs of terms, of each of which the
type IS , m ( 1l2 \(m) ftm A ~(m)
_ . m
dx m
to these let the theorems of Leibnitz and of equation (207) be
applied : then
d m .A. m (uz) m
dx m
l>L|<*> + ^;^
J. J. . <v *
................... + **>h (222)
- < w > +y zV"'-
also
m .
1.2
the last term of which expression is the same as that of (222),
and therefore in the subtraction both disappear ; and these are
the only integral terms ; hence (220) consists of a series of de-
rived-functions, and therefore u, which is made up of a system
of terms satisfying the distributive law, is also a derived-func-
tion, and therefore udx is integrable immediately by virtue of
its form.
247-3 Upon an examination of the series (222) it appears
(1) that there are terms of the form
6? nt
M.
where M is a constant, and A, z (k > are functions of x, aud in
which therefore the order of the derived of u is the same as the
index of ^-, which affects the whole subordinate subject ; and
T t 2
324 CALCULUS OF VARIATIONS. [247.
(2) that there are other terms, the general type of which is
plainly,
1.2.3...* 1.2.3. ..#' dx m ~ k
and of these, if k and k' correspond to any particular term, so
must there also be another to which k' and k correspond, and
which is, therefore,
l)...(m V + 1) m(m-l)...(m +1) d^
1.2.3... k' 1.2.3... k dx m ~ K
so that there are pairs of terms of the form
- k .cu^-^ ,_ k d m - k '.cu( m -u 1
dx**-* dx~ J ;
M
where M is substituted for the coefficient of (225) and (226),
and where
c = ^nZ^zW, (228)
and is therefore a function of x.
Now in A.rt. 245 it has been shewn that a pair of terms, such
as (227), can be expressed in a series of terms of the form
, d.b lU ' . d*.b z u" t , rf-.ft,*-?.
b U + -^ + -^ + -'- + d^->
and we shall suppose all the terms of (222) to be so expressed ;
and by a similar process all the terms of (223), so that ultimately
by addition
' d.z n uW
+ - + -~'
but B = 0, because it has been proved in the preceding Article
that u dx is an exact differential : and therefore
u
+ " "*" + + n ; (231)
<&? rfa? 2 dx n
and therefore, finally,
.+ rfw "^ B :r';(232)
- JOll* -f ? JO I I ,7 M
dx dx z dx n
where B b B 2 , B 3 , . . . B n are functions of x : the general form of
these may be found, but in the general case it is too compli-
cated to be available for any useful purpose, and it is better to
determine them, if necessary, in each particular case. It is
plain however that the only term in (220) which will give
248.] JACOBl'S DISCRIMINATING CONDITION. 325
d n B u^ ^n
V^ is z -j - > and that the only term of this latter
eta?" dx n
expression when expanded, as in (222), which is of the required
form, is rf. A.***"'
te" '' (233)
whence it follows that
B M = A n * 2 , (234)
- (235)
248.] On a review then of the two preceding Articles, it
appears that
= I bHbydx
=IV
dx dx* dx n
M, on the right-hand side of the equation, being the symbol for
by, and therefore not to be confounded with u in b 2 u: it also
appears that if z be a function of x such that, when substituted
for u in the expression for SH, the whole vanishes, then
" (237)
is an exact differential by virtue of its form, and independently
of the value of u ; and that its integral is of the form
d n -\* n uW
+ - + -> (238)
and from this we infer as a corollary, that as 8H by or ubH
when expressed in the form (236) is a particular case of (237),
so will the integral of bHudx be of the form (238) : but as 8H
and (237) contains u, u',...u (n) } and uz, (uz)', (uz)", ... (uz) {n} in
corresponding places, so in the integral of bHudx, when ex-
pressed in the form (238), u , u", . . . u (n) must be replaced by
u /\' fu\M ,
-> \ ~ />(- ; and therefore
z \z> \xJ
/ :
Jo
, oam
(239)
Now the process to be pursued is as follows : we must find
a value of z ; that is, we must investigate a certain expression,
which, when substituted for u, will satisfy 8H = 0: hereby we
326 CALCULUS OP VARIATIONS. [249.
shall be able to integrate by parts the infinitesimal element-
function of the second variation, viz. 8H u dx, and to express it
in the form (239) : and in the general case, by the repetition of a
similar process we shall ultimately arrive at an expression con-
sisting of two factors, of which one will be a complete square,
and the other, which is easily determined, will by its sign de-
termine the sign of the second variation of the definite integral,
and hereby give the required criterion of the critical values.
249.] Suppose that the integral of the equation H = is
y = /(#i, Ci, c 2 ,... c 2n ), (240)
which contains 2 n arbitrary constants : suppose also that each
of these arbitrary constants receives a variation, so that y is in-
creased by 8y, wherein
and H becomes H + 8H: then, as the varied value of y differs
from the original value only in the arbitrary constants, it must
also satisfy H + 8H; and as the variations of the arbitrary con-
stants are arbitrary, we may replace them by new constants
GI, 2, ...Can? so that the equation 8H = becomes satisfied by
and as y contains 2n arbitrary constants, so will by also contain
them : but 8H = cannot involve derived functions of an order
higher than the 2%th; and therefore the above value of by is
the complete integral of the equation 8H = 0; the right-hand
side therefore of (242) is available as a value of z satisfying the
equation SH = 0.
250.] Now to apply these conditions ; before proceeding to
the general case, wherein the element-function involves derived-
functions of the ftth order, it is better to consider the simple
case wherein />, /.
v = /(*, y, y ),
therefore, by equation (182),
ft v'
H = Y -^ = '
.-. 8H = Ao + ^, (244)
where u = by, and AI =
-srT
\ay I
250.] JACOBl'S DISCRIMINATING CONDITION. 327
Let Ci and c 2 be the arbitrary constants contained in the in-
tegral of H = 0, and let
then z satisfies the right-hand member of (244) ; and therefore
(246)
is an exact differential; and therefore as u or 8 y is arbitrary^
z 8H dx is also an exact differential, and its integral is by virtue
of (239) d u
Bl ^
Now suppose that
u = by = zb'y, (247)
... -, = =,
where b'y is a new function of x ; then, substituting in the second
variation of the definite integral, we have
I"*/ ^ /"H 1 f 1 d iu\d.tfy ,
= 8'yBi-r- (-) / BI-J- (-) T-^fo? (248)
L 5 l dat\zJjt Jo dx\zi dx
And, to take the case most free from difficulty, let us suppose
the limiting values to be fixed, so that 8y, and therefore 8'y
vanishes at both limits. Then, replacing B X by its value given
in (235), (249) becomes
' ' /OKA\
(250 >
ite integral /
Jo
Therefore the definite interal / v dx will be a maximum or a
minimum according as -r-^ is negative or positive ; provided
ay
that it does not change sign nor become infinite between the
assigned limits ; and provided also that the constants GI and c 2
in (245) and u are not such as to make zu' uz' vanish or
become infinite.
328 CALCULUS OF VARIATIONS. [251.
It is worth remarking, that if zu' uz'=0, then u = z = 8z/ ;
in which case 8 H = 0, and therefore the second variation of the
definite integral vanishes ; and this is plainly inconsistent with
the possibility of our deducing from it the criteria of maxima
and minima.
For an application of the preceding, let us consider the case
of the longest or shortest line between two given points :
v- 1-f* 2 *
which is always positive if the radical in v is affected with a
positive sign.
Also, since the complete integral of H = is, see Ex. 2, Art. 243,
y =
-
'
by Ci# + c 2 = z
/ZUUZ'\ 2 _
> Z ' ' '
Ci and c 2 therefore must not be so assumed as to make
= for any value of x between the assigned limits.
251.] For a second example of the criteria, let
(254)
Let Ci, Cz, Cz, Cj, be the four arbitrary constants which enter
into the complete integral of H = ; then the value of z, which,
substituted for w, satisfies (254) is
so that, as before, zbHdx is an exact differential, and its inte-
gral is by virtue of (239)
d /u\ d d
25 I ] JACOBl'S DISCRIMINATING CONDITION. 329
Let - = = b'y ; (257)
where b'y represents a new variation of y ; then, integrating the
expression for the second variation of the definite integral, and
observing that the terms at the limits vanish, we have
r
Jo
= f
Jo
f l
= -J.
d.Xy d d 3 .b'y ) d.b'y
- - --
(358)
if Ml = y = = = . (260)
fife </a? 2
Now, let
where c\, c' 2 , c' 3 , c' 4 are other new arbitrary constants employed
like the former ones in (255) to represent arbitrary variations of
the constants c\, c 2 , c 3 , c 4 : and therefore z\ is a value of u which
satisfies 6H = 0.
Also, since from (256)
it appears that any value of M, which makes 8H =0, will also
satisfy the right-hand member of the equation ; but 8 H = 0, if
t z \'
H = ZI, therefore f J is a solution of the right-hand member :
(263)
Let Ul
and siibstituting this in (259) we have
whence, integrating by parts, and omitting the integrated part
which vanishes at the limits, we have
PRICE, VOL. II. U U
330 CALCULUS OF VARIATIONS. [252.
P d f M! 1 d.X'
/ c 2 I i-y -5-
J aH /i\ f dx
I \ z ' J
,
(265)
where, by virtue of (234),
(**\*
C 2 = Ba-
d
Also, -r-.fftf =
c? zu'uz
dx ' zz\ z'z\
z {(zz l "-z l 'z' f )u + (z"z 1 -zz")u+(zz 1 'z'z 1 ) u"}
2 ~~ ^ D 'J
>??" \ 2
' ^^ / ^ " f J /f\/~*O\
-, oy + oy f ax. (^oo)
And therefore for a maximum or minimum value of the definite
integral it is requisite that ^ //2 j should be respectively nega-
y
tive or positive for all values of the variables between the limits ;
also the second factor must neither vanish nor become infinite :
the arbitrary constants therefore must be so determined as to
fulfil these conditions.
252.] If the infinitesimal element-function of the definite in-
tegral contains derived-functions of y up to the wth, the process
to be pursued is exactly analogous to those of the two particular
cases discussed above ; and therefore I need give no more than
an outline of it.
Let z, Zi, z 2 , ... z u -\ be n values of by expressed in the forms
(242) and (261) and containing n different series of arbitrary
constants : then the second variation is
of which the integral becomes, by neglecting the quantities at
the limits, reduced to
2 53'1 JACOBl'S DISCRIMINATING CONDITION. 331
and so on ; until ultimately
8 2 w = ( ) n s n ( ' y ] dx; (269)
\ ft V* I
. I 1 1 .1
(/2 y \
-=- ~\ , and another factor which, as in
(266), is of the form of a complete square ; and where
z =
,
dx
and so on. It appears therefore that the maximum and minimum
/ d 2 v \
value will depend on the sign of ( . 2 j ; and that it is neces-
y
sary that this latter quantity should not change its sign for any
value of the variables between the given limits ; and the arbi-
trary constants must not be such as to allow the other factor in
(269) to vanish or to become infinite.
253.] We need not enter at length on the determination of
criteria for relative maxima and minima, because we have shewn
above that such cases are by means of an indeterminate multi-
plier reduced to those of absolute critical values, and the criteria
determined for this latter case are therefore applicable to the
former one. Let us however shew that the solution given in the
fourth example of Art. 243 is a maximum :
u =
Jo
v =
7O
C.
Also, since the curve is determined by the differential equation
rv ^ Y l
H = = Y 3-2 ;
dx
u u a
332 CALCULUS OF VARIATIONS.
y
and therefore the answer gives a maximum or minimum value
according as y" is negative or positive. Suppose the origin to be
at the centre of the circle ; then, .since, as shewn by the value
of u, the curve is taken in the first quadrant, y" is negative,
and therefore the solution corresponds to a maximum.
SECTION 5. Investigation of the critical values of a double
definite integral.
254.] It only remains for us now to investigate the con-
ditions of the critical values of a definite double integral, of
which the variation has been calculated in Art. 202. On re-
ferring to equation (77) Art. 202, it appears that the expression
for bu consists of three parts ; viz., two partially integrated
terms whose value depends on the values which CD and its de-
rived-functions have at the limits which are assigned by the
given limiting equation; and the third term, which is wholly
unintegrated, and cannot be reduced unless o> receives a de-
terminate value. Now let
z _cM_dz, d z z" d*z,' d z z ti __ Q
dx dy dx* dxdy dy*
Then as bu = 0, by reason of u having a critical value, it
follows that Q, = ; and from this differential equation is the
required function to be determined. (270) is plainly a partial
differential equation of the fourth order ; the general integral
of which is in most cases beyond the present powers of the
integral calculus : we can in many cases however deduce from it
some geometrical property which is sufficient to define the re-
quired surface.
255.] To find the surface the portion of which enclosed by a
given curve has a minimum area.
In this problem the limits of integration are given by the
given curve : and
256.] CRITICAL VALUES OF DOUBLE INTEGRALS. 333
= rr$+j(*+z*}*fy&i (271)
'O ''O
u
and therefore (270) becomes
whence there manifestly results
z ,
+ -
\ n (dz\i
r 2 (5) (
and on comparing this with equation (77), Art. 359, Vol. I, it is
seen that the two are identical, and therefore the geometrical in-
terpretation of (273) is, " The surface of minimum area is such
that the sum of the reciprocals of its principal radii of curvature
at every point vanishes : " hence we infer that the principal radii
of curvature at every point are equal and of opposite signs.
256.] Let the problem be "To determine the form of the
surface which being of given extent, and terminated by a given
curve, includes the greatest volume between it, the plane of any,
and the right cylinder whose director is the projection of the
given curve on the plane of xy : in this case
the content = / / / dz dy dx
1 1 z dy dx
the surface = 1 1 {I-\-z f2 + z, 2
therefore, if A. be an undetermined multiplier,
and therefore the equation, H = 0, becomes
dx dy
whence by developement, as in the last example, we have
and to interpret this geometrically; let p\ and p% be the principal
334 CALCULUS OF VARIATIONS.
radii of curvature at any point on a surface, then by equation
(27), Art. 347, Vol. I, we have
-- 1 -- - 3 -- , \i I )
pi p 2 u2
and- if these symbols are expressed in terms of the derived-func-
tions of z, it will be seen by comparison with (276) that
+ - = i; (278)
Pi P2 *
and therefore the surface which under a given superficial area
contains the greatest volume is such, that the sum of its prin-
cipal curvatures at every point is constant: and this result is
usually expressed as " The mean curvature is the same at every
point of the surface."
The equations (276) and (277) have never yet been directly
integrated, but Mr. Jellett has in Liouville's Journal*, shewn
indirectly that the sphere is the only surface which satisfies
them.
Neither in Art. 254 nor in the examples have we said any-
thing about the definite part of the expression for bu, which is
written at length in equation (77), Art. 202 ; the extreme diffi-
culty of the subject obliges us to omit it : if however the equa-
tions (273) and (276) could be completely integrated, it would
enable us to determine their arbitrary constants.
In the general case too of a multiple integral of any number
of variables, the formulae are so complicated, and almost neces-
sitate new symbols of abbreviation, that they are beyond the
scope of the present work : the student however will find them
expressed at length in the Memoir of M. Sarrus cited in the
foot-note of page 234.
* Tome XVIII, p. 163. 1853.
INTEGRAL CALCULUS.
PART II.
INTEGRATION OF DIFFERENTIAL FUNCTIONS
OF TWO AND MORE VARIABLES.
CHAPTER XIII.
INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE
FIRST ORDER.
SECTION 1. General considerations of Differential Equations.
257.] The infinitesimal element-functions, which have been
the subjects of investigation in the preceding pages, are for the
most part explicit functions of one variable only ; and those
which involve more variables, and their differentials or derived-
functions have only been incidentally noticed, as has been the
case in Art. 101, and in the last two preceding Chapters on the
Calculus of Variations. We have now however arrived at a
point in our treatise where it is necessary systematically to con-
sider other and more complicated differential expressions ; and
although such an expression as dy =f'(ad) dx may correctly be
called a differential equation, yet the name is technically given
to such expressions as those written below, wherein there exists
a relation between two or more variables, and their differentials
or derived-functions of any order ; and our object is to inte-
grate them, that is, to determine the functional relation between
the variables in their integral form ; or at least to discover some
new relation between the variables and their differentials, and
in which the highest differential is lower by at least unity than
the highest one in the original equation.
336 DIFFERENTIAL EQUATIONS. [258.
It is convenient to classify such expressions : the principles
on which we shall make our classification are (1) the highest
order of the differential or derived-function which is involved,
and (2) the degree or index to which the highest differential
or derived -function is raised; order is predicated of a dif-
ferential equation as to the former, degree as to the latter ; thus
if a differential equation contains x, y, dx, dy, d 2 x, d 2 y, or x, y,
dii d^i/
--, -y-V, it is said to be of the second order; and if the high-
dx dx 2
est differentials or derived-functions enter in only linear forms,
or to the first power, such an expression is said to be of the first
degree; but an equation containing x, y, \-jr)> (j~^j * s f
the second order and of the second degree; and so of other
similar expressions.
Another variety of differential expression is that which con-
tains partial derived-functions; and these may be of the form
corresponding to either an implicit function of many variables, or
to an explicit function of the form z =/(#, y, ...) : this last case
may however, as we shall see in the sequel, be reduced to the
former one by means of the principles of Art. 50, Vol. I. Thus
the usual forms of differential equations are the following :
F (x, y, dx, dy,... d m x, d n y) = 0, (2)
dz
35
idz\ (d 2 z\ I d z z \ /d' 2 z\
()' () () - = '
of the first three forms we have already had instances in the
last Chapter. Equations of the forms (1) and (2) are called
total differential equations, those of the forms (3) and (4) partial
differential equations.
258.] As it is important to have a correct notion of the
meaning of a differential equation, I will at the outset consider
it firstly from a geometrical point of view, and secondly as it
originates analytically by means of the elimination of arbitrary
constants and arbitrary functions, according to the methods of
Section 7, Chap. Ill, Vol. I.
258.] GEOMETRICAL INQUIRY. 337
To fix our thoughts, let us take a differential equation of the
first order and first degree, and suppose it to be in the form
5 =/<>' ^
let x and y be the rectangular coordinates of a plane curve ;
and let T be the angle between the axis of x and the tangent to
dii
the curve at the point (x, y) : so that tan r = -jf- = f(x, y) ; as x
UJu
and y are general in (5), let us take x y to be particular, al-
though arbitrarily chosen, values of x and y, and let r be the
corresponding value of T : so that
tan TO = f(x Q , y ) ;
and through the point {X Q , y ) let a line be drawn cutting the
axis of x at the angle T O : on this line let there be taken a point
(a?!, yi) contiguous to (XQ, y ) } and through it let a line be drawn
cutting the axis of x at an angle T\, so that
tan TI = /(#i, yi) ;
and on this line let there be taken a point (a? 2 > yz) contiguous to
(#1,^1) and through (,r 2 , / 2 ) let a line be drawn making an angle
r 2 with the axis of x, where
tan r 2 = /(# 2 , y z ) :
and let a similar process be repeated n times, until at last we
arrive at the point (x n , y n ) ; hereby we shall have formed a series
of short lines inclined to each other at different angles, and
abutting at the points (ae Q , y ) and (x n , y n ) ; let now every two
successive points be infinitesimally near to each other, and also
let the number of times that the process is repeated be infinite,
then the distance between the extreme points is still finite, and
the broken line which joins them becomes a continuous curve,
and the distances between each two successive points become
arc-elements of the curve : and hereby the curve between the
two points will have been constructed from the given differential
equation. Now from the process thus conducted it is manifest
that the position of each point of the curve depends on that of the
immediately preceding point, the law of dependence being given
by the differential equation (5) ; the nature of the curve there-
fore is given by the differential equation : but it is also equally
manifest that the position of every point, and so of the curve,
depends on that of the first assumed point, viz., on (XQ, yo), and
PRICE, VOL. II. X X
338 DIFFERENTIAL EQUATIONS. [258.
the position of this point is arbitrary : although therefore the
nature of the curve remains the same, whatever be the values
of XQ and y 0} yet the position of it alters, and therefore the dif-
ferential equation expresses a property common to a series of
curves, the particular one of which is determined by means of
the arbitrary values x$ and y : but as a complete integral equa-
tion determines both the nature and position of the curve which
it represents, it is plain that the coordinates of the first point
must enter into the integral equation; and therefore the inte-
gral of (5) must contain these, and cannot be complete without
them ; the integral therefore of (5) must be definite ; but it is
convenient to leave the superior limits in the general form
(x, y), so that they may refer to any point on the curve. It is
plain also, from the theory of definite integration, that if F (#, y)
be the indefinite integral of (5), the definite integral is
v(x, y) F(a? , y ) = 0; (6)
and as X Q , y are arbitrary constants, we may replace F (X Q) y )
by an arbitrary constant c, and thus the integral equation of
(5) is of the form F( ^ y) = c . (7)
or, more generally, F (x } y,c) = ;
that is, in the process of integration one arbitrary constant c
has been introduced.
Again, suppose the given differential equation to be of the
second order and of the form
and let the inferior limits of integration be the coordinates
#o, yo from the point corresponding to which we shall consider
the curve to begin : this point is of course arbitrary : and also
d z v
since -~= involves three consecutive points, see Art. 202, Vol. I,
dx*
and as there is only one relation, viz. (8), between the three
points, the second, as well as the first, is arbitrary ; but not so
the third ; its position with reference to the two others becomes
fixed by means of equation (8) ; and similarly will every other
consecutive point on the curve, and thus the whole curve, become
fixed; in the complete integral therefore the coordinates to
these first two points must enter, and, by the theory of definite
259-] GEOMETRICAL INQUIRY. 339
integration, in the form of two constants, which will be arbi-
trary, because the first two points on the curve are arbitrary :
and so if c\, c z are two arbitrary constants, the complete integral
of (8) is of the form
* to y, c i} c 2 ) = 0. (9)
This is also otherwise manifest : the first indefinite integral of
d*u
(8) will be a function of x, y and -~ ; and therefore the definite
CL30
integral will be of the form
and replacing the second term by an arbitrary constant c l} the
first integral will be of the form
and the integral of this will, by reason of what has been said
above, involve another new arbitrary constant c 2 . It appears
therefore that the complete integral of a differential equation of
the second order requires the introduction of two arbitrary
constants.
By a similar process may we shew that n arbitrary constants
enter into the complete integral of a differential equation of
the nth order. We have already had a case of this in Art. 96,
equation (6).
It may perhaps be superfluous to remark that in thus taking
the definite integral of a differential equation, the differentials
or derived functions must not become infinite or discontinuous
for any value of the variables between the limits.
259.] And to form a correct notion as to the meaning of a
partial differential equation, let us consider the following ex-
ample of a partial differential equation of the first order :
or, as it may be otherwise and equivalently expressed,
Equation (10) is the general equation of a tangent plane of a
surface, which passes through a given point (a, b, c) ; or, what is
X X 2
340 DIFFERENTIAL EQUATIONS. [260.
equivalent, (10) implies that all the normals to the surface are
perpendicular to straight lines which pass through a given point :
and it is not for one surface only, or for one particular species
of surface, that this property is true ; it is not only for a given
cone or for circular cones that the property holds good ; but it
is true for all conical surfaces whose vertices are at the given
point : and therefore a symbol expressing so general a condition
must enter into the final integral equation : in other words, the
complete integral must contain the law of the director-curve of
the conical surface; and such can be the case only when an
arbitrary function is introduced: the complete integral there-
fore of a partial differential equation of the form (10) or (11)
must contain an arbitrary functional symbol : in fact we know
that the integral of (10) or (11) is either
._. t= \ = 12
\Z C Z C'
Z C
2 C Z-C
Hence it appears that the integral of a partial differential equa-
tion of the first order requires the introduction of one arbitrary
function.
260.] Again, let us consider the processes of the differential
calculus whereby differential equations originate.
Those involving differentials and derived-functions arise in
the elimination of constants from given integral equations and
theit successive differentials or derived-functions ; now, as ex-
plained in Section 7, Chap. Ill, Vol. I, each successive differen-
tiation or derivation gives a new equation, and therefore if an
equation be differentiated n times, there will be on the whole
n + 1 equations, by means of which n arbitrary constants may
be eliminated, and the equation which finally results from such
elimination will be a differential equation of the nth order ; the
reverse process therefore of integration ought to reintroduce
these : and hence on passing to a complete integral from a dif-
ferential equation of the nth order, n arbitrary constants must be
brought in ; thus the complete integral of a differential equation
of the first order requires the introduction of one arbitrary con-
stant ; that of the second order requires two ; and it is only
when a sufficient number has been introduced that the integral
26 1.] ANALYTICAL INQUIRY. 341
is complete; and also we are hereby supplied with a test of
such completeness. It may be that determinate functions will
also be introduced as well as constants, as appears from the
reverse process of Art. 87, Vol. I.
Those involving partial derived functions, whether of the im-
plicit or the explicit forms, arise from the process of eliminating
one or more arbitrary functions ; this is manifest from Art. 88
91, Vol. I, and from the differential equations of surfaces,
Chap. XVI, Vol. I ; and therefore in passing by means of inte-
gration from a partial differential equation to its complete inte-
gral, the arbitrary functional symbols must be introduced.
261.] We may also thus prove that the complete integral of
a differential equation of the nth order and first degree involves
n arbitrary constants.
Let us suppose the differential equation to be of the form
dy d n y
-. ... -~ = 0,
doe' dx n l
and to admit of being put into the form
d"y .1 dy d n ~ l y
~
and let us suppose that it, and all its integrals up to the last,
satisfy the conditions which are requisite for developement in
Taylor's series.
Let (14) be differentiated successively, and let the necessary
(fn + ly
eliminations be performed, so that we can determine n+ ^,
d n+2 y dy d n ~ l y
-j ~, ... in terms of x. y, -f , ... ^-: and let the limits for
dx n+2 ' " dx dx n ~ l
which the integral of (14) is to be taken be XQ, y Q and x, y ; then,
by equation (14), Art. 119, Vol. I,
where the subscript cyphers indicate that particular values of
the symbols are to be taken, those, namely, which correspond
to the inferior limit. Now from the preceding remarks it is
plain that all the differential coefficients after [-= ^-) may be
Voa?*" 1 ^
(dy\ /d n ~ l y\
expressed in terms of # yo, {-f-l , ( , n _^) , so that the
* O/30 ' Q * (HOG o
342 INTEGRATION OF DIFFERENTIAL EQUATIONS. [262.
series (15) will involve n and only n undetermined quantities,
viz. the term independent of x, and the several coefficients of
x, x*, ... a? n-1 , which are n in number and may be expressed by
n constants, c\, c- 2 ,...c n ; and therefore into the complete inte-
gral of (14) n arbitrary constants enter.
Of course it is supposed that none of the quantities y ,
. //7/ \ / //**~""1?/ \
( -f- } . ...( . v ) is infinite or discontinuous between the limits :
Vrfff'o \d**~ l / 9
as however we have not given any criteria for determining
whether these conditions are satisfied or not, let the above be
taken to establish an a priori probability that the theorem, as
stated, is true. A rigorous proof of a particular case is given in
Sect. 7 of the present Chapter, and might be extended generally.
As an example of integrating by this process, let us take
- + ay + bx* = 0,
Let the inferior limits be y , and XQ = ;
therefore (15) becomes
y=
[.2.3.4
CUK CL Ou CL 00
2b
which involves only one arbitrary constant, viz. y .
262.] When the integral of a given differential equation con-
tains n arbitrary constants, and these in their most general form,
2,63-] INTEGRATION OF DIFFERENTIAL EQUATIONS. 343
it is called the general integral; and conversely, if an equation
in terms of x and y satisfies a given differential equation of the
wth order and contains n arbitrary constants, it is the general
integral : and if particular values are given to one or more of
these arbitrary constants, as, for instance, if any of them is
zero, then the integral is called a particular integral ; and some-
times it happens that we are able to replace one or more of the
arbitrary constants by a particular function of x and y, and
render the equation such as will satisfy the given differential
equation, when at the same time such a result cannot be ob-
tained by giving any particular constant value to one or more
of the arbitrary constants of the general integral : in this case
the integral is called a singular solution. Our chief work is the
discovery of the general integral, by means of which particular
integrals evidently may be determined : and we shall investigate
as far as possible the general properties of singular solutions,
and also indicate some specific forms of differential equations
which admit of such solutions.
And in most cases we shall be obliged to leave the arbitrary
constants undetermined ; the complete integral of a differential
equation requires that the integral should be definite, and there-
fore the constants ought to be expressed in terms of the limits ;
but it is manifest that this can be done only when the physical
conditions of the problem are given, as in the geometrical appli-
cations of the calculus. Differential equations, however, for the
most part arise in mechanics and other applied mathematics, on
the investigation of which we have not yet entered : the con-
stants therefore which are introduced in the process of integra-
tion must in most cases be left arbitrary, at least for the present.
263.] A simple form of differential equation which admits of
integration immediately, or, as it is commonly said, by simple
quadrature, is that where the variables are separated : the general
form of it in the case of two variables is
f(x}dx + <$>(y)dy = 0; (16)
whence we have for the definite integral, XQ and y Q being cor-
responding values of x and y,
(
/(*) dx + $ (y) dy = ; (17)
o ^vo
and if the integrals are indefinite
344 INTEGRATION OF DIFFERENTIAL EQUATIONS. [264.
= c, (18)
where c is an arbitrary constant.
And if there are three variables, the general form of the equa-
tion is f(x} dx + <t>(y) dy + x (z) dz = 0, (19)
(z)dz = 0. (20)
Ex.l. -^ + dy =0,
(l-# 2 )* (1 y 2 ) 5
sin -1 x -f sin" 1 y = sin^c,
c being an arbitrary constant ;
.-. x(\-y*Y- + y(l-xrf = c,
which is the general integral ; and if c = we have a particular
integral y=x.
264.] Another form in which the variables immediately admit
of separation is , ,
XYxdtf + YXi^y = 0, (21)
where x and xi are functions of x only, and Y and Y A are func-
tions of y only ; for dividing through by Xx Y! we have
dx + dy = 0, (22)
*i Y!
... [^dx + ll-dy = c, (23)
J Xi J YI
which is the general integral of (22).
Ex. 1.
dx dy
.'. + = 0,
x y
log x + logy = logc 2 ,
.-. xy = c 2 ,
c being an arbitrary constant.
Ex. 2. dyx^ + dxy* = 0,
dy dx
where a is an arbitrary constant.
These methods however are so simple that it is unnecessary
to add other examples.
265.] INTEGRATION OP EXACT DIFFERENTIALS. 345
SECTION 2. Integration of exact total differentials of two and
more variables.
265.] We will first consider the case of two variables : Sup-
pose a differential expression to be
vdx + ^dy = 0, (24)
where p and Q are functions of x and y : it may be that (24) is
the exact differential of some integral function of the form
u = F (x, y} = c ; or it may be that some factor common to the
two terms has been divided out, and that (24) will not be an
exact differential until this factor, or some other factor, has been
introduced; this latter case is reserved to Section 6 of the
present Chapter.
Suppose however that (24) is the exact differential of a func-
tion of two variables
M = ?(x,y) = c, (25)
then P dx + Q dy = DM
<*>
and as dx and dy are arbitrary though infinitesimal increments
of x and y, (26) can only be true when
Hence we have a criterion whether (24) is an exact differential
/ d z u \ I d z u \ . ._,,
or not: for since I-; r-J = (-3 r-)i " (*') are true, we have
\dydx' \dxdyi
< 28)
.. . .
and therefore if it is not on inspection plain whether (24) is an
exact differential or not, we may apply the condition (28) ; and,
if it be fulfilled, we are assured that (24) is an exact differential.
The equation (28) is commonly called the condition of inte-
grability. Let us suppose it to be fulfilled. Since p dx is the
^-partial differential of , the ^-integral of P dx will give us the
whole function of x which enters into the general integral ; and
similarly the y-integration of Q dy will give us the whole func-
tion of y : the addition therefore to the ^-integral of P dx of
those functions of y which the ^-integral of Q dy contains and
are not in the <r-integral of P dx will give the whole variable
PRICE, VOL. u. Y y
346 INTEGRATION OF EXACT DIFFERENTIALS. [266.
part of the general integral of (24), and the addition of a con-
stant, or the determination of the definite integral, when the
limits are given, at last gives the general integral of the dif-
ferential equation.
Hence we have u I p dx + Y (29)
x, (30)
where Y and x are functions severally of y and x only, and
which are added to the partial integrals of P dx and Q dy ; and
where Y is the sum of all the y-functions which are in (30) and
are not in (29); and where x is the sum of the ^-functions which
are in (29) and are not in (30) .
266.] Ex.1. (2ax + by+g)dx + (2cy + bx + e)dy = 0,
p =
therefore the criterion of integrability is satisfied. Now let us
symbolize by u x and by u y the x- and y- partial integrals,
u x = (2ax + by+ff)dx = ax^ + bxy+gx
J r \", (31)
u y =J(2cy + bx + e)dy = cy 2 + bxy + ey
and let Y and x be functions of y and x respectively, which are
added to the partial integrals as above ; and therefore
Y = cy z + ey,
x = ax 2 + gx,
by means of either of which we have from (31)
u = axP + bxy + cy
where k is an arbitrary constant.
Ex.2. y^
P--1-
' ' \dy
Q =
and therefore the criterion of integrability is satisfied.
267.] INTEGRATION OF EXACT DIFFERENTIALS. 347
ydx .x
- 1
Cx
= /-g
J x^^
* xdy y
Uy ~ / r g Tan
, X TT
36
.'. u = tan- 1 - -f e.
Ex. 3. {$>(%y} + xy $'(xy) } dx + x 2 $(xy) dy = ;
p =
and therefore the criterion of integrability is satisfied; and
we have
= x
u =
267.] It will be observed that, if the variables are separated
as in (22), P and Q are functions severally of x and y only, and
therefore that
hence the criterion of integrability is satisfied ; and therefore
by two single integrations the general integral is determined.
The unknown function Y in (29) may also be determined in
the following way without the integral (30) : Take the definite
integral of (29) and differentiate it with respect to y,
rx
u = / P dx + Y, (32)
<du\ d C x
) i j
\dy> dyJ Xo
and since (-r- ) = Q, we have
p dx + -r- ;
dy
-=- = Q / ~:~dX
dy J Xo ay
vy 2
348 INTEGRATION OF EXACT DIFFERENTIALS. [268.
fl?Y r^j u t /OQ
= Q / -7- dx, by reason of (28).
dy J Xo dx
= Q Q + Qo>
representing by Q O the value of Q when x = X Q :
(33)
r* /v
.-. u= rdx+ Q dy; (34)
Vd *^o
or representing p and Q by/(#, y) and </>(#, y) we have
/*# />
w = / /fo lO*M- f 0(^0, y) <fy (35)
^n * Vo
A/ T^
= / $(x,y)dy + l f(x, y Q ) dx. (36)
' J *
268.] Next let us consider the case of a differential equation
of three variables, and of the form
p dx + Q dy + R dz 0, (37)
where P, Q and B are functions of x, y and z. Now of course it
may be that either (37) is an exact differential, or that some
factor common to all the terms has been divided out, and that
the expression can be made exact only by introducing this, or
some other equivalent, factor : this latter case we shall, as here-
tofore, reserve to Section 6 of the present Chapter, and shall first
consider the case where (37) is an exact differential. If we
recognise immediately the general integral of (37), it is of the
form = p(ar,y,*) = c, (38)
and we need apply no criteria of integrability : and this is mani-
festly the case in such an example as
/Vt ftj g
-2 dx + -^ dy + ~ dz = ;
a 2 o 2 c 2
where k is an arbitrary constant ; and in
($+z)dx+(z+a;)dy+{y+y)d2 = 0,
.;. yz + zx + xy k 2 = u = 0.
268.] INTEGRATION OP EXACT DIFFERENTIALS. 349
Let us however suppose that (37) is the exact differential of a
function of three independent variables of the form (38) ; then
= DM
\ . tdu\ , fdu\ ,
)*+(j^* *.(}*' (39)
and as da?, dy, dz are arbitrary though infinitesimal, this equa-
tion can be true only when
du
Hereby we have criteria whether (37) is or is not an exact
differential ; for since
/ d 2 u \ _ / d 2 u \ i d 2 u \ _ / d 2 u \ , / d z u \ _ i d z u \
\' ~ \i' \dz~d~xi ~ \Wd~zl y a \d7di ~ \d~}'
dydz' ~ \dzdyi \dzdxi ~ \dzl y \ddyi ~ \dy~dlz
if (40) are true, we have
which equations are called the conditions of integrability of
(37) ; and if they are fulfilled we can integrate as follows :
Since P dx is the ^-partial differential of the general integral,
the a?-integral of vdx will give us the whole function of x which
enters into the general integral ; similarly will the y-integral of
0,dy give the whole function of y, and the ^-integral of n,dz, the
whole function of z : if therefore we add to the <r-integral of
p dx those functions of y which are contained in the y-integral
of Qdy and are not in the .r-integral of ?dx; and if again we
add to the sum those functions of z which are contained in the
^-integral of ndz, and which have not already entered, the re-
sult will evidently contain all the variable part of the general
integral, and therefore by the addition of an arbitrary constant
will the general integral be obtained.
Hence we have ,.
u = / vda? -f Y -f z
u =
= r&
(42)
where x, Y, ... are severally functions of x, y } ... only ; and are
determined in the manner explained above.
350 INTEGRATION OF EXACT DIFFERENTIALS. [269.
269.] Ex.l. yzdx+zxdy + xydz = 0.
/dR\
(dx) = y >
and the conditions of integrability are satisfied : and let u x , u y) u t
represent the several partial integrals ; then, taking indefinite
integrals, /.
u x = I P dx
= lyzdx
= xyz;
similarly, u y = xyz,
u z = aeyz;
. . u = xyz A: 3 = 0,
where k is an arbitrary constant, is the general integral.
_ xdx-\- ydy + zdz zdxxdz
Ex.2.- h ; - 5 +3ax?dx + 2bydy + cdz = Q.
(x* + y* + zrf ** + **
On applying the criteria to this equation, it will be found that
it satisfies them : and representing by u x) u y , u z the x-, y-, and
s-partial integrals,
X
=/k
Z
Similarly, from the other partial integrals, we have
U = x* + 2 + z z
therefore, if k is an arbitrary constant, the general integral is
u = (a? 8 + y a + *)*+ tan- 1 - +ax 3 + by 2 + cz k 0.
It will be observed, that, if the variables are separated in the
general form (37), the conditions of integrability are satisfied,
for then
270.] INTEGRATION OP EXACT DIFFERENTIALS. 351
dyi ~ \dz' ~ W ~ W " W ~ \dy) ~~
and therefore if u x , u y , u z are the three partial integrals,
u = u x + Uy + u g + k = 0,
where k is an arbitrary constant, is the general integral.
270.] We may also express the general integral in terms of
definite partial integrals as follows : as the process is similar to
that of Art. 267, it is unnecessary to repeat every step of it ;
let the equation be
/i fo y, z) dx 4/2 (x, y, z) dy 4/3 (x, y, z) dz = ;
.' u = I fi(ar,y,z)dx + V', (43)
Ar
where v is a function of y and z ;
x d
Cy
.-. v = f 2 (xo,y,z)dy + w; (44)
-Vo
where w is a function of z only.
[x Cy
.-. u - I f l (x,y,z)dx+ /2(zo,y,z)dy + w,
J XQ ^O
du * d . y d dw
/ 3 (*, y, z) = / 3 (x, y,
dw
= I Mxo,y Q ,z}dz;
Jz
rx r y rz
u = f l (x ) y,z}dx + ] f z (xQ,y,z}dy + \ MxQ,y*,z)dz, (45)
J *o Jy<> J*o
As an example of this form let us take the simple one
_
352 IMTEGRATION OF EXACT DIFFERENTIALS. [271.
dz
f
.'. M = /
^Xn X
= log (a? 3 + y 2 + *) -log (# 3 + */ 2 + z)
+ log (# 3 + y 2 + *) - log (# 3
+ log (^o 3 + y<? + 2} log (# 3
= log (tf 3 + y 2 + xr) log (# 3
271.] Lastly, let us briefly consider a differential equation of
n independent variables of the form
where p l5 P 2 ,...p n are functions of the n variables Xi, Xi,...x n \
in order that this may be an exact differential of a function
u = F (xi, x%, ... # n ) = c, (47)
we must have
du
dxi
(48)
and that these equations should be true, it is necessary that
" \dx n >
(dvz\__ (dr n
> ' " \ ~j / \ ~j
N ax n ' V ax?,
dx n ' \ dx$
; (49)
dx n
the number of which conditions is the same as that of n things
taken two and two together, that is, is equal to
__ 1 \
; and
when these are satisfied, and the n partial-integrals are found,
the general integral may be determined from them by a process
similar to that employed in the cases of two and three inde-
pendent variables.
We may also express the general integral in terms of definite
partial integrals in the following manner. Let the coefficients
of the differentials in (46) be/i (x lf <z? 2 , ... #), / 2 (x\, # 2 , ... x n ),
.../(#!, # 2 , ...Xn); and let the inferior limits of integration
272.] INTEGKATION OF HOMOGENEOUS EQUATIONS. 353
be x 1? x 2 , ... x n : then by a process similar to that of the last
Article
/*.T 2
i(xi,x 2 , ...x n )dxi + ft(Xi,Xs...x
/X
*
/n(X 1} X 2 , ...X n _!,^)^ n . (50)
SECTION 3. Integration of Homogeneous equations of two
variables.
272.] Differential equations of the first order and degree
can generally be integrated only when they satisfy the criteria
of integrability ; and therefore when an equation does not fulfil
these conditions, our first object is to investigate, if it be possible,
some mode of so transforming it that its equivalent may be in
the required form : the principal means which are useful for
such a transformation are (1) an introduction of new variables
by way of substitution, (2) the multiplication of the equation
by a factor which will render it an exact differential, and which
is commonly called an integrating factor : these processes we
go on to examine.
First, let us take the case of two variables, x and y, and
suppose the equation to be
vdx + Qdy = 0; (51)
now suppose that the criteria of integrability are not satisfied,
but that P and Q are homogeneous functions of x and y, of n
dimensions : then dividing through by x n , so that x n may stand
as a common factor, (51) takes the form
= 0. (52)
Let y = xz, . . - = z,
x
dy = xdz + zdx,
and neglecting the factor x n , (52) becomes
dx} = 0,
)dz = 0,
dx <f>(z)dz
^ + f(z)+z<t>(z)-
an equation in which the variables are separated, and therefore
PRICE, VOL. II. Z Z
354 INTEGRATION OF HOMOGENEOUS EQUATIONS. [273.
the conditions of integrability are satisfied ; and thus the inte-
gration depends on that of two simple differentials of one
variable.
Instead of arranging the equation (51) in the form (52), where-
in x n is the common factor, it might equally as well have been
put in the form
y n f(l}dx + y n 4>( X -}dy = Q', (54)
/ J
3C
and if x = yz, or - = z be substituted, the variables will be
separated, as in the former case, and the criterion of integrability
will be satisfied.
273.] Ex. 1 . y z dx + (xy + x 2 ) dy = 0.
Let x = yz, .-. dx = ydz + zdy,
2 )dy = 0;
= 0,
dy dz n
, 2
x
Ex. 2. x*ydx(x* + y*}dy = 0.
Let x yz, .-. dx = ydz + zdy;
.'. ^ =*&,
y
X*
y = ce 3y3 .
Ex.3, xdyydx (x z + y 2 )*dx.
This is an homogeneous equation of one dimension.
Let y = xz, dy = xdz-\-z dx;
.'. x(xdz + zdx) xz dx -f (x z + # 2 z 2 )^ dx ;
dx dz
which is the general integral.
Although either of the substitutions y = xz or x = yz will
274-3 INTEGRATION OF HOMOGENEOUS EQUATIONS. 355
produce the same result, yet a judicious choice will frequently
shorten the subsequent processes : the student however must
in this matter be left to his own skill.
274.] Let us also consider homogeneous equations of the
form (52), and the introduction of the new variable, from a geo-
metrical point of view ; (52) may evidently be put in the form
^=
dx
dii ?y
and since -~- = tan T, and - = tan B, T and 6 being the angles
CLOG 00
respectively at which the tangent to a curve, and the radius
vector are inclined to the axis of x, the above equation, inter-
preted geometrically, expresses that a relation is given between
these two angles. Thus, suppose that T = 26, then
dy _ 2xy
dx x'ty 2 '
Let x = yz, . . dx = y dz + z dy ;
dy 2z dz
_ 7 I _ M .
7 + ^Ti-
.-. Iog^ + log(z 2 +l) = 0;
.-. x 2 = 2cy-y 2 -,
which is the equation to a circle, whose radius is the arbitrary
constant c.
And to take another example, see fig. 50 : to find the equa-
tion to a curve such that a perpendicular MS, let fall from M,
the foot of the ordinate on radius vector OP, shall cut the axis
of y at the point T', where it is cut by the tangent PT'.
tan s o M = tan o T'M ,
_
x ~ dy'
yx~-
dx
.- . xydy-\ (x*y*)dx = 0;
275.] By the introduction of the new variable z the original
expression (51) has been so transformed as to admit of the
variables being separated; let us examine the process more
closely : take the form (52), then
z z 2
356 INTEGKATION OF HOMOGENEOUS EQUATIONS. [2,76.
P=
(55)
Q -
and (53) has been found by dividing (52) by x n+l
which is manifestly equal to vx + qy; hence the equation (51)
satisfies the criteria of integrability when it is divided by
p# + Q2/; therefore (px + o,y)~ l is an integrating factor of
p dx + Q dy = 0. Let us apply this process of integrating homo-
geneous equations.
Ex. 1. xdx + ydy m(xdy ydx).
(x + my) dx + (ymx) dy = ;
therefore the integrating factor is
and the equation becomes an exact differential, of the form
(ae + my} dx+(y- mx) dy _ _
/
"
Ex. 2. Again, let us take Ex. 2, in Art. 273.
x 2 y do? (x s + y s ) dy =0;
the integrating factor is y~ 4 : therefore
a? 2 y cfo? (# 3 -f y 3
= DM;
'** -f
x z dx
276.] And that the factor (p^ + Qy)- 1 renders (51) an exact
277-] INTEGRATION OP HOMOGENEOUS EQUATIONS. 357
differential may also thus be proved ; the condition of integra-
bility becomes satisfied : for multiplying by the factor, we have
_
and the condition of integrability is
dp do,
dy p# + Qy dx
( (dp\
QI-^-T- -f
;
WPQ WPQ
= 0;
since by Euler's Theorem, P and Q being homogeneous func-
tions of n dimensions,
and therefore the criterion of integrability is satisfied.
277.] A form of differential equation which is easily reduced
to the homogeneous form is
(diX + biy + Ci) dx+(a 2 x + b 2 y-}- c 2 ) dy 0: (57)
let a^ + ^iy + ci = g, .-. dg =
rj, drj
j : - ,
i o 2 a 2 0i
i .
a 2 dg aidrj i '
aib 2 j
and substituting in (57) and reducing,
(6af + aai?)#-(ft 1 + fli'?)di7 = 0, ( 59 )
an homogeneous equation, which is integrable as above.
This transformation is manifestly equivalent to that of a sys-
358 INTEGRATION OF THE FIRST LINEAR EQUATION. [278.
tern of rectangular coordinate axes, in which the origin and the
direction of the axes are changed : and it is always possible,
unless ,
-^ =-- = * (say), (60)
a 2
for in this case d and drj are infinite : but (57) becomes
(ka->,x + kb2y-{-Ci)dx + (azX + bzy-{-Cz)dy = 0,
( 2 #+022/) (dy + kdx) -\-c\dx-\-Czdy = 0, (61)
in which, if we put a^x + b z y = z, and eliminate x or y, the vari-
ables will be separated, and the integration can be performed.
Also by a similar substitution may the variables be separated
in the equation 7 f , , . , ,/, ,.
dy = f(ax + by) dx. (62)
The reader is referred to a paper by Professor Stokes of
Cambridge, in Vol. IV of the Cambridge Mathematical Journal,
for an investigation of certain properties connected with the
integrability of homogeneous equations.
SECTION 4. The integration of the first linear differential
equation.
278.] Another form of differential equations in which the
variables admit of separation is
= 0, (63)
where P I? P 2 , PS are functions of x only ; and which is called the
dii
linear equation of the first order, because -^- and y enter in
ttiOO
only the first degree, and there is a vague analogy between it
and the equation to a straight line.
Dividing through by PX and making obvious substitutions, the
equation becomes
dy+f(x)ydx = v(x)dx. (64)
Let y = zt-, .-. dy = zdt + tdz; (65)
. . zdt + tdz +f(x) ztdx =^ P (x) dx
zdt + t{dz+f(x)zdz] = v(x)dx. (66)
As we have introduced two new variables z and t, and have
made only one supposition respecting them, we may make an-
other ; therefore let
dz-\-f(x)zdx = 0,
279-] INTEGRATION OF THE FIRST LINEAR EQUATION. 359
dz // N T
- = -f(x)dx,
2
.-. log* = -ff(x)dx, (67)
z = ?-//<*>**; (68)
thus (66) takes the form
dt ef^ x]dx (x)dx,
.-. t = c + eSsw dx i?(x)dx;
.-. y = zt
e -//(* * j c + / e ff(x ) d# F (^ ^ f (gg ^
No constant has been introduced in (67), because it is desir-
able to keep complex formulae in as simple a form as possible :
the generality also of the final result is not aifected by the omis-
sion, because such a constant would disappear in (69) by reason
of the form of the result.
In terms of definite integrals (69) is
y = e-S r * Sw d *$y + reS^^v
279.] Ex. 1. dy -\-ydx = e x dx;
in this case /(#) = 1 ; .' . lf(x)dx = x,
y = e ~ x {c + le- x e x dx}
e ~ x {c + x}.
Ex.2. (1 x 2 )dy + xydy = adx;
dy x a
-//(*)* = (1
ax
360 INTEGRATION OF THE FIRST LINEAR EQUATION. [280.
280.] Another form which admits of reduction by means of
substitution to (64), and therefore of having its integral deter-
mined in the form (69), is
, (70)
and which is generally known by the name of Bernoulli's linear
equation of the first order ; for dividing through by y n we have
y- n dy + y~ n+l f(x}dx = v(x}dx. (71)
Let y~ n+l = z, .'. (n \)y~ n dy = dz,
and therefore by substitution
dz (n \)zf(x) dx (n \)v(x}dx', (72)
and by the formula (69)
y n ~ l J e
The explanation of the failure of the above substitution when
n = 1 is too obvious to require more than a passing remark.
Ex. 1. -/- + - = xy^ ;
dx l x 2
x = xdx.
Let
~~ 30
dit
x dx xdx
therefore, according to the notation of (69),
= log (1-a? 8
Another form of differential equation to which the above
method of solution is immediately applicable is manifestly *
28 1.] PARTIAL DIFFERENTIAL EQUATIONS. 361
SECTION 5. The integration of partial differential equations
of the first order and degree.
281.] We must now consider differential expressions of an-
other character ; those, viz., wherein a relation is given between
partial derived-functions and the variables : the general forms
of these are (3) and (4) in Art. 257. I shall at present take the
simple case where the partial derived-function enters linearly,
and where the coefficients are functions of the variables, includ-
ing of course the case where they are constant.
First let it be observed, that a partial differential expression
which arises from an implicit function of two variables of the
form M = *(*,y) = c, (75)
and the general form of which is
' (76>
where P and Q are functions of x and y, although involving
partial derived-functions, is in fact a total differential expression ;
for differentiating (75) we have
<du\
/du\
dy _ \dx' _ Q
dx~ idu\ " *'
\dy>
.-. <zdx ?dy = Q, (77)
which is a total differential equation of the form (24).
Now, from the explanation of partial differential equations
given in Articles 259 and 260, whether from an analytical or
from a geometrical point of view, it follows that the integral
of a partial differential equation of the first order and degree
requires the introduction of an arbitrary function ; and although
the integral may be particular, yet it is not general without it.
Since then a total differential equation of the form (77) may by
an inversion of the process followed above be changed into a
partial differential equation, so does its general integral require
an arbitrary function : the method of determining it will be
explained in Section 6 of the present Chapter : thereby also we
PRICE, VOL. ii. 3 A
362 PARTIAL DIFFERENTIAL EQUATIONS OF [281.
shall be led to a solution of total differential equations still more
general than that of the preceding Sections.
Let us now consider a partial differential expression of three
independent variables #, y, z, and of the form
where P, Q, R are or may be functions of x, y and z, and in
which z has been considered a variable dependent on two inde-
pendent variables y and x. To consider it however in the most
general form, let us suppose the original function to be of the
form u _ p,/^, y 2 \ __ c /70\
where F symbolizes the arbitrary function, which the complete
integral requires ; then, by the process of Art. 50, Vol. I,
fdu\ tdu\
(dz\_ \dx> ( dz \_ W.. (80)
' du
iau\
\dz'
substituting which in (78) we have
and this is the most general form which a partial differential
equation of the first order and degree can have : and it is of
this that we shall investigate the integral.
Now of (79) the differential is
*+<$*+*-
thus from this and (81) we have
tdu\ idu\ tdu\
\> \> \
dz'
' rdz vdy qdx
and let us moreover assume that
~dx = dy = dz'
either of which equations, it will be observed, involves the other
by reason of (83) ; and let us suppose that two independent
integrals of these equations are found, and that they contain
two arbitrary constants c\ and c 2 , and are of the form
/i to y, *) = <?i, f to y> *) = <*, (85)
where c\ and c 3 are arbitrary constants.
282.] FIRST ORDER AND DEGREE. 363
Now from these we have
dy i \ dz
dx ' ^ dy
from which, by reason of (84), we have
dx i \ dy ' ^ dz
dy i \ dz
on comparison of which with (81) it appears that either fi or/ 2
satisfies (81) : and so also will any arbitrary function of/i, /g:
for let F represent an arbitrary function of/i,/ 2 , that is, of Cj.
and c 2 ; then multiplying the members of (87) severally by
^Te~) y an< ^ Adding, we have
a/2 '
and therefore F (f\,f%) satisfies (81) : and therefore we have from
the general integral _ / f / \ _
= F (ci) c z ) =
or, as it may be expressed,
and either (88) or (89) is the general integral, because each con-
tains an arbitrary function in its most general form.
282-3 The process requires further development and illustra-
tion : but it will be better first to consider and solve some
particular examples.
Suppose that the given equation is
/(^); (90)
then z = jf(x, y)dx + $ (y), (91)
where < (y) is the arbitrary function which enters into the general
integral, and which has y only for its subject. Similarly, if
3 A2
364 PARTIAL DIFFERENTIAL EQUATIONS OF [282.
Thus if f*V-i-*f
dx' x + y
dz dx
z+y
log (x + y) =
and this is the complete integral.
We may also thus prove (91) : replacing (-T-J in (90) by its
value from (80), we have
tdu\ idu\
and therefore from (84)
dx _ dy dz
'
.'. dy = 0, y =
dz = dxf(x, y} = dxf(x,
z
f(x,
J
ff(x,y)dx
or, what is equivalent, by means of the substitutions of (80)
Now by the conditions (84) we have
dx dy dz
- = -/ =
a b c
ay=iCi~\
1; (94)
az = c 2 J
dx dy dz
- = - = (93)
cx
and therefore by reason of (89)
bxay f(cx az); (95 )
or u v (bxay, cxaz) 0; (96)
282.] FIRST OEDER AND DEGREE. 365
either of which is the general integral and involves an arbitrary
functional symbol.
It is useful to observe the geometrical interpretation of the
process :
Let (95) or (96), viz., u = F, represent a surface : then from
(92) it appears that the normal to the surface at every point of
it is perpendicular to a straight line, whose direction-cosines are
proportional to the quantities a,b,c: but as these determine the
direction and not the position of a line, we can only conclude
that every normal is perpendicular to one of a series of parallel
straight lines : and the successive positions of these lines may
vary according to any law ; which law however is not given by
the differential equation, but is required for the integral equation
of the surface : in fact the insertion of it is absolutely necessary ;
for otherwise the equation cannot represent a surface > and the
geometrical form of the law is the equation to the director curve
along which the parallel straight line moves, and generates the
surface; and this surface is cylindrical, This is also manifest
from (88) and (94) ; (94) are the equations to two systems of
parallel planes respectively parallel to the axes of z and y : and
the intersection of two, viz., one of each system, will give the
generating line of the surface ; and the line of intersection will
of course vary according to the functional relation between c\
and c 2} the particular values of which determine the particular
intersecting planes.
Ex.2. (x-a)(^
\dx>
The equivalent of this in the most general form is
= 0; (97)
and therefore by (84)
dx dy dz
xa y b zc
ya) =
a) = log (2 c)+logc 2 ;
x a x a
- 7 ) ^2 i
y b z c
x a
~y^b '
366 PARTIAL DIFFERENTIAL EQUATIONS OF [282.
ix a xa\
u F ( JT, ) = 0;
\y b zc'
which may also be expressed as follows :
^b zc x
u =
c xa\
, f ) = 0.
a y o'
Z C X-
Observe the geometrical meaning: (97) indicates that the
normal to the surface is perpendicular to a straight line which
passes through a given point (a, b, c), and therefore the surface
is generated by a straight line which passes through the given
point and moves according to a given law : and this is a pro-
perty of conical surfaces, of which therefore (98) is the general
equation, and the arbitrary functional symbol contained in it
expresses the law of the director-curve.
Ex. 3. (mz ny)(-j-} + (nxlz)(-j-} = ly mx.
y
The equivalent to this in its most general form is
!) = 0; (99)
let -*L_ = -Ji- = * ; (100)
mzny nxlz lymx
multiplying the numerators and denominators severally by
x, y, z, and adding ; and again operating in the same way with
/, m, n, the sum of denominators in each case is zero : therefore
the sum of the numerators must also vanish : therefore
xdx + y dy + zdz = 0,
Idx + mdy + ndz = 0;
= c 2 ; (102)
)-, (103)
or u = v(x z + y 2 + z 2 , Ix + my + nz) = 0;
either of which is the equation of surfaces of revolution, and in
which the origin is on the axis of revolution ; and equation (99)
implies that all the normals of the surface pass through the
axis: also from (101) and (102), which are respectively the
equations of a sphere and a plane, it follows that all plane sec-
tions of the surface which are perpendicular to the line whose
direction-cosines are proportional to /, m, n are circles.
2-83-] FIRST ORDER AND DEGREE. 367
(104)
= F (--i, ---) = 0; (105)
\X II X Z>
or u
X Z
or M - F --, -, - = 0. (106)
\y z z x x y'
283.] The supposition made in Art. 281, by which (84) is
assumed from (83), requires further elucidation ; and that our
notions may be definite, I shall consider it from a geometrical
point of view. Suppose the integral equation to be that to a
surface; then, from (81) and (82), it appears that the normal
to the surface at a certain point is perpendicular to the line
whose direction-cosines are proportional to the values which
p, Q, B have at that point, and also to any line of which the ele-
ment on the surface is ds, the projections on the coordinate
axes of ds being dx, dy, dz ; and combining these two conditions,
as in (83), it follows that the normal to the surface is coincident
in direction with the normal to the plane containing these two
lines (P, Q, R), (dx, dy, dz). Now the direction (p, Q, R) is fixed for
any one point, and the direction of ds is indeterminate; in order
therefore that we may leave the most general condition to be ful-
filled hereafter, we may suppose these two directions to be the
same, which fact is expressed mathematically by the equations
(84): so that now (-j-\(-r-}> ry-) are indeterminate, as appears
from (83), and therefore the normal is only limited to being in
the plane which passes through the point under consideration,
and is normal to the line (P, Q, R). Thus far it appears that
PARTIAL DIFFERENTIAL EQUATIONS OF [284.
two consecutive points on the line (P, Q, R) will be on the surface,
but nothing is determined as to consecutive points in other
directions.
Now suppose that the integrals of the two equations (84) are
found and are (85) : these are manifestly the equations to two
surfaces, and, being simultaneous, express a line which is their
line of intersection, and lies on the surface, and it is for all points
along it that equations (84) are satisfied. Now the forms of
these surfaces depend on the forms of P, Q, R ; and as the equa-
tion of each of them contains an arbitrary constant, Ci or c 2 , so
by the variations of these do systems of surfaces arise, and by a
relation which is arbitrary, but which we may assume to exist
between their constants, do we obtain a series of lines, all of
which lie on the surface u = 0, and therefore by which, in their
several and successive positions, the surface is formed ; and this
relation between c\ and c 2 may be expressed by a functional
symbol which will enter into the final equation ; and although
this function may be arbitrary, yet for any one surface it will
be determinate ; and hence will the values of ( -^-L ( -r- )> IT-]
\dx> \dy' \dz>
become determinate, and the position of the points contiguous
to (x, y, z) be fixed in other directions than along (p, Q, R), that
is, in other words, the resulting equations will express a con-
tinuous and determinate surface. Although then the assump-
tion of (84) may appear to restrict the generality of (81 ), inas-
much as it causes the conditions expressed by it and (82) to be
satisfied along only a line on the surface, yet it leaves us free
to introduce the general functional symbol of relation between
c\ and c 2 , and thereby are we enabled to express the class of
surfaces of the greatest extent which satisfies the condition of the
given partial differential equation.
The reader will perceive the agreement between the method
here explained and the process of solution applied to the exam-
ples of the last Article.
284.] A similar method may also be applied to the integra-
tion of partial differential equations of the first order and first
degree of any number of variables.
Let the partial differential equation involve n variables,
#1, %?,, ... # n , and suppose the required integral to be of the form
284.] FIRST ORDER AND DEGREE. 369
U = v(a?i, # 2 ...#n) = 0;
and suppose the differential equation to be
du \ I du \ i du\
(108)
where all the variables are supposed to be independent ; for if
such were not the case, but if one were supposed to be a func-
tion of the other (n 1), the equation might be changed into
the form (108) by means of equivalents analogous to (80).
Now the total differential of (10?) is
and let us assume that the following (ft 1) relations exist be-
tween (108) and (109),
d&i dx?, dx n
TT 17 ==M(say).
Suppose now that we can determine the integrals of the
(w^-1) different equations which are involved in (110), or can
by any means (as in Ex. 3 of Article 282) determine (n 1)
different relations between the n variables; and suppose them
to be of the forms
#2, ...#) = ci, / (*i, #2, ...*) =c f , *)
// \ t > ("*/
_l(4?i,J?, ...*>) = C n -i, J
where Ci, e 2 , ... c n ^ represent n 1 arbitrary constants.
These arbitrary constants however must be related to each
other by a functional symbol, such as
fcta,^,...**.!) = 0, (112)
or *(/!,/* .../-i) = 0; (113)
where/i,^, ... are used as abbreviations for /i (MI, #2, t x n)>
in (111), as may thus be shewn : let (111) be differentiated, and
we have
and multiplying these severally by l-yyh I-TZ-)* * (j3
V dji> v /2 / x /n-
PRICE, VOL. II. 3 B
370 PARTIAL DIFFERENTIAL EQUATIONS OF [284.
and adding, the coefficients of dxi, dx z , dx n are evidently
( ), ( ), ... (-: ), and therefore we have
V ///*>, I \ rf.r* I \ f/.r., '
*daei'
and replacing dx\ y dx z , ... dx n by their proportionals from (110),
we have
comparing which with (108) it is manifest that (with the excep-
tion of an added constant, which is immaterial) 4> = u ; and
therefore, from (113), the general integral is
=*(/i,/ 8 ,.. ./-i) = 0; (117)
or, as it may be written,
/!= <M/2,/3,.../n-l). (H8)
Also if we operate on the several equations of (114) with the
series of equalities (110), by comparing the results with (108) it
will be manifest that the functions f\,fz, ...f n -\ are all such as
when substituted for u satisfy (108) ; and are therefore solutions
of the given equation : each however will be less general than
(117), because (117) combines them all under one other arbi-
trary functional symbol.
The student desirous of further research into one of the most
difficult parts of the Integral Calculus must have recourse to
(1) the "Analyse appliquee" &c. of Monge, 5th edition p. 421 ;
(2) to papers of Jacobi in Crelle's Journal; especially to that
" de Determinantibus Functionalibus," Crelle, Vol. XXII, and in
which he will see the subject discussed in all its depth. I may
however mention that although I have shewn that (117) is such
as to satisfy the given equation, yet I have not proved that it is
the necessary solution; the question is, are any and what re-
strictions introduced by the hypothetical assumptions (110)?
But these inquiries are beyond the range and scope of the
present work.
Let this be changed into its equivalent
285.] FIRST ORDEE AND DEGREE. 371
therefore, by the assumptions (110),
dt dx dy dz
'x + dy + dz _ dt dx
xt
log (t + x + y + zY = tog'
J
For convenience of notation, let t + x -(- y + z = &> 3 ; and there-
fore, by (117), the general integral is
u= < {(x t) <o, (y t)di } (z t)(a} = 0.
Ex. 2. Determine the form of a function of n variables which
will satisfy the differential equation
du\ i du \ i du
dx\ dx<i dx n
x\ &2 mx n '
Xi #3 #n-l X n
' T, = C2 ' T,~~ c *>~ ~^T Cn ~ l > ~x^ :
therefore the general integral is
, *,1 5tl, -5L) = 0; (119)
Offi Xi Xi m '
and if a?i, x z , ...#,,_ i are independent variables, and x n is de-
pendent, it may be expressed
the right-hand member of which is an homogeneous function
of (n 1) variables and of m dimensions : the above is manifestly
a proof of the converse of Euler's theorem.
SECTION 6. On integrating factors of differential equations of
the first order and degree.
285.] "We return to total differential equations, with the
object of investigating the conditions, subject to which differen-
3 B 2
372 THE DETERMINATION OF [286.
tial expressions of two or more variables, which do not in them-
selves satisfy the criteria of integrability, may yet be made to
do so by means of an integrating factor; and first we shall
consider an expression of two variables of the form
rdx + Gtdy = 0; (121)
where p and Q are functions of x and y ; and we shall shew
that there is always a factor p, which is generally a function of
x and y, which will render (121) an exact differential of u, so
that p. (P d#+ Q dy) = D u = 0. (122)
For suppose the general integral of (121) to be put into the
f rm f(x,y) = c; (123)
where c is an arbitrary constant : then we have, by differentia-
tion,
(124)
'
dy
and from (121) we have
therefore, equating (124) and (125), we have
dy
dii
to both sides let us add ~, and then reducing, we have
CLOC
(126)
but the left-hand member of this equation is an exact differen-
tial ; such therefore is the right-hand member ; and thence we
1 J '/*
conclude, that any factor p, which is equal to - ( ) , will render
(121) an exact differential.
286.] We may also prove that not only is there an integrating
factor, but also that the number of them is infinite.
287.] AN INTEGRATING FACTOR. 373
"I >J-/*
For let us represent by jx, the quantity - \-jt-) , then, if u
is the general integral,
/* (p dx -f Q dy} = DW, (127)
and therefore if both sides are multiplied by < (u), we have
<f)(u) DM = p.<f)(u) {fdx + Qdy} ; (128)
and therefore, as <j>(u)i>u, is a simple integral, it follows that the
right-hand side of (128) also admits of integration ; and as <f)(u)
is an arbitrary function of u, it follows that there is an infinite
number of factors which will render (121) an exact differential.
287.] Suppose /z to be one of the integrating factors ; then
p,v dx + JJ.Q. dy = = DU (129)
is an exact differential ; and therefore by (28), Art. 265,
(130,
which is a partial differential equation of the first order and
degree in ju, and is therefore to be integrated by the methods
of the last Section.
Of this equation let the general integral be
v = $(x,y,ij.) = 0, (132)
or in the explicit form, p. = <i> (#, y} then (131) becomes
= 0, (133)
and therefore by virtue of the hypothesis made in (84)
dx _ dy dfj,
Q p ( , ,/-,, , J^ * "\ * ^ /
by the integration of which equations p must be expressed as
an arbitrary function of x and y. I know of no method of find-
ing the integrals of the general form (134), although, as we shall
shew, there are many cases which admit of integration.
We may also infer, as a corollary, that since the integrals of
(134) involve an arbitrary function, there is an infinite number
of factors which will render (121) an exact differential; the theo-
rem which was proved in the last article.
374 THE INTEGRATING FACTOR OF [288.
For suppose that the criterion of iutegrability is satisfied, then
(134) give
= 0,
of which let the integral be u = e 2 ,
.'. /A = ci
where F' expresses an arbitrary function, and which is supposed
to be the derived of F ; then, since
u,
and contains therefore an arbitrary function.
Suppose that /x is a function of x only, so that {-J- j = 0; then
from (131)
dp
du, ,
.-. ? - dx; (135)
/* Q
and as the left-hand member is a function of x only, such must
also the right-hand member be ; and this condition may be
satisfied in two ways ; either
(1) Q may contain x only, and P may be of the form
*i y + *2, where x x and x 2 are functions of x only ; or
(2) The functions of y which enter into (-r-) ( ) and Q
\dy' \dx>
must be the same in both, so that they may divide out in
(135). And thus finally
-!-
ffjj
- dx. (136)
Q
A similar result is of course true if ^ is a function of y only.
288.]| It is required to determine an integrating factor of
p dx + Q dy 0, where p and Q are homogeneous functions of x
and y of n dimensions.
289-] HOMOGENEOUS EQUATIONS. 375
From (134) we have
dx_ _ dy _ dp _ _ ydx-xdy
dv :
the last member following from operating on the first two mem-
bers of the equality ;
dfj,
/A vx + Qy
Now multiplying out the numerator, and replacing y (-T-) and
c?o\
r-) in terms of their equivalents deduced from the equations
fi,y* /
C?P>
) + yl^-l = nv I
\ //7/ /
(138)
which express Euler's Theorems of homogeneous functions, we
have
and therefore, since vdx + o,dy = Q, writing d? and do, for the
total differentials of P and Q, (139) becomes
d\i.
P.T + Q?/
(140)
and therefore - - is an exact differential: and we may write
= p
hereby then can u be determined, and an integral be found.
Let these results be compared with Articles 275 and 276.
376 THE INTEGRATING FACTOR OF [289.
Next let both members of (141) be multiplied by v'(u), and
) du = * ; (142)
and therefore if one integral of (141) can be found, an infinite
number may also be determined.
289.] As an example let us consider the equation
ydxxdy Q; (143)
the condition of integrability (28) is manifestly not satisfied :
in this case the formula (141) fails, because p<z > + Q^ = 0; and
the reason is obvious : (139) is in the indeterminate form ^ 7
because the numerator of (137) = : let us therefore return to
first principles. Suppose ft to be the integrating factor of (143),
yuK dy = = DU,
dx _ dy dfj.
x " y ' 2fj.'
where f represents any arbitrary function, and is the derived
of/; whereby (143) becomes
ydx xdy ix_\
* ' =
'i (144)
and therefore the general integral of (143) involves not only an
arbitrary constant c, but also an arbitrary function; and to it
any form may be assigned : thus all the following quantities
satisfy the equation (143) :
i ( x \ \ x - n x
w=log(-), =sm~ 1 -. =sm2-. ...... ;
V y y
(143) might also have been solved as a partial differential equa-
tion : for supposing its integral to be of the form
i = F(a7,y) = c, (145)
290.] HOMOGENEOUS EQUATIONS. 377
then
from which and from (143), equating the values of ^ , we have
of which the integral is
u = F(-) = c. (147)
Ex. 2. It is required to determine the most general form of
the integral of , , , 2 , , n /tAQ\
Axy aa?+(y* x* 1 ) dy = 0; (148)
__ /y>2\ /y-j/
x J^y = DM = 0, (149)
Zxydx
U x =
f y z -
<z? w
.-. = log ii- c,
y
where c is an arbitrary constant.
Therefore by reason of .(142) the most general integral of
(148) is -
(150)
The same result might also have been arrived at if we had
transformed (148) into a partial differential equation, according
to the method of the last example.
In the same way will the most general integral of an exact
total differential always involve an arbitrary function.
290.] It is required to determine the integrating factor of the
linear differential equation of the first order which is of the form
(y/(a?) -*(#)} dx + dy = 0, (151)
see equation (64), Art. 278.
Let p. be the factor; therefore
PRICE, VOL. II. 3 C
378 THE INTEGRATING FACTOR OF [290.
.'. - f(x)dx,
r"
*, (155)
which is a particular integral of (153), because p is a function
of x only, and therefore (-J-J = 0: a more general integral we
tS
are at present unable to find : multiplying (151) by this value
of j. we have
0, (156)
x = f*//W dx { yf(x) F (#) } c?^
= y eSf (x}d * I ef f(X}dx v(x) dx,
y = \eff^ dx dy
= y eff (x]dx ,
u = yeff^ dx \eff( x)dx v(x'}dx = c, (157)
where c is an arbitrary constant.
This function has been found by means of only a particular
value of the integrating factor ; there are therefore many other
integrals; and the above equation will enable us to determine
them.
From (154) we have
(y/(#) *(#)} dx + dy = 0,
and therefore if c% be the second arbitrary constant which is in-
troduced in the integrals of (154)
c 2 = yeffw<te_ l e
therefore from (155)
where <f>' is the symbol of the functional relation c^ = 3>'(c z ), and
LINEAR DIFFERENTIAL EQUATIONS. 379
is an arbitrary symbol, although supposed to be the derived
of <. And applying this most general value of p,, we have as
the general integral of (151)
e//<*)** le^^ dx Y(x')dx\ = c. (159)
u =
291.] It appears then that the equation (151) may be inte-
grated when multiplied through by e/JK**: this is also mani-
fest as follows : , /., x , , N -,
ay + yf(x) ax F (so) ax,
... e ffw d xdy + yeffW dx f(x)dx = efsw dx T?(x)dx. (160)
Now the left-hand member is evidently an exact differential,
and by integration we have
y e fA*)d* - \ e ff(x)dx P (,p) dx + C} ( 161 )
which result; it is to be observed, is the same as that of Art. 278.
Ex. 1. dy -\-ydx = ax n dx.
f(x) = 1, /. jf(x)dx = x;
.-. fj. = e x ,
e x dy + ye x dx = ae x x n dx\
.-. ye x = al e x x n dx
l)... 3.2.1 }e*
.-. y = a{x n -nx n - l + ...( ) n n(n-l) ...3.2.1} + ce~*,
where c is an arbitrary constant.
n adx
Ex. 2.
.-. ff(x)dx =
fv
^)*}n = c;
3 c 2
380 INTEGKATING FACTORS. [292.
and therefore the most general integral is
u = 3>\y- -{x + (I+x 2 )*}"\ = c. (162)
292.] We have thus shewn that homogeneous, and linear
equations of the first order can be rendered exact differentials
by means of a multiplier, and that hereby the integrals can be
put in a more general form than our previous processes author-
ised : we proceed to determine the integrating factors of a few
particular equations.
Ex.1. a(xdy-\-%ydx) = xydy.
Zaydx + x(ay}dy = 0.
Let fj. be the integrating factor :
dx dy dp.
og p. =
whence of course the most general value of p. may be determined;
but as it takes a complicated form, let us suppose that the rela-
tion between c\ and c 4 is such that c\ c 2 = : then
1
n* = ;
any
and we have ,
2aydx + x(a-y)dy
- z= DM ^ wj
xy
.-. u x = 2log.r,
u y alogy -y;
.. u = 2 log # + logy y = c. (163)
Ex. 2. ydx + (ax*y n 2x)dy = 0.
293-] INTEGRATING FACTORS. 381
dx dy d/j.
dy_ _
= i (164)
3 2axy n
d(M dy 2 dx
fjL y x
log/x = logy log x* + log <?i;
.* . M )
and this is a particular value of the integrating factor ; using
which, the given differential equation becomes
~ -^2 = DM = 0; (165)
ai/n+2 j.2
y y fi ca\
u = = c ', (lot))
n + 2 x
and as this integral is that of the first two members of the
equality (164), we have
ay n+2 y 2
, o ~ = C 2 5
and therefore the general integral is
2
ft ( 167)
293.] We proceed now to a differential expression of three
independent variables, of the form
vdx + qdy + ndz = 0; (168)
where P, Q, n are functions of x, y, and z ; and suppose p to be
a factor, by which, when multiplied, it becomes an exact differen-
tial of a function .,, cos
M = v(x,y,z) = c, (169)
and thus to become
ekr = 0; (170)
IMTEGRATING FACTORS. [293.
where p generally is a function of all the variables ; then the
conditions of its being an exact differential are, see equations (41),
* +
-*(!)+ * +'()-{(=)-(=)}
* -{-(=)}
(172)
multiplying the first of which by p, the second by Q, and the
third by R, we have
which condition must be satisfied, in order that (168) may
admit of being made an exact differential by means of a multi-
plier : we shall return hereafter to the meaning of the necessity
of this condition.
Now it is manifest that the three equations (172) are equiva-
lent to any two of them together with (173) ; and if of these,
three integrals involving three arbitrary constants can be found,
the most general integrating factor may be determined : if how-
ever we can integrate only one or only two, we may use the
resulting expression as an integrating factor, although it may
not be the most general.
Also from (172) in many cases, by various combinations, may
other forms of differential expressions be found, whereby inte-
grating factors may be determined. Thus one form may be
obtained in the following manner : multiply the second of the
group (172) by dz, and the third by dy, and then subtract the
third from the second, and we have
* +()*->}
294-1 INTEGRATING FACTORS. 383
and therefore by (168)
similarly,
and the general form of the integrating multiplier will be de-
termined by the equation
*(ci,c a ,c s ) = 0; (179)
where <I> expresses an arbitrary function, and c\, c z , 3 are to be
expressed by their equivalents determined as above. The most
general form of the multiplier of course requires the integrals
of all three equations ; I know of no method of finding the
integrals of all in their above general forms ; in many cases
however, as the following examples shew, the integration is
possible.
294.] Ex.1. zydx-zxdy + y z dz = 0. (180)
(173) in this case is yz(x + 2y)+a?yz + 2y*z, which is equal
to 0, and therefore the condition is satisfied; and from (176)
= log- +logci,
y
which gives us a particular value of /A. And multiplying
zy dx zx dy + y z dz _ _
y z z
u = -+logz = c. (182)
y
384 INTEGRATING FACTORS. [294.
Again, from (178),
X
= -
.-. p = ^e; (183)
~ "
and multiplying (180) by this, and integrating, we have
X
u = zei> + c', (184)
and therefore either this or (182) is an integral of the given
equation ; and therefore the general integral is
u = v (z e~y) = 0. (185)
We can also in this case by means of (181) and (183) find the
value of fj. which is deducible from (177), and therefore deter-
mine the most general value of /x: but the process is rather
long, and leads to no useful result.
Ex.2. (bz cy)dx + (cxaz)dy + (ay bx)dz = Q; (186)
(173) in this case becomes
which is equal to 0, and therefore the condition is satisfied.
The equations for determining p become
fj,(bz cy) 2 = ci -I
n(cx az) 2 = c 2 L , (187)
and therefore any value of p, which will satisfy the equation
* {n(bz-cy)\ \i.(cx-azf, n(ay-bx) 2 } = (188)
may be used as a multiplier to render (186) an exact differential.
Let us however take one of its particular forms, say the first
of the group (187), and we have
(bz cy) dx + (ex az) dy + (ay bx) dz
(bz-cy)*
= DM = 0, (189)
... .. = o, (190)
bz cy
and by taking the other values of p. we might obtain other,
although equivalent, values of u; and thus the most general
form of the integral is
M /-) = Q
294-] INTEGRATING FACTORS. 385
Ex. 3. (y 2 + yz+ z 2 ) dx + (z 2 + zx + x 2 ) dy + (x 2 +xy+y 2 )dz = 0. (192)
The condition (173) is satisfied; and to determine p we must
have recourse to first principles :
d d
* . / fyu i w I 'y2\ it ( / >>2 i wti I ?/2\
7 * r V ~i '***' \~ / ^"^ 7 r" \*^ ~T~ ^ 7 "l j /I
dz dy
\dy' \dz/
whence we have
dx dy dz dp
dx + dy + dz dx + dy + dz
~ yi + xyxzz* ~ (yz)
dx + dy + dz
x + y+z
log =
I*
and multiplying (192) by this we have
(y 2 + yz + z 2 ) dx + (z 2 + zx + x 2 ) dy + (x 2 + xy + y 2 ) dz
~~(x + y + z) 2
(194)
(x + y + z) 2
y 2 + yz+z 2
x + y + z
yz + zx + xy
/y> I nj _l_ # *^
cc/ ~|~ (J J~ &
Uy = - - - - Z + X,
u z = - - - - x + y,
x + y + z
and thus the general integral becomes
(196)
(197)
x+y+z '
the arbitrary functional symbol F including the arbitrary con-
stant of integration.
PRICE, VOL. n. 3 D
386 INTEGRATING FACTORS. [ 2 95-
295.] Equations (172) also admit of combination into a more
simple form when r, Q, R are homogeneous and of n dimensions :
for multiplying the second of (172) by z, and the third by y, and
subtracting, we have
dv\ (dp
+
whence we have
Similarly,
and therefore multiplying by dx, dy, dz, and adding,
= 0, (200)
, (201)
where c is an arbitrary constant: we subjoin an example in
which the method is applied.
It is required to integrate the partial differential equation
() +*()-" <*>
or, which is equivalent,
idu\ tdu fdu
y I T- ) + * 1 --
y \dx> \
Hence by (84) we have
295-1 INTEGRATING FACTORS. 387
dx _ dy _ dz
y " z " x
y(zx)+z(xy)+x(yz)
2 yz) dx + (y 2 zx) dy + ( a xy ) dz
and as the denominators of these last equations are equal to
zero, the numerators must also vanish : and therefore
(x 2 yz} dx + (y z zx) dy + (z z xy) dz = 0, (204)
.-. * 3 + y * + z3 -xyz= Cl ., (205)
(zx)dx + (xy)dy + (yz)dz = 0, (206)
and this expression satisfies the condition (173); and as the
equation is homogeneous we have by means of (201)
(z-x) dx+(x-y) dy + (y-z) dz _ _
' --- -- "---- - - ___ \J ( __ \J (*VV// I
(x + y
and by integration
1 e Ti , 1
u x - loz + zz + x* + tan- 1
3* 8*(y )
1 , . 2y z x
--- + ptan- 1 -^ - ,
t
u z = ..... --I - tan-
3*
and as the difference between the circular functions contained
in u x , u y , u z respectively is a constant, it follows that either one
is an integral of (206), and that therefore another particular
integral of the equation is
1 , , 2xyz
tan- 1 = - = c 2 ;
and therefore the general integral of the given differential
equation is
log (yz + zx + xyY H r tan~ l -^ - - = F(<r 3 + y 3 + ^ 3 xyz}, (208)
where F is the symbol of an arbitrary function.
I may by the way observe that the solution of homogeneous
equations is often facilitated by a substitution similar to that
3 D 2
388 INTEGRATING FACTORS. [296.
made in Art. 272. Thus we may integrate Ex. 3 in Art. 294 by
assuming
* = fa y = VZ,
in which case the equation (192) becomes
dz fra-Hy -
296.] It is manifest from the examples worked in the last
Articles, that the difficulty of determining the integrating factor
is the chief obstacle, and is in most cases insurmountable : there
is however another mode of solution less direct than that given
above, but of which it is desirable to give a brief description,
because it is the only one which has hitherto been generally
applied.
Since the differential equation pdx + o,dy + ndz = is a
function of three variables, we may consider one of them to be
dependent, and the other two to be independent ; let the inde-
pendent ones be x and y, so that the integral is assumed to be
of the form z = f(x, y} ; now we may consider x and y to vary
separately, and therefore z to vary owing to the variation of x
or of y, when the other does not vary : suppose that in the dif-
ferential equation we consider y to be constant, and therefore
the variation of z to be partial and to be due to that of x : in
which case the equation becomes
?dx + K.dz 0; (209)
let p. be an integrating factor of this equation when y is con-
sidered constant : and let us suppose
(p dx + R dz) = F (x, y, z) ',
then in the integration of this equation, since y has been con-
sidered to be constant, a function of y must be introduced, and
therefore if Y represents an arbitrary function of y, the integral is
F (x, y, z) - Y : (210)
now (210) is manifestly such as to give the right value of \-j-j
in (209) : it remains for us to determine Y so that it shall give
idz\
the right value of ( -7- ) : and it is also evident that if (210)
satisfies these conditions it is an integral of the given differential
296.] INTEGRATING FACTORS. 389
equation. Of (210) let the total differential be taken, and be
. , 7 7
)dx + (^-}dy + ( -j-jdz = ~r dy ;
dx> \dyl \dz> dy '
but \-j-jdx + ( Jdz = [j, {v da? + n dz] ,
y + c; (211)
whence may Y be determined.
But in order that Y, as assumed in (210), should be a func-
tion of y only, it is necessary that (-*-) uo. should be inde-
If
pendent of x and z : and if this is true, the x- and ^-differentials
of it vanish ; and therefore
d ( /</p>
(212)
=
(213)
_ (W) o
but since UP = (-r-K and UR (-7-), (214)
= P\.} + *(^r, ,
(215)
\ f
substituting which in (213) we have
dy ' 1>; (216)
and from (214) we have
(/rfp\ /?Q\K
^M/T\r
390 GEOMETRICAL INTERPRETATION OF [ 2 97
and therefore we have
which is the condition of integrability before arrived at, Art.
293 : and therefore, if this condition is satisfied, the method of
integration may be employed.
Ex. 1 . (yz + xyz) dx + (zx + xyz) dy + (xy + xyz) dz = ; (218)
the condition (173) is satisfied : let y be constant ; then (218) be-
comes
(z -f xz) ax + (x + xz) dz = 0;
\+x , 1 + z ,
.-. L dx + ^dz = 0.
X Z
u logxz + x + z + Y = 0;
x
DM =
x
from (218):
Y = logy + y +
u =
297.] I propose now to return to the consideration of the
expression (173), which contains the condition that (168) should
be integrable by means of a multiplier ; and to give clearness
to our notions, I shall consider it from a geometrical point of
view, and especially with reference to certain properties of sur-
faces connected with curvature.
Let it be observed that the equation
vdx + Qdy + ndz = 0, (220)
expresses the condition that the line (P, Q, R) is perpendicular to
the line (dx, dy, dz), that is, to the line joining two consecutive
points ; but P, Q, K are generally functions of x, y, z, and there-
fore vary as we pass from point to point ; if therefore (220) ex-
presses the property of a surface, that surface cuts orthogonally
the system of straight lines whose direction-cosines are propor-
tional to P, Q, R : it is of course easily conceivable that straight
lines (P, Q, R) may be drawn at random, and that there cannot
298.] CONDITION OF INTEGRABILITY. 391
possibly be any surface which cuts them orthogonally, or, in
other words, that (220) cannot express a property of a surface,
as we shall see in a subsequent article ; but we shall now take
certain general properties of surfaces, and prove that the exist-
ence of these necessitates the condition (173).
For this purpose I shall take the theorems of Monge and of
M. J. Bertrand; the former of which is proved in Article 348,
Vol. I ; and the latter is given in Liouville's Journal, Vol. IX,
p. 133, and of which the enuntiations are,
(1) At every point on a surface there are two directions per-
pendicular to each other, along which the first two consecutive
normals intersect.
(2) If at any point P on the surface the normal PG is drawn,
and two lines P Q and P R of equal infinitesimal length are drawn
on the surface, and perpendicular to each other, the normal at
the point Q shall make the same angle with the plane P G Q that
the normal at the point R makes with PGR ;
Or in other words, " The radii of torsion of two geodesic lines
on a surface intersecting at right-angles are equal at their point
of intersection."
And I shall consider these in order, and thereby obtain the
meaning of the condition (173).
298."] First then as to the theorem of Monge : suppose that
the line (P, Q, R) is a normal to a surface ; and that its equations
are ..
where P, Q, R are functions of x, y, z, and vary for each point of
space ; and where x, y, z are the coordinates to a point on the
surface.
Then the condition that (221) should intersect its consecu-
tive line is
rdx,} -\-dz{?do. QC?P} = 0; (222)
and let it be observed that (222) when developed involves dx,
dy, dz in quadratic forms, and as these quantities are the same
in (220) and (222), the combination will give rise to two values
of each of the ratios dx : dy : dz; and thus it follows that there
are two directions along which the adjacent intersecting lines of
(221) cut the plane (220) which is perpendicular to the line (221).
392 GEOMETRICAL INTERPRETATION OF [298.
Now by Mongers theorem these directions are to be at right-
angles to each other : let us therefore develope (222), and in it
and (220), for convenience of notation, substitute , rj, C for dx,
du, dz : also let 7 ,
Q() -*()-*, (223)
\dx> \dx>
fdp\ fdv\ /dq\ /dn\
"U -"(35) + p y-"U = D ' (224)
and let B, c, E, F be symbols for the symmetrical quantities ; then
we have = 0, (225)
R = 0; (226)
let 1 77! d, ^2 >?2 ^2 De the values of rj corresponding to the two
lines of intersection ; then
aCi = 0,
= 0,
-_; (227)
But from (225) and (226) we have
_,,!Zl _cf ElS2 + D,f+, + ,f, = o,
P Q R
{QRD BR 2 CQ 2 } -f Q (RPE cp 2 AR 2 }
+ TJR{PQF AQ 2 BP 2 } = 0; (228)
or, making obvious substitutions,
= 0; (229)
N f" = 1], (230,
1 N C2 ??2 == " J
TT, (231)
and by reason of (227) and because
^fc + ^a + M = 0, (232)
i+.f + i..4>, (233,
and replacing the quantities for which substitutions have been
made, we have
(P 2 + Q 2 + R 2 ) (A + B-f C) AP 2 BQ 2 CR 2 QRD RPE PQF = 0;
and therefore
A 4- B + c = 0,
299-1 CONDITION OF INTEGRABILITY. 393
which is the general condition : and therefore implies that if
(220) expresses a property of a surface, it must satisfy the
equation (234).
299.] Next let us consider the Theorem of M. Bertrand ; of
which however it is desirable to insert a proof, as such does not
at present occur in any of the ordinary text books.
Let the equation to the surface be
and let u, v, w, Q be employed as symbols in the same meaning
as in Art. 346, Vol. I ; let (#, y, z} be the point under considera-
tion, at which let the normal PG be drawn : then u, v, w are pro-
portional to its direction-cosines ; let PQ PR = da be the equal
infinitesimal lengths on the surface, originating at p and per-
pendicular to each other: and let l\m\n\, l^m^n^ be the direc-
tion^cosines of these lengths ; so that
and therefore the ^-direction cosine of the normal at Q is
u , ( d u d u d u
Q ( 1 dx Q 1 dy Q 1 dz Q
which after reduction becomes
u , dcr S i l dv \ , . ( dv \ ,
- -- + m l -
Q Q ( \dx> \d>
and the other direction-cosines are
v dv (, /dv\ /dv\ id\
Q (
w
/OOQ ,
' (338)
Let 0! be the angle between the normal at Q and the plane
PRICE, VOL. ii. 3 E
394 GEOMETRICAL INTERPRETATION OF [300.
PGQ; then, because / 2 , m^, n 2 are the direction-cosines of PK
which is normal to the plane PGQ, we have, omitting the terms
which vanish,
sin <fr = 1 2 (236) + m 2 (237) + n 2 (238)
da- ( dv\ /dv
dv . /dv
Again, let < 2 be the angle between the normal at R and the
plane PGR; then the value of sin^ is of the same form as that
of sin</>i, and the positions of l^m^ni, and of I 2 m 2 n 2 are inter-
changed : but since u, v, and w refer to the point p, and since
_
&'-~ \fatl'
this transposition will not change the value of (239), and there-
fore 0! = fa ; and the proposition is proved.
I may, by the way, observe that as sinc^ and sin</> 2 are of
the same sign, the normals at the points Q and R lie either both
towards, or both away from, the angle QPR; and therefore by
the principle of continuity there is some direction intermediate
to those of PQ and PR, where the angle (f> will vanish, and the
normal corresponding to which will be in the plane containing
PG, and will therefore meet PG; that is, for that direction two
consecutive normals will intersect : and by the theorem just
proved there is of course another direction perpendicular to this
last, of which the same property holds good : these directions
are manifestly those of the lines of curvature; and thus the
Theorem of Monge is only a particular case of the more general
Theorem of M. J. Bertrand.
300.] And I have also one other property of surfaces to
notice : Suppose the origin to be transferred to be the point p,
and the axis of z to coincide with the normal PG, and suppose
Q and R to be on the axes of x and y respectively : then in (239)
u = 01 /! = n / 2 = 01
v=ol m 1 = ol W2=lL (240)
- = 1 I % = [ rh = |
Q J
301.] CONDITION OF INTEGRABILITY. 395
and therefore
(241)
Q
and therefore by Bertrand's Theorem
) ; (242)
that is, If at any point on a surface taken as origin the axes of
x and y are drawn in the tangent plane, and normals to the sur-
face are drawn at points on the axes of x and y at equal infi-
nitesimal distances from the origin, the angle which the normal
at the point on the axis of x makes with the axis of y is equal
to that which the normal at the point on the axis of y makes
with the axis of x; and these angles become right when the axes
are the lines of curvature. Hereby then we have arrived at a
geometrical interpretation of the analytical proposition
/ d 2 u \ i d 2 u \
( ) = ( -I. (243)
\dxdy' \dydx''
301.] And the Theorem of Bertrand yields an easy interpre-
tation of the condition that (220) may be made an exact dif-
ferential by means of a multiplier. If (220) expresses a property
of a surface, then that surface cuts orthogonally all straight
lines whose equations are
^ = ^^ = ^. (244)
P Q R
where P, Q, R are functions of x, y, z ; and therefore the direction
of the line varies from point to point. But if the equations
(244) are those to a normal of a surface, then, by the method of
the last Article, and in accordance with the same notation, if
p 2 -f Q 2 + R 2 = s 2 , (245)
da-
da-
3 E 2
396 CONDITION OF INTEGRABILITY
dv ( , , /dp
dcr
i
s
and by Bertrand's Theorem, if (244) are the equations to a
normal of a surface <pi = </> 2 ; therefore equating (246) and (247)
- +
-4".) { () - (5
but since the lines (l i} mi, HI), (1 2 , ra 2 , n^) are perpendicular to
(p, Q, R), we have
R (249)
whereby (248) becomes
-' (2o0)
which is the condition of integrability ; and therefore we infer
that if (220) does not satisfy this condition, it does not express
the property of a surface, and that its integral cannot be of the
form __ , .
Also if the origin and the coordinate axes are those explained
in Article (300), then
da- (do.\
J d f >' < 251 )
da- /dp\ '
sm< 2 = l-y-l
s \ nu I
and therefore
and if the axes be transformed into another system of rectan-
gular coordinates, it is easily shewn that (252) becomes
where the accented letters express in reference to the new system
similar values to those of the unaccented letters in the old system.
302.] DIFFERENTIAL EQUATIONS OF THREE VARIABLES. 397
302.] To resume the analytical investigation : it appears that
an equation of the form ?dx-\- o.dy = can always be rendered
an exact differential by means of a multiplier, and that its inte-
gral involves an arbitrary functional symbol. Also it appears
that -pdz + Qdy + ndz = is not always capable of being made
an exact differential by means of a multiplier, and can be made
so only when the condition (173) is satisfied.
Suppose however that (173) is not satisfied; but that on in-
spection we can separate vdx + o.dy + T>idz into two parts, which
are respectively exact differentials multiplied by factors, so that
it becomes
/o-^\
= 0; (2o4)
and therefore is satisfied by
u = c, Ui = Ci; (255)
then u and U! are so related that one is constant when the other
is: and therefore
,OK*\
(256)
the . form of <p being at present undetermined : but from
'u; (257)
substituting which in (254), we have
M + / x 1 4/(u) = 0; (258)
which equations are sufficient for determining the form of < :
and the result (255) becomes
u = c, ux = <f)(c); (259)
each of which is the equation to a surface ; and the two when
taken simultaneously, as is necessary in this case, express the
curve of intersection of the two surfaces: the differential equation
therefore expresses a property of a curve and not of a surface.
Or again suppose that we cannot by inspection separate the
differential expression into two parts of the form (254) ; yet
by the following process we can shew that it expresses a pro-
perty of a curve and not of a surface, that is, if (x, y, z) be a
point on a surface, it is possible to draw through the point and
on the surface an infinite number of lines, the consecutive points
of which shall satisfy the differential equation, although the
equation to the surface does not.
For suppose the equation to the surface to be ?i(oe } y,z) = c\ }
whence we have , , 7 /or\
= 0; (260)
398 DIFFERENTIAL EQUATIONS [302.
then multiplying this last by v, and adding it to the given dif-
ferential equation,
(p + vu)efo? + (Q-f j>v)flfy + (a-f rw)efo = 0, (261)
and suppose v to be so determined that this shall satisfy the con-
dition (173) : and let the integral of (261) be 2 (x,y, z) = c 2 ;
then FI and r 2 taken together satisfy the differential equation ;
and therefore all the curves in which these two surfaces intersect
satisfy the equation : now F 2 will manifestly contain an arbitrary
function, and therefore there will be an infinite number of lines
of intersection ; although therefore no one surface satisfies the
conditions of the given equation, yet through any point on the
surface F! may an infinite number of lines be drawn along which
we may pass without violating the conditions, but we are unable
to pass from one line to another across the others.
Another way of considering the matter is this ; assume
y = 4> (a?), and substitute for y in the given differential equation,
whence we have
(p + a <'(*?) } &K + R dz = ; (262)
4>(x) having been substituted for y in p, o. and R. Suppose the
integral of (262) to be
F (x, z, c) = 0,
where c is an arbitrary constant : then the intersection of the
cylinders whose equations are y =/(#), and F(#,,C) = 0, satisfies
the requirements of the given differential equation.
Ex.1. zdx + xdy + ydz = 0. (263)
The condition (173) becomes in this case xyz, which is
not equal to 0.
Let y = </>(#), .'. dy = $'(x}dx:
and (263) becomes
= 0,
= 0;
the integral of which, and the equation y = $ (x), together satisfy
the differential equation. Thus suppose that y = x + c, then
we have
(x + c) dz -f z dx + x dx = 0,
d. (264)
303-] Or THREE VARIABLES. 399
Thus as y x + c expresses a plane perpendicular to that of
xy, and as (264) is the equation to a hyperbolic cylinder per-
pendicular to the plane of xz, and as each of these equations
involves an arbitrary constant, it follows that the series of lines
of intersections of these two surfaces satisfy the given differen-
tial equation.
I have said nothing as to the means of determining the inte-
grating factor of a differential expression of more than three
variables, because I am unwilling to enlarge the volume by in-
vestigations which are not necessary aids in our subsequent
applications of pure mathematics to physics.
SECTION 7. On singular solutions of differential equations of
the first order.
303.*] Thus far we have investigated general and particular
integrals of differential equations of the first order; but in some
cases there are functions of x and y which satisfy the given
equation, and yet cannot be deduced from the general integral
by any particular value of the arbitrary constant : such func-
tions are called singular solutions, as was noticed in Art. 262,
and we now proceed to investigate their properties and modes
of discovery; and as the inquiry is one of the most difficult in
this branch of our subject, our best course is to recur to first
principles of definite integration, and thus to state the question
and its conditions in the most elementary form.
Let the differential equation be exact, and be of the form
p dx + Q dy = DU 0; (265)
P
where p and Q are functions of x and y ; and let us replace
tyf( x ,y}, so that f ,
dy = f(x,y)dx; (266)
and let us suppose the integral to be definite, and the limiting
values of the variables to be XQ y Q) x n y n ; we shall find it conve-
nient in some cases to replace one or the other of these sets by
general symbols x y : also let us suppose an integral of (266) to
be y = p(*)j (267)
* For much of this Section I am under obligation to M. Cauchy, and to
M. Moigno : see Moigno's Calcul Integral, Lec.on XIX.
400 SINGULAR SOLUTIONS. [304.
.-. dy = v'(x)dx; (268)
then f(x} must be subject to these conditions,
(269)
F>) = /{*,*(*)}
and also a condition similar to the former at the superior limit.
Suppose the interval x n XQ to be divided into n infinitesimal
parts, and x\, x 2 ... x n _\ to correspond to the n 1 points of di-
vision ; and the corresponding values of y to be yi, t/ 2 ... ?/_i;
and f(X) y} to be finite and continuous for all these values : then
y y_i = f(x n -\, y n -i) (x n a? n _i) j
adding the right- and left-hand members severally of these
equations, the sum of the right-hand members is, by preliminary
Theorem III, Vol. I, the product of x n XQ and of some mean
value of the other factors, and therefore if 6 is a general symbol
for a positive and proper fraction,
y yo = (%n ffo)/{ffo+0(# #0), yo + 0(j/n y<>)}. (271)
Let us also express the greatest of the values of /(#<>, /o)>
f(x\> yi), /tow yn) by A, then
y n yo= (# - a?o) ^ A, . (272)
and therefore (271) becomes
yn yo = (^n ^o)/{^0+^(^n d? ), ^0 + ^A (^ n ^ )} (273)
Whence we know y n in terms of y$ ; and if for x n and y n are
written the general values of x and y, we have
y = y Q + (x X )f{x + 6(xx ), y Q + 0(xx Q )}. (274)
This Theorem is, it will be observed, a particular case of
(15), Art. 261.
It may be shewn, by a method similar to that employed in
Art. 6, that the truth of (274) does not depend on the particular
mode of division of the interval x XQ, provided that the parts
of it are infinitesimal.
304.] And there is another condition to which (274) must be
subject : we have supposed XQ to be a constant ; but as the dif-
ferential equation does not assign any values at either limit to
305-] SINGULAR SOLUTIONS. 401
the variables, y 0} although particular, must be arbitrary; and
therefore y and y must both be continuous variables, and one
may be considered to vary continuously with the other. Equa-
tion (274) which gives the relation between y and y Q must be
consistent with such continuous variation ; and this can only
be the case when the ^-differential of the coefficient of (x # )
on the right-hand side of (274) is not infinite; that is, when
d ~f(ffi 77 ^
, ' - does not become infinite for any value of the variables
between the limits.
Hence if x x = h, since y F (x},
} ; (275)
d f(x y}
and we infer, that if f(x,y} and ' J 3if are finite and con-
ay
tinuous for all values of the variables between x and X Q) there
is always a function of x, viz. F(#), capable of satisfying the
given differential equation, and of becoming a definite value,
viz. y = (a? ), when # = # - And as these are the conditions
required for a general integral, it is also proved that every dif-
ferential equation of the first order has an integral.
305.] And consistently with these conditions we can shew
that there is only .one general form of function which will satisfy
the given equation ; for suppose that y = F (x) is a general
form of function which satisfies the equation dy = f(x, y} dx ;
and that there is also another ; and suppose it to be of the form
y = F (a?) + *(*); (276)
then we have simultaneously
=f{x, F(#
(277)
} (278)
and as this is to be true for all values between the limits, it is
true when x = x , in which case 4>(^o) = 0, and 4>'(#o) does not
generally vanish ; and therefore we have
PRICE, VOL. ii. 3 F
402 SINGULAR SOLUTIONS. [306.
'(#) = x -7-
which is inconsistent with the given conditions. The general
integral therefore involves only one general form of function.
Of the truth of this theorem we have had instances in the last
Section.
306.] Let us suppose then y = F (x) to be a function of x
which satisfies the given differential equation dy =/(#, y) dx;
then if each of the functions f{x, F {%)}, -j-f(x, y) is finite and
ay
continuous for all values of x, or, at least, for all values of x
between certain assigned limits, we may take any one of these
for that which we have represented by XQ; and thus y = F(#)
will be a function of x which will satisfy all the conditions
stated in Art. 304, and therefore will be either the general or
a particular integral of the given differential equation.
But if, on the other hand, f{x, F(#)} or -j-f(x } y) is infinite
ay
or discontinuous or indeterminate for all values of x, then the
conditions necessary for a general or particular integral cannot
be fulfilled : the case of discontinuity we may at once discard
as beyond our province ; and f{x } v(x)} } which is equal to F'(#),
cannot be infinite for all values of x unless v'(x) is, and there-
fore unless F (#) is, and thus this is another case which we may
exclude : and therefore the cases which remain are
/{*,*(*)}=, (280)
!/<*>*> =
~/<*,jf) = co, (282)
and when y = F (x) is such as to satisfy the given differential
equation and at the same time to satisfy one or other of these
conditions, the integral is not general.
Yet such a value satisfies the differential equation, and is
therefore either a particular integral or a singular solution;
and to determine these it is necessary to investigate the rela-
306.] SINGULAR SOLUTIONS. 403
tions between x and y which will render f(x,y}, or -j-f(x,y),
<*>y
indeterminate, and which will render -r-/(^, y) oo ; if they
satisfy at the same time the given differential equation, they are
either singular solutions or particular integrals ; if the general
integral is known, there is no difficulty in determining whether
any particular constant value of the arbitrary constant will
reduce the general integral to the form y = v(x), but if only
the differential equation be given, we must apply the criterion
of Article 308.
The last of the above-mentioned conditions may be conve-
niently applied in the following way: let us use Lagrange's
notation of derived functions : then, since
/(.,) = . (283)
dy'
If therefore -p be found from the given differential equation,
y
and be equated to oo , and a functional relation between x and y
be thereby determined and of the form y = F (x), this is a sin-
gular solution if it satisfies the given differential equation. Of
this method of discovering singular solutions some examples
are added.
y dx^
dy , y
' := W =
fj tvt " O fyt
UUi Aili
. ay_ _ J_ ^ y
dy 2x~
if y 3 = 4m#;
and as this satisfies the given differential equation, it is a sin-
gular solution.
Ex.2.
due ~ "
404 SINGULAR SOLUTIONS. [307.
dy 1
___ " .
if y = 0, or if x = ;
and as either of these satisfies the given differential equation,
they are singular solutions.
Ex 3 =
the singular solution is (x 2 + y 2 a 2 ) = 0.
307.] If the differential equation be of the form
ft -- ft ( 284 >
then, considering at, y, and y' as three independent variables, we
have
(285)
and this is oo , if , /.
and -pj is not simultaneously equal to : and if we eliminate
y between (284) and (286), the resulting equation in terms of
x and y will be a singular solution, if it satisfies (284) ; but if
7 /
simultaneously (-J- j = 0, then the condition (284) is satisfied,
t7
and the result may be a particular integral or a singular solution.
x+ya
30 8.] SINGULAR SOLUTIONS. 405
substituting which in the given equation we have after reduction
,
i /
which satisfies the differential equation; and as \-f-} does not
vanish, it is a singular solution.
Ex. 2. y-xy'-^(y') = /(*, y, y') = ;
Therefore the elimination of y' between this and the given dif-
ferential equation will give the singular solution, if the result
satisfies the equation.
Ex. 3. x* + 2xyy'+(a*-^y'2 = Q=f(x,y,y'). (287)
between which and (287) eliminating y' we have
y* + x* = a 2 ;
j /i
and as \-r-} is not equal to 0, and as the above satisfies the given
differential equation, it is a singular solution.
Ex.4. y'* + yy' + x=f(x,y,y') = 0', (289)
V+y = o, ify'=-, (290)
between which and (289) eliminating y', we have
y 2 = 4#, (291)
but as this does not satisfy the given differential equation, it is
not a solution at all.
308.] We proceed to the investigation of a criterion for de-
termining whether y = F(#), which satisfies a given differential
equation dy =/(#, y) dx, is a singular solution or a particular
integral.
406 SINGULAR SOLUTIONS. [308.
In the first place be it observed that if the general integral
of the equation is known, we can thus determine whether
y = F (x) is a particular integral or a singular solution. If it be
a particular integral, the substitution of F (x) for y in the general
integral will yield a particular value for the arbitrary constant :
but if it be a singular solution, the arbitrary constant will be
equal to a function of the variables : to this subject we shall
return in the following Articles, and therefore I merely subjoin
an example.
Ex. 1. The general integral of the equation y\-]p} + 2#-^
* CL3C CltX/
y O is y 2 = 2ea? + c 2 ; other solutions are (1) y 2 = 2a?+l;
(2) y 2 + # 2 = 0; are they singular solutions, or particular in-
tegrals ?
(1) Comparing y 2 = 2e,r + c 2 and y z = 2# + l, it is manifest
that c = 1 ; the solution therefore is a particular integral.
(2) y 2 =
C "^ X y
and therefore (2) is a singular solution.
In the case however where the general integral is not known,
we can determine whether y = F (#) is a particular integral or a.
singular solution by an inversion of the process of Articles 306
dv f
and 307 ; that is, by inquiring whether ~ is not or is rendered
fff
infinite by the substitution of F(#) for y; or whether (-/-,}
dy
becomes zero by the same substitution. Of this process we
subjoin some examples, and shall first take that which has just
been considered.
Ex. 1. The equation yy' 2 + 2xy' y =/(%, y,y') = Q is satis-
fied by (1) y a = 2#-f 1; (2) by y a +a? 2 = 0; are they singular
solutions or particular integrals ?
which does not vanish for the relation (1), but does vanish for
(2): therefore (1) is a particular integral, and (2) is a singular
solution.
309.] SINGULAR SOLUTIONS. 407
Ex.2. The equation y' 2 + yy'+ x = is satisfied when
y 2 + (x I) 2 = 0; is this expression a singular solution or a
particular integral?
and as this does not vanish when y z + (x l) a = 0, this function
is a particular integral.
Through the preceding Articles we have considered the dif-
(J9/
ferential equation to be a function of x,y,-f-, and have deduced
CMP
our results on this supposition ; we might however just as well
ft y
have considered it to be a function of y, x } -=-, in which case
the conditions for a singular solution would be
* *? = oo, and -$L = 0; (292)
dx dy , dx
.-j
dy
and the resulting equation x F(y) must of course satisfy the
differential equation.
309.] Thus far we have investigated singular solutions with
reference to the differential equations of which they are solu-
tions ; we proceed now to deduce them from, and to point out
their properties in reference to, the general integrals of the dif-
ferential equations ; and herein we shall recur to the property
of them which was made characteristic at first, viz., that they
are particular forms of the general integral when the arbitrary
constant of integration is replaced by a function of the variables,
whereby the solution becomes a function of x and y only, and
also is such as to satisfy the differential equation.
Let us suppose the general integral of a differential equation
F(*,y,c) = 0, (293)
where c is an arbitrary constant introduced in integration ; then
the differential equation has been formed by the elimination of
c between (293) and
dt/
Now let us consider whether the same value of -/- can be
ax
obtained from an equation of the form (293), if we consider c
to be replaced by a function of x and y, say of the form </>(#, y),
408 SINGULAR SOLUTIONS. [309.
which we shall abbreviate into </> for convenience of notation ;
because if this is possible, the function hereby obtained is a
singular solution.
Suppose then the integral to be
* (*,y, 40 = 0, (295)
That the values of -j- deduced from (294) and from (296) should
be identical, it is necessary that
/C?F\ dx
\dy)
and this may be satisfied in three different ways :
(1) -p = 0, .-. (fr c; and we fall back on the general in-
tegral if c be general, and on a particular integral if c has a
specific value.
(// TO \
j- ) = 0, and if we eliminate <b between this and (295),
U(f>'
or, what amounts to the same thing, if we eliminate c between
p(a?,y.c) = 0, and (-?-) = 0, (298)
\ J ^ / / 7 \ _/ A I 3 \ /
the resulting equation will be a relation between x and y which
will satisfy the given differential equation, and will not be a
particular integral, unless c should chance to be equal to a par-
ticular constant previously involved in the differential equation.
Before we proceed to examine the third case, I will give two
or three examples of this second condition, and also consider it
from another point of view.
310.] Ex. 1. The general integral of a differential equation
is y = c(x + c) 2 ; it is required to find the singular solution.
=
= 0,
C = X,
3II-] SINGULAR SOLUTIONS. 409
of which values the former makes y = 0, and as the same result
is obtained if c == 0, it gives a particular integral.
The second value gives
4
which is the singular solution.
Ex. 2. The general integral of a differential equation is
c^x cy + a = ; it is required to find the singular solution.
v(as,y,c) = c 2 xcy + a = 0,
whence y 2 =. 4<ax;
and as no particular value of the constant can give this equa-
tion, it is a singular solution.
Ex. 3. The general integral of a differential equation is
y ex (A 2 + a 2 c 2 )* = 0; it is required to find the singular so-
lution. f , /z.9 , 9 2\4 rv
v(x,y,c) = y ex (6 2 + fl 2 <? 2 ) 2 = 0,
bx
a (a 2 -a? 2 )*'
x* y 2
whence we have ~ + ^ = 1,
o
a
and which is the singular solution.
311.] The geometrical relation of the curves represented by
the general integral and by the singular solution is worth no-
ticing. The singular solution manifestly represents the envelope
of all those expressed by the general integral when the constant
is made to vary : this is evident by the process of determining
such envelopes, which is explained in Vol. I, Chap. XIII, Sect. 2,
and which is identical with that for determining the singular
solution, when the general integral is known.
Let us consider this by the light of a particular example.
Suppose that we have a differential equation
PRICE, VOL. ii. 3 G
410 SINGULAR SOLUTIONS. [3 1 2.
()'-+=<" , <*">
the general integral of this is, as will be explained in the fol-
lowing section,
c 2 x cy + a = 0, (300)
y = cx+-, (301)
which is manifestly the equation to a straight line; and to a
series of straight lines, if c be considered a variable parameter ;
and the envelope of all these is the singular solution, and is a
parabola whose latus rectum is 4 a, as appears from Ex. 2 of the
preceding Article ; it will be at once seen that (301) is the
equation to the tangent of a parabola in what is sometimes
called the magical form.
If (299) and (300) are compared, it will be seen that -j- in
the first is replaced in the second by c : the c-differentiation
therefore of (300) produces the same result in terms of c as the
-^-differentiation of (299) produces for -~. Hence this method
\AJuU tttXs
of deducing the singular solution is, in this form of equation at
least, the same as that investigated and applied in Art. 307 : we
have not therefore hereby obtained any more general method.
312.] Let us now consider the third condition by which
(fj~\? \
j = oo ; and
this requires further and closer consideration. And to put the
conditions in a more general form, let us consider that, not
~ only, but also -T-, is to have the same value, whether it be
dx dy
deduced from (294) or (296), in which case we shall find, by a
process similar to that of Art. 309, that (-, - J = oo ; and there-
fore as it is immaterial in a question of identity whether we con-
sider a quantity or its reciprocal, we must have simultaneously
(*)-,. (*)-.,
and these conditions, we may by the way observe, are identical
with (280).
And in reference to this let it be noticed, that it is incon-
312.] SINGULAR SOLUTIONS. 411
sistent with the very first principles of differentiation that the
derived-functions should have infinite values : if they have, the
rules according to which they have been found fail. Now in
differentiating a function of one variable only, say of x, it may
be that its derived-function becomes infinite for a particular
constant value of the variable : thus, for instance, if y (x 2 a 2 )*,
y' = oo , if x = + a ; but in a function of two variables, as, for
. , , o , 2 0.4 n (du\ (du\ . f
instance, u = (x* 1 + y a l y = 0> ( -7- ) = and I -r- 1 = oo . it
' \dx> \dyi
x 2 + y 2 = a z : that is, the total differential of u is infinite for this
relation between x and y. Here then we have met with a case
which is beside the common rules of differentiation : in reference
to the singular solutions it is important : for suppose that u in-
volves other functions of x and y which are not infinite for the
particular relation which makes the above values infinite, and
suppose moreover that it contains a function of an arbitrary
constant, and that the derived-function of it with reference to
this arbitrary constant does not become infinite for this relation
between x and y, then all these quantities must be neglected
in comparison with those which become infinite, and therefore
the function between x and y which renders them infinite satis-
fies the differential equation, and is independent of the arbitrary
constant which the general integral contains : and this last
property is characteristic of a singular solution. Hence we infer
that if a relation between x and y which is independent of the
arbitrary constant of integration renders infinite (-7-) and (-7-)*
and at the same time satisfies the differential equation, it is a
singular solution, provided that it cannot be obtained by giving
any particular constant value to the general constant of inte-
gration.
Ex.1. u = x + 2(y xY c = 0,
idu\ _ 1
W ~ " (y- X} t
> = oo it y = &
du\ 1 r
and in this case c = <r, that is, the arbitrary constant receives a
variable value, and therefore we have a singular solution.
302
412 DIFFERENTIAL EQUATIONS OF [313.
Ex.2. c 2 -2/ + a 2 -# 2 =
in which case c = y, and therefore the solution is a singular.
The same singular solution may also be found by the methods
previously investigated. The differential equation of which the
given equation is the integral is
therefore in accordance with (283)
^ = , if * + y = a,
d"U 3C
and as this relation gives -jf- = -- , it satisfies the differential
citX/ y
equation given above, and is a singular solution.
Also if we find the c- differential of the general integral, and
then eliminate c according to the method of Art. 310, we have
and thus all the methods for finding singular solutions lead to
the same result.
SECTION 8. Differential equations of the first order and of any
degree.
313.] Order of differential equation, as has been observed,
depends on the index of the symbol of differentiation with
which the highest differential or differential coefficient is af-
fected, and degree on the power to which such highest differen-
tial or differential coefficient is raised. Thus a differential ex-
pression of the first order and wth degree is that which involves
(dii \**
~\ and no higher power : the general form of such is
313-] FIRST ORDER AND HIGHER DEGREE. 413
where PI, F 2 ,...F n are used as abbreviated symbols for functions
of x and y.
Let us suppose that the equation (303) admits of being re-
solved into n factors of the form
%L- fl = o, *L - /2 = 0, . . *L-f n = 0, (304)
dx ' dx dx J
where /!,/ 2 , .../ are the roots of (303) and are generally func-
tions of x and y : let each of these be integrated separately,
and let their integrals be
0i (#, y, Ci) = 0, 02 (x, y, c 2 ) = 0, ... n (x, y, c n ) = 0, (305)
where c\, c z ,...c n are n arbitrary constants. Then the equation
0i (# y, Ci) x 02 (#, y,c z )x ... x n (x, y, Cn) (306)
will contain all the integrals of the equation (303). And the
generality of this final equation will not be affected if the ar-
bitrary constants are equal, that is, if Ci = c z = . . . = c n = c,
because c is arbitrary, and therefore will pass through the values
Ci, c 2 , ... c n , if it receives all the values of which it is capable:
of this method some examples are subjoined.
Ex. 1.
--
dx
dy
y = ax + c 2
which may be combined into the single equation
(y - ax* ci) (y + ax 2 - c a ) = 0, (307)
and if Ci e 2 = c,
(y-c)*-a 2 x* = 0. (308)
The singular solution of this equation is x = ; and considered
geometrically the general integral represents two parabolas
which have a common axis, viz. that of y, and a common vertex
on the axis of y at a distance c from the origin ; and the sin-
gular solution represents a point on the axis of y.
Ex. 2. -
414
dx
dx
= a,
CLAIRAUT S FORM.
.-. y = ax + Ci
y = ex 3 + C 3
[3*4.
(309)
and the integral is
(y axCi) (y bx z c 2 ) (y ex* c$) ; (310)
and which may be simplified if c\ = c 2 = c 3 = c ; and in this case
the singular solution is a point on the axis of y.
a z x 2 } (dx \x
dy 1
dx
(a 2 -
y = sin -1 -
y a
y =
(311)
and the integral is
(y - sin- 1 - - d) (y - cos- 1 - - c a ) (y* - a?* - c 9 ) = 0. (312)
which are homogeneous, and may be integrated by the methods
explained above.
314.] Certain forms of differential equations of the first order
and any degree may be conveniently integrated without being
resolved into their factors : of these the most prominent is that
which is called Clairaut's form, and is
(313)
where / is the symbol of a given function. Now differentiate
this, and we have
3 1 4.] CLAIRAUT'S FORM. 415
=.
which may be satisfied in two ways ;
(1) = 0, .-, - = C, (315)
da? 2 dx
and substituting this in (313) we have
y = cx+f(c), (316)
which is the general integral, containing the arbitrary constant c.
We might of course integrate (315) immediately, whereby we
have
where Ci is a new arbitrary constant: but as (317) is to satisfy
(313), GI =f(c). This result is also manifest from the fact that
(313) is a differential expression of the first order, and therefore
its integral must contain only one arbitrary constant.
-*
and substituting this value of -^ in the equation (313), an
ax
expression results which of course satisfies the differential equa-
tion, and is independent of c the arbitrary constant, and is
therefore either a particular integral or a singular solution ; and
it is manifestly the latter, because c, which is equal to -~ } is re-
placed by a variable, viz. <(#), in virtue of the equations
(315) and (319).
therefore by differentiation
dx dx
which is satisfied either by
_ a
y- v + ->
and this is the general integral ; or by
416 CLAIRAUT'S FORM. [315.
and this is the singular solution, since it involves no arbitrary
constant, and is not a particular integral, because the constant
is replaced by f ] , which is not a constant value.
.
dx
ax ax t do?
therefore the general integral is
y = ex c z ,
dij x
and the singular solution is deduced from -/ = ~ , an d is therefore
dx A
x z = 4y.
dy , dy
Ex.3, y = x-^ + sm- 1 -^-,
dx dx'
therefore by differentiation
dx
d jL = d y_ . j j. . dx \
dx dx \ (dx z dy*)* )
and therefore (1) = 0, ~- = c,
dx z dx
y = cx+ sm- 1 ^;
or (2) x^dx^-x^dy^^dx^,
dy_ _ (x* 1)*
dx x '
.-. y = (a? 2 l^ + sec" 1 ^,
and this is the singular solution.
315.] The differential equation which is integrated by the
above method admits of an easy geometrical interpretation.
dii
Since tan -1 -^ is the angle between the axis of x and the tan-
dtV
gent to a plane curve at the point (x, y), equation (313) is that
to the tangent of a curve, in which the intercept by the tangent
dy
of the axis of y is expressed as a function of -~- t or, in other
dx
315.] CLAIRAUT'S FORM. 417
words, rjo = f (-} ; and the general integral is the equation of
* CLOO '
any tangent line, the arbitrary constant contained in which is
the tangent of the angle between it and the axis of x ; and the
singular solution is the curve-envelope of all such tangents, and
is found in fact by a method which is identical with those of
the last section : viz. by eliminating either c between the general
integral and its c- differential, or -~ between the differential
dx
Ull
equation and its -^-differential.
The most general geometrical problem which involves a dif-
ferential equation of Clairaut's form is that wherein the length
of the perpendicular from the origin on the tangent to a curve
is a function of the angle which the perpendicular from the
origin makes with the axis of x : and as problems of this kind
are numerous, and often elegant, two or three are subjoined.
Ex. 1. To find the equation to the curve, the perpendicular
from the origin on the tangent of which is of constant length a.
The differential equation of the curve is plainly
ydx xdy = ads, (320)
dy (, /dy\ 2
V = x-~- + a^l + (-r-)
dx ( \dx'
Differentiating which we have
r a *y
_ dy__
' ~
dx* ' dx
y = cx + a(\ + c^ } (322)
which is the general integral ; and is the equation to a straight
line inclined at tan" 1 c to the axis of x } and the perpendicular
distance from the origin on which is equal to a.
Also from the second factor in the right-hand member of
(321) dy _ x
dx (a 2 -# 2 )*'
substituting which in (320) we have
% 2 + y 2 = a 2 ,
PRICE, VOL. II. 3 H
418 CLAIRAUT'S FORM. [315.
the equation of a circle whose radius is a, and which is the
singular solution, being the envelope of all the lines whose
equations are (322).
The following process is worth noticing : differentiate (320),
= ad 2 s
dy
and since d*x and d 2 y are arbitrary,
dx
y a-j- =
ds
dy
# + -f =
as
x 2 + y 2 = a 2 .
Ex. 2. The product of two ordinates of the tangent of a curve
drawn at two given points on the axis of x is constant ; it is re-
quired to find the equation of the curve.
Let the origin (see fig. 51) be taken at the point of bisection
of the line A B which joins the two points A and B at which the
ordinates AQ, BR are drawn : and let x and y be the current co-
ordinates to the tangent line : let OA = OB = a ;
dy
.. tan RTB = -p-:
dx
and let AQ x BR = k 2 : then the equation to RQ is
ydx-xdy={k 2 dx 2 + a*dy 2 }*. (325)
Now from (324) by differentiation we have
dx
dn* + - -rm!> = o, (326)
' ' dx 2 -* ' dx ~ C>
.-. y =
which is the general integral : also from the second factor of (326)
3 1 6.] CLAIRAUT'S FORM. 419
dy kx
dx ~~ a {a* -at*}*'
and therefore from (324)
which is the singular solution; and is an ellipse of which AB is
the major axis.
Or thus ; differentiate (325), and equate to zero the coeffi-
cients of (fix and d 2 y : then
y kdx
ady
and squaring and adding,
Ex. 3. The triangle contained between the rectangular axes
and the tangent of a plane curve is of constant area (--); shew
that the equation of the tangent of the curve is
ydx xdy k(dydx}^,
and that the singular solution is
X V = 4--
Ex. 4. TRQ being a tangent to a curve in fig. 51, perpendi-
culars AY, BZ are drawn to it from the points A and B, and the
included area ABZY is constant: find the equation to the line
TRQ, and shew that the singular solution of it is the equation to
a parabola.
316.] I propose now to consider other forms of equation
(303) ; and let us first take the case where (303) does not admit
of resolution into factors such as (304), but where we can re-
duce it to the form
- /(*, y'), (327)
if we use Lagrange's notation of derived functions ; then dif-
ferentiating (327) we have
3 H 2
420 DIFFERENTIAL EQUATIONS OF
(328)
which is a differential equation involving two variables x and y ',
the integral of which will be of the form
F(ar,y',c) = 0, (329)
where c is an arbitrary constant : and if y' can be eliminated by
means of (327) and (329), the resulting equation will contain
x, y, and c, and will be the integral of (327).
A particular form of (327) is
*(y f ), (330)
df \ dy
df\ /rfFX
_ dy> \dy'l
~ ~
which is a linear equation of the first order, and can therefore
be integrated by the methods of Art. 278 or Art. 291.
Clairaut's form (313) is plainly a particular case of (330),
and one in which y ', determined by equation (331), is equal to a
constant.
Similarly, if equation (303) is capable of being put into the
f rm * = *(y,!0, (332)
then by differentiation
< 333 >
and since y" = %L = y' ^j- , (334)
therefore (333) becomes
1 = (^
and involves only y and y' ; whence the integral of it is of the
form
', c) = 0, (335)
and thus if y' be eliminated by means of (332) and (335), the
result will be the integral of (332).
Some examples illustrative of these processes are subjoined.
i _ /
Ex.1.
x
3 1 6.] FIRST ORDER AND HIGHER DEGREE. 421
but dx = -nr ,
y
2y' 2 dy'
~ 2
= {# (1 -#)}*- tan- 1
Ex.2. = x'* + 2'.
* I _
which is a linear equation, and of which the integrating factor
is (y I) 2 ;
y'+ C ,
between which and the given equation we may eliminate y', and
so obtain the required result.
Ex.3. 4 = x z + ' 2 .
which is homogeneous ; and of which the integral is by Art. 288
and between which and the given equation y' may be eliminated.
Ex.4, y = xy' + ax 2 y' 2 + bx 3 y' 3 + ....
Let xy' = ,
.*. y =
.. ...
but x--
I MIL iJU f . . .
-
f . . . \n - 77; - .
y y'
.-. y(y'duudy'} = (1 + 2au + 3bu 2 + ...)y' 2 du,
dif
.-. ~ + 2adu + 3budu+ ... = 0,
DIFFERENTIAL EQUATIONS OF [3 1 ?-
from which and the given equation y' may be eliminated, and
the resulting expression will be the integral required.
317.] Another form of differential equation of the first order
and nth degree which admits of solution is
(dy\ n (dy\ n ~ l {dy\ n -* dy
MrfD+Ms) + 'yp + -+'"-> + * = - < 336 >
where FI, YZ, - F are homogeneous functions of x and y ; so
that the equation admits of being put in the form
+ ...+,._, + ,. = .(337)
^x> dx \x'
Let - = ^, .'. y = tx, (338)
5?
and thus (337) becomes
)=0; (339)
also differentiating (338)
c?y = tdx -\-xdt,
dt
(840)
(341)
(342)
either of which equations reduces the integral of (337) to a
single integration ; and one or other is to be employed accord-
ing as by means of (339) y' is more easily expressed in terms
of t, or t in terms of y.
Ex. 1. ydx
. . ydx = x (dy 2
Let y xt, .-. t 2 = l + y' 2 , (343)
* *=-*-=
ORDER AND HIGHER DEGREE. 423
.-
but from (343) y' = (t 2 -!)*
x _ - -.- . ,
5 ~ 2 + g
which is the general integral.
The geometrical interpretation of the given equation is " To
find the equation to a plane curve such that the projection of
its ordinate on the normal is equal to the abscissa."
Ex. 2. To find the curve the arc of which commencing at a
given point is a mean proportional between the ordinate and
twice the abscissa.
s 2 = 2xy,
ds = *
1\*,
2
2t-l
and therefore from (341)
Ex. 3. To find the curve such that
* 2 = mx z -f ny 2 ,
424 PARTIAL DIFFERENTIAL EQUATIONS OF [318.
. l > 2 _ (m
' 2
+ y
whence we can easily find y in terms of /; and by substituting
in (341) we can determine the equation of the required curve.
Ex.4. s 2 = x* + y 2 , .-. t = y';
dy y
/ = - y = ex.
dx x
318.] Partial differential equations of the first order and
higher degree sometimes offer themselves for solution in pro-
blems of solid geometry ; and it is incumbent on us to consider
them so far as they are subject to integration ; but here we are
close on the boundaries of our knowledge ; and it is often neces-
sary for the complete investigation of functions satisfying given
differential expressions to have recourse to considerations which
belong to integral calculus as applied in mechanics, &c., and
which are therefore beyond and extraneous to the fundamental
principles of the pure science : it will be for this reason that we
shall in the sequel omit some subjects which are to a certain
extent within our present grasp ; but which I believe it to be
more advantageous for the student to defer to a future part of
our course, in order that we may have at our disposal all the
materials which are available for the complete investigation.
This course too is also historically preferable. Such equations as
I allude to have arisen in physical investigations of light, heat,
&c., and they express properties referring to peculiar constitu-
tion of the physical material of the theories which pretend to
account for the phenomena ; and therefore it has been with
reference to these suppositions that they have been made sub-
jects of inquiry, and it is in respect of these that their integrals
become interpretable. Of some few partial differential equations
of the first order and higher degree it is desirable to seek the
integrals.
319.] In the integration of these equations it is convenient
to represent ( j by p, and ( j by q, according to a received
y
notation ; and suppose the equation which is proposed for inte-
gration to be of the form
319-] FIRST ORDER AND HIGHER DEGREE. 425
f(x,y,z,p,q) = U, (344)
where z is a dependent, and x and y are two independent vari-
ables ; so that the integral is of the form
z = F (a?, y}, (345)
. . ckr = pdx + gcfy ; (346)
but as this is an exact differential,
d_ d_
~dy P '" ~dx q '
/&A /<fc\ fdz\ fdq\ idq\ (dz
\> "*" W/ V' " h
and if from (344) we determine in terms of x, y, z, and p,
q (-i), f-^i, (347) becomes of the form
' " \dz'
= -
and by the assumptions (110), Art. 284,
d* =i dy = dz = dp
P Q R S
Suppose that the integrals of these three differential equations
can be found, and are
fi(^y^P) Ci, fi(x,y,*,P) = c z , f*(x ) y,z,p) = cz, (350)
where GI, c 2 , c 3 are arbitrary constants, then, as shewn in
Art. 284, the most general solution is,
*(/i,/.,/s) = 0: (351)
but the generality of this solution must be restricted, because
the functions p and q must satisfy the exact total differential
(346) : and therefore if we can determine p by means of (349),
and thence q by (341), we may substitute in (346), and thus
determine z in terms of x and y. The determination of p will
involve one arbitrary constant, viz. c\, and the integration of
(346) will involve a second, viz. c 2 , which, by virtue of the argu-
ment of Art. 284, must be a function of the other constant.
Ex.1. ^ 2 -f q*= 1; ... q= (1 -J9 2 )*,
~ P dp
PRICE, VOL. II. 3 I
426 PARTIAL DIFFERENTIAL EQUATIONS OF [319.
and (347) becomes
dx _ dy _ dz _ dp
~J == (1_2 ) 4-T : = "OT
. . dz
z -
Ex.2. =
and thus (347) becomes
dx dy dz dp
(352)
dz =
where <^> symbolizes an arbitrary function.
Ex. 3. The differential equation of a tubular surface gene-
rated by a sphere of radius a, and the centre of which moves
along a director-curve in the plane of xy is z 2 (l. +p z + q 2 ) a 2 ;
it is required to find the equation to the surface.
and (347) becomes
32,0.] FIRST ORDER AND HIGHER DEGREE. 427
... * = * = _|!* = =4. (3 53)
j9 q a z z* a z p
from the last two of which
2 2~ == 1 ' * * P ~~ 2 '
zdz
(a 2 z 2 ')
but this is the equation to a right circular cylinder, the axis of
which is in the plane of xy : and it is therefore only a particular
integral of the differential equation : hence we must return to
other terms of (353).
d P^dz qz*dz
~ ~
squaring and adding which
(*-c 2 ) 2 + (y-c 3 ) 2 = a
and if we replace c 3 by < (c a ), we have
which is the general equation to tubular surfaces, and wherein
c 2 = <|> (c 3 ) is the equation to the plane director-curve.
The Memoirs of Jacobi referred to in Art. 284 remove some
difficulties which beset the integration of partial differential
equations of the higher degrees, but it is beyond the scope of
the present work either to raise the difficulties or to solve them.
SECTION 9. Particular methods of integrating differential equa-
tions of the first order and of any degree.
320.] In the present section various methods will be indi-
cated for integrating particular differential expressions, which
have historical or other interest, and I shall also take the oppor-
tunity of shewing the application of this branch of the integral
calculus to one or two other problems.
3 i a
428 DIFFERENTIAL EQUATIONS.
The integral of a differential expression may sometimes be
found by substitutions different from any of those of the pre-
ceding articles, and which are suggested by the form of the
expression. The following example indicates the kind of sub-
stitution.
-p x i xydy + y z dx _ d.f(y)
tan- 1 - -
321.] It is often convenient to change a differential equation
in terms of x and y into its equivalent in terms of polar coor-
dinates r and 9, and especially when it involves expressions of
the forms xdy ydx, xdx + ydy, (x 2 + y 2 )^. Thus, for example,
it is required to integrate
m(xdy ydx) = xdx + ydy,
x r cos 8, dx = dr cos r sin d&,
y = r sin 0, dy = dr sin + r cos dd,
.'. xdy ydx r z dQ,
xdx + ydy = rdr,
and the equation becomes
mr z dO = rdr,
dr
mdd = ,
r
* _ r* nWu
Let this process be compared with that of Ex. 1, Art. 275.
Again, the integral is required of
xdy ydx =
the right-hand member of which is integrable by known methods.
322.] DIFFERENTIAL EQUATIONS. 429
322.] Sometimes the integrals of the sum of two or more
expressions can be found in finite algebraical terms, although
the integral of each separately would involve an elliptical or
other transcendental function ; the reason of course being that
the transcendental parts neutralize each other : of this we have
had instances in Fagnani's theorem as to elliptic arcs, and in
Ex.1, Art. 263. The following example is a remarkable illus-
tration of this. It is required to integrate
dx dy
. (354)
Let
(ooo)
and let each term of (354) = dt : so that
*=*=*; (356)
and therefore if the new variable t is equicrescent
2dxd z x 2dyd 2 y
d * = -di*-> ^ = -^' (358)
but dx = (ai-t-2a 2 # + 3a 3 ,z >2 + 4a 4 <r 3 )a t #-
(o59)
^-y 4 ), (360)
^+y 3 ). (361)
Let xy = z, x + y = s, (362)
therefore (360) and (361) become
whence by subtraction
dzdszd^s
zdt 2
(2dzds2zd 2 s)ds
430 DIFFERENTIAL EQUATIONS. [322.
ds 2
where c is an arbitrary constant.
ds
.'. -^ = 5r{a s
and therefore by substitution
(363)
y 4 }*
; (364)
and this is the integral of the given equation (354).
Another and equivalent form of the same equation is
(365)
.
{l-e 2 (sin0) 2 }* {l-c 2 (sin4>) 2 }*
/ in this and in the former cases being an elliptic function :
hence we have
-jj2 = e 2 sin0cos0, -~
d(j) 2 .
-"" "
= e 2 sin(0 <) sin
Let -\- <p = (T } (^> = 8j
-TTa = e 2 sino-cos8,
-37 fi = c 2 sin o- sin 8 ;
dt dt
. . d z a- ^ da- db
... Sm8 __ _ COS 8^^=0,
_ d<r da
cosecS = c rr = csm8.
dt dt
dd d<f>
{l-e 2 (sin^) 2 } i +{l-e 2 (sin</)) 2 } i = c sin (0 -(/>), (366)
which is the integral of (365).
323.] DIFFERENTIAL EQUATIONS. 431
The above equation and some more general cases of a similar
form are discussed by Prof. Richelot of Kb'nigsberg, in Crelle's
Journal, Vol. XXIII, p. 354. The following expressions may
also be integrated by a similar process :
{a + a 2 y
by assuming x z = , y 2 = 17 ; and more generally
dx _ dy
by assuming x n = g, y n = 77.
I can only refer the student to the solution of other particular
equations by Jacobi in Crelle's Journal, Vol. XXIV, p. 1, and
to an extension by Hesse in the same Journal, Vol. XXV, p. 175.
323-3 Some functional equations are conveniently solved by
means of integration and differentiation, as the following exam-
ples shew.
Ex. 1. Determine the form of z =/(#), so that for all values
of* and y /(*)+/fy) =/(* + y), (368)
Take the ^-differential ; then
whence y being independent of x, we infer that/'(#) is constant
whatever value x has ; therefore
f'(x) = c,
f(ae) = cx + d, (369)
substituting which in (368)
and therefore the most general form of f(x) which satisfies
< 368 > is
Ex. 2. Determine the form off, so that
/(a?)+/(y)=/(a?y). (370)
Take the ^-differential
f'(*) = yf(xy). (371)
Again take the y-differential
432 THE GENERAL EQUATIONS [324.
.-. */'(*) = y/'(y),
therefore xf'(x) is a constant : let us suppose
#/'(#) = <*,
f(x) = a log-,
c
substituting which in (370)
. x y xy
alog- + alog- = a log - ;
= alogx.
Ex. 3. If /(a?)/(y) =f(x + y), f(x) = e*
324.] It is required to find a general property of rectifiable
plane curves : in other words, to integrate the equation
ds z = dx z + dy z . (372)
This may be put into the form
ds z = (dx cos a dy sin a) 2 + (dx sin a + dy cos a) 2 , (373)
where a represents an arbitrary angle : which equation is satis-
ds = dx cos a dy sin a
= dx sin a + dy cos a;
whence integrating
s = xcosa ysina+f(a
= x sin a -f- y cos a
where /(a) and < (a) are two arbitrary constants of integration.
Now to combine these so that they may form an envelope and
thus a curve, let us take the a-differential of each ; then
= <z?sina y cosa+/'(a)
( 375 )
= x cos a y sin a
and therefore we have
s = <rcos a y sina+/(a) "I
= .z-sina + ycosa /'(a) \\ (376)
= x cos a ysina /"(a)J
where / represents an arbitrary function ; and hence we have
325.] OF RECTIFIABLE CURVES. 433
x = sina/'(a) + cosa/"(a) ^
y = cosa/'(a)-sina/"(a) I; (377)
the values of which manifestly satisfy (373).
There are also other forms which satisfy (373) : such as
y = *-a
where $ and \j/ are symbols for arbitrary functions : and taking
the a-differentials we have
= s<}>'(a) + l -,
(.)'(.) L
l-ai
(379)
hence we have the system of equations
S 2 = {*-}* + {y-^r(a)} -I
= (*-a) + {y-^(a)}^'(a) -', (380)
= -l-{Vr'(a)}+{y-^(a)}*''(a) ->
which are plainly equivalent to those by means of which the
equation to an Evolute is determined from that to the Involute :
and which is accordant with the fact that all E volutes are rec-
tifiable.
It is worth observing, that if the second and third equations
in (376) are those to a plane curve in terms of x and y, the
length of the curve is given by the first.
In Vol. XIII of Liouville's Journal is a Memoir by M. J. A.
Serret which contains a solution, by a process somewhat similar
to that above, of the equation
ds 2 = dx 2 + dy 2 + dz 2 ;
and observations on the mode of solution will be found in
Art. 8 of Note I, appended to Liouville's Edition of Mongers
Application d' Analyse &c.
325.] Integration of Riccati's Equation.
The differential equation P\y' + Pay + ^a = 0, where PI, P 2 , PS
are functions of x, has been completely integrated; the form
which next suggests itself is PI y" + Pa 2/' + 1*3^ + ^4 = 0, where
PI... are functions of x\ but this has never yet been completely
integrated, and will not be until the properties of certain tran-
PRICE, VOL. ii. 3 K
434 KICCATl'S EQUATION.
scendents, which are in the form of definite integrals, have been
more completely investigated : a particular form however of it is
j- + ay z = bx m , (381)
which is known by the name of Riccati's Equation, having
been first discussed by Riccati in the year 1775 in the Acta
Eruditorum, and of which, in particular cases, solutions can be
found : these I proceed to investigate.
First suppose m = 0; then (381) becomes
+ * =
in which the variables are separated.
Again, let y z n ; (381) becomes
nz tl - 1 dz + (az zn bx m )dx = 0;
and this will be homogeneous if
n 1 = 2n = m\
that is, if n 1, m = 2 ;
therefore the equation
' '
<383 >
becomes homogeneous if for y there be substituted a?- 1 ; and
the integration can be performed.
And to investigate general conditions of integrability ; let
y AXP + VZ, (384)
then the equation becomes
x q dz -f (qx<*- 1 + 2A.axv +< i + ax z ^z) z dx
+ (p\xP- l + aA 2 x z P)dx = bx m dx; (385)
in which, let
.-. p -1, A = -, q = 2;
therefore (384) becomes
-^ + ^' (387)
and (385) becomes
(388)
326.] RICC ATI'S EQUATION. 435
in which equation the variables are separated if m = 4; and
we have ,j~ fj T
-P-, + - 2 = 0. (389)
az 2 b x*
Again in (388) let x = - j then
If
dzaz 2 du = bu- m -*du, (390)
which is of the same form as (381) ; and therefore if (381) is
integrable for any particular value of m, say // = /, it is also
integrate when m = _^_^ (391)
326.] Again in (381) let y = -; then
ss
dz = dx bz*x m dx. (392)
Let (m +1) x m dx dv, then (392) becomes
7 1 -1 TO
^+ -^z* = L- v "^i ; (393)
rfvm + 1 m+1
which is of the same form as (381) ; and therefore if (381) is
integrable for any particular value, say /u, of m, it will be inte-
grable also when
(394 >
Now we have seen above that (381) is integrable, when p = 4,
therefore the equation is also integrable when
_4
Also from the conclusion of Art. 325 we infer that the equation
is integrable when . ~
and therefore from (394) it appears that
_8
5'
and thus substituting successively in the two formulae
u,
afn - .^^ j. __ /J 1 AOJ
we have the following series of values :
_8 _12 _16
~3 J "T' ~~Y'
_4 _8 _12 _16
3' 5' ~~T' ~~9'
3 K 2
436 RICCATl'S EQUATION. [327.
A t yj
the types of the general terms being respectively = and
-,-_ /I, IV\
=- ; and in which if n = 0, and if n = <x> , we have the two
<vW -}- J.
values of m, viz. 0, and 2, which on inspection render (381)
integrable.
Ex. 1. dy + y 2 dx=x~* dx.
As this form is one of those which fall under the series (396)
we must put y = z~\
.-. dz dx=z z x~*dx.
Let x~* dx =r dv, .*. 3x~$ = v;
,. dz-\-z*dv = (-} dv.
1 u
Let, according to (387), z = | 5- ;
v v
dv du 2udv
dz = 1
v 2 v 2
C?M M 2 C?t? 81
du dv 1 . M 9 1
and substituting for u and for v,
3
u = 3-^, v =- i,
yx* x*
327.3 Thi 8 example, and it is one of the easiest, sufficiently
indicates the tediousness of the process, and the succession of
the substitutions. If m has a value corresponding to the first
term of the series (395) the method is of course that of Article
325 : but if m has any other value, then we shall have to pass
successively by alternate processes from one series to the other,
until at last we shall arrive at a form wherein m will have the
value 4.
The above process is unsatisfactory, because although it points
out certain cases where the variables are separable, still the
number of them is limited ; and they are obtained by particular
327.] RICCATl'S EQUATION. 437
artifices, and the investigation does not prove that they are the
only possible ones. M. Liouville, however, in the Vlth volume
of his Mathematical Journal, has proved by a rigorous investi-
gation that the cases comprised in the above series are the only
ones where the integral can be expressed in an algebraicalj
logarithmic, or exponential form. The argument is not simple
enough for insertion in an elementary treatise, and therefore
the reader desirous of further information must have recourse
to the original Memoirs.
There are also other forms which are capable of reduction to
Riccati's Equation. Thus, if
dy + ay 2 x n dx = bx m dx ; (397)
let x n dx = dz, x n+l (n + 1) z,
m n m n
dy + ay 2 dz = b(n + l) n+1 z n+l dz, (398)
which is of the form (381).
The Equation of Riccati also admits of transformation into
the form of a differential equation of the second order, under
which it is often convenient to consider it.
Let ,= -, (399)
_
dx ~ az 2 dx 2 az dx 2 '
d 2 z
= abx m z. (400)
Ui&-
Let ab = k, .-. -^ = kx m z. (401)
ft >Yt&
m + 2
Again, in (401) let x 2 = t, and we have
d 2 z m 1 dz 4>k
~di 2 + m + 2 l~di~ (m + 2) 2 Z>
and if we substitute
nn/i ^L If
n = ~ / = -' . (402) becomes
2(w + 2) (m + 2) 2
438 RICCATl's EQUATION. [327.
and if z = ut~ n ,
d 2 u n(n-\)
- 7r -u=lu. (404)
It appears therefore that (400), (403), (404) are all equivalents
of Riccati's Equation, and that the properties which are true of
any one are also true of each of the others. If therefore we
can determine either a particular or a general integral of either,
that of Riccati's equation will be determined by the equation
log 2 = lydae, (405)
A Memoir by M. Malmsten of the University of Upsala, and
inserted in Vol. XXXIX. of Crelle's Journal, p. 108, on the
various forms and properties of Riccati's Equation, may be con-
sulted with advantage by the reader who is desirous of further
information.
328.] DIFFERENTIAL EQUATIONS OF HIGHER ORDERS. 439
CHAPTER XIV.
INTEGRATION OF DIFFERENTIAL EQUATIONS OF ORDERS
HIGHER THAN THE FIRST.
SECTION 1. General properties of differential equations of
higher orders.
328.] WE are now just on the outskirts of our science, and
are unable to give any general theory for the integration of
differential equations of higher orders ; almost all that deserves
the name of philosophical treatment has been exhausted ; and
thus it only remains for us to insert such discussions on iso-
lated topics as are useful either in the way of extending the
boundaries of our knowledge, or for the purposes of subsequent
application.
The most general forms of differential equations of the nth
order are (1) (2) (3) (4) in Art. 257, the last two of which are
partial : and the discussion of these is reserved to a future Sec-
tion of the present Chapter : and we shall confine our researches
to an equation of the form
v dy d * y dny \ - o m
y '^' d&'"'d&> - '
which contains only two variables, and wherein one of these is
equicrescent. Of such equations we have in Art. 258 pointed
out the geometrical meaning ; and in Art. 261 have shewn that
the general integral involves n arbitrary constants. If a func-
tion satisfies the equation (1) and does not contain n arbitrary
constants, it may be either a particular integral or a singular
solution ; and either the one or the other of these according as
one or more of the arbitrary constants has been replaced by
particular constant values or by functions of the variables : and
it is manifest that such substitutions may take place, at any one,
or at more than one, of the successive integrations.
440 DIFFERENTIAL EQUATIONS OF HIGHEK ORDERS. [32,9.
329.] Now with reference to general properties of differential
equations of the form (1), I will first observe that if (1) admits
of being expressed explicitly,
and, the limits of integration being (X Q , y Q ) (x^, yi), if (2) as
well as all its derived-functions remain finite and continuous
for all values of the variables within the limits, then (2) can
be integrated in a series, that is, approximately, by the method
of Art. 261 : and that its general integral will contain n arbi-
trary constants.
And next, I will observe that a differential expression such
as (1) may admit of integration by reason of the form of the
expression, and independently of any specific relation between
an and y : the conditions that this should be the case have re-
ceived much consideration from Euler, Lagrange, Lexell, Pois-
son ; and lastly from M. J. Bertrand*, and M. J. Binet, as quoted
in Moigno's Calcul Integral, Vol. II. p. 551 : and it is to Euler
and to the last two that we are indebted for most of our know-
ledge of the subject. In the following articles the conditions
requisite for such a case are investigated by means of the Cal-
culus of Variations.
Suppose the integral of (1) to be definite, and the limits of
integration to be those particular values of the variables which
carry the subscripts and 1 : and let the definite integral be
expressed according to the notation of Art. 185. Now our
object is to determine the conditions which (1) must satisfy, so
as to be the ^-derived function of some other function of the
form
ay
independently of any relation between x and y ; that is, so that
(4)
o o
and that this equation should subsist independently of the
functional connexion of x and y.
Suppose then that this functional relation undergoes a small
variation, and that the values of the variables and of their
* See Journal de 1'Ecole Royale Polytechnique, Cahier 28, Paris 1841, p. 249.
330.] DIFFERENTIAL EQUATIONS. 441
(n 1) derived functions at the limits do not change; then by
reason of (4) the value of the integral will not be altered, and
therefore ri
&./ j?(x,y,y',y",...yW)dx = 0; (5)
Jo
Let us employ the notation introduced in Article 192 : then it
is manifest that if we replace the left-hand member of (5) by its
value given in equation (31) of Art. 192, (5) cannot be true
Unle8s -
and this therefore is the condition requisite that (1) should be an
exact differential independently of any relation between y and x.
Let it be observed in (6) that Y, Y', Y"... are partial derived
functions ; but that the subsequent #- differentiations are made
on the supposition that all these quantities are implicit func-
tions of x : and therefore they do not vanish, although x may
not enter explicitly into them.
330.] Let us pass to the converse of the above. Suppose that
?(x,y,y', ...y (n} ) satisfies the condition (6), then I say that its
integral is capable of being expressed in the form of (4), and
independently of any relation between x and y : or what is
tantamount, if (6) is satisfied, the integral can be expressed in
terms of the limiting values of the variables and of their derived
functions ; and this is what we mean by definite integration.
For in this case, by virtue of equation (31) Art. 192, the varia-
tion of the integral on the left-hand side of (5) will be expressed
in terms of the limiting values of the variables and of their
derived-functions, and in terms of these alone, and therefore the
integral must be a function of these quantities only. Hence
also, if these limits are fixed, their variations disappear, and the
variation of the definite integral also vanishes. Some examples
are subjoined.
Ex. 1 . Let p be a function of x and y : it is required to de-
termine the condition that F dx should be integrable independ-
ently of any relation between x and y.
In this case (6) becomes
therefore r must not contain y.
PEICE, VOL. ii. 3 L
442 CONDITION OF INTEGRABILITY OF [331.
Ex.2. Determine the condition requisite that (p + Qy')dx,
where P and Q are functions of x and y, should be integrable
independently of any relation between x and y.
(6) becomes in this case
---*
. .
(7) becomes
which is the same condition as that before found in Art. 265 :
hence also we may infer that the complete integral of the dif-
ferential equation of the first order and degree contains an un-
determined functional symbol.
It is good also to exhibit a posteriori the criteria of Euler
vv
given in equation (6). The ^-differential of - - is
and as this is an exact differential independently of any func-
tional relation between x and y, it ought to satisfy (6) ; now
Y =
dy
dy'
*
,-.().,
\dy i x
' _ W 2y 2y
dx x 2 x 3 x x 2 '
_ y" 2y f 2y
dx 2 ~~ x x 2 x 3 '
Y _^ d ^" _o
dx dx 2
331.] We may also by a similar process determine the con-
ditions that rdx m should be integrable m times successively,
33 1 -] DIFFERENTIAL EQUATIONS. 443
and independently of any particular relation between x and y ;
m being not greater than n which is the index of the highest
derived-function contained in F. Let
v = F(a?,y,y',...y<>)J (8)
then it is manifest by the principles enuntiated above that (in
accordance with the notation of Art. 96) the variation of the
definite integral of
r
must not involve terms containing signs of integration. Now
using the symbols of Art. 192, and supposing 8# = 0,
m f"
vdx m = I
f
= /
bvdx m
(9)
Of this series let us take a typical term, say Y ( *> by (k) , which we
may write in the form
(10)
Now, by the theorem proved in the foot-note of page 319,
(k) d k by_d k _ k d k -* d?w k(k-l) d k
dx k ~ dx k '^ y 1 da*- 1 ' dx y * 1.2 dx k ~*' z y
.k d d k - l ?w.
/m iJlt^ii fm-k If fm-k+l ff-yik)
T <*> ^3 dx m = / Y<*> by dx m ~ k - y / - by dx
UX J I J U/X
k Cm-l ffk-l y (k)
-(-''-'iJ ![-*.**
l-k+l
and therefore the right-hand member of (9) consists of a series
of terms of which (12) is the type ; and wherein k receives all
integral values from k = to k = n, both inclusive ; and where
Y= Y.
/
v dx m is to be free from terms
under signs of integration, the coefficients of by under the
/m fm-l f2 r
, I ,.../,/ must vanish of themselves ;
whence we have
3 L 2
444 CONDITIONS OF INTEGRABILITY OF [332.
dv' dV
~~i r
dx dx 2 ' dx>
y' Q I Q - / \n_l n _ f|
dx dx 2 ' dx-- 1
v" __ -J- _ nJL_ / \n-2'"V -"^ **
1.2 dx * 1.2 d.r 2 1.2 - -
(13)
This series of conditions must be continued so long as the inte-
gration-signs have positive indices; for when the indices are
negative, and when they vanish, the corresponding terms have
their limiting values : of the general form (12) therefore we
must take the last m terms; that is, the terms corresponding
to values of the indices of the integration-signs until k =. m 1 ;
in which case we have
y(m 1) _ m __ (_
_
dx 1.2 da*
n- m+ i Y ;n)_
~ l '
so that we have m equations of condition ; and if these be satis-
fied the given differential expression will be integrable m times
successively.
332.] A similar process enables us to determine the condi-
tions necessary that
F (^, y, y', y", y (n] , *, *. *", (n) ) &*, ( 15 )
in which we have used the notation of Art. 197, should be in-
tegrable independently of any relation between #, y, and z : for
if the variation of the integral of (15) does not contain a quan-
tity under the sign of integration and depends only on the
limiting values of the variable quantities, then
^ .
Y _ _ I _ _ _ ( _ \n _ f) I
dx H dx 2 } dx"
dz' d 2 z" d n z ln)
y _ I / _ \n _ _ A
dx + dx* " ( > dx- ' J
and similar conditions must be fulfilled if the element-function
contains any number of variables ; and also conditions similar
to (13) and (14), if such an element-function be capable of m
successive integrations: thus suppose \dx n to involve m vari-
333-1 DIFFERENTIAL EQUATIONS. 445
ables besides x, then the number of conditions requisite that
v dx n should be integrable n times successively is mn.
It is beyond the scope of our work to investigate the cor-
responding condition in the case of a multiple integral: the
student, however, desirous of pursuing the inquiry will obtain
the necessary aid from Jellett's Calculus of Variations referred
to at the foot-note of page 234.
333.] There is a particular form of differential equations
of the wth order called the linear, many properties of which
will be investigated in the following sections, but which it is
convenient to consider at once in reference to the conditions
(13) and (14). Suppose p n , p n _i,...p 2 , PI, PO, Q to be functions
of x and y, then the equation is
d n y d n ~ l y dy
That this should be integrable once without any specific rela-
tion between x and y, it must satisfy (13) ; and therefore
that it should be integrable twice, it must also satisfy the condition
</p 2 ^ 2 P 3 vn-i rfn " lp " o. no\
Pl - 2 ^ +3 ^-- ( " ) rf^- = 0;
and so on. Thus if P! is a function of x only, PJ-Z- 4. y
CLOG OiOO
satisfies (18), and is integrable immediately. Again, suppose
that P 2 , PI, PO are functions of x only ; and let it be required to
determine the value of PO in the equation
so that the equation should be integrable once; in this case
(18) becomes , , 2
d 2 y dy
=
dx d
is an exact differential expression ; and of it the integral is
446 CONDITION OF INTEGRABILITY OP [334-
suppose again that (20) is integrable twice ; then, in addition to
(21) we must have from (19)
(23)
and this condition might also have been deduced from (22), by
applying to it the criterion (18), that (22) should be integrable
once.
There is also one other point that deserves notice. Suppose
that (20) does not satisfy (18), but can be made to do so by the
introduction of a factor ; let JLI be the factor, then we have
MP2 + fXPl ^^ /XP y = 0; (24)
so that (18) becomes
z
0; (25 >
and if from this any value of //, (general or particular) can be
found, then (20) may be integrated directly. It will be observed
however that (25) is a differential equation of the second order
in terms of jz, and that therefore the difficulty of solution (as
far as the order is concerned) is not lessened.
SECTION 2. Investigation of properties of linear differential
equations.
334.] As we do not know any general method of solving
differential expressions of the second and higher orders, we are
obliged to have recourse to such particular forms of them as
have yielded to the powers of analysis ; and amongst these the
most remarkable is that known by the name of the linear equa-
tion : into which the independent variable and its derived-func-
tion enter in only the first degree, and where the coefficients
are functions of the variable x only. Thus the most general
form is
d n y d n ~ 1 y d n ~ 2 y dy
^ + P1 ^ + P2 S^ + -+ P "-'S + P "* = X ' (26)
where PI, P 2 ,...p n , x are functions of a? only. Of this equation
we shall prove some general properties, and then proceed to the
solution of particular examples.
334-] LINEAR DIFFERENTIAL EQUATIONS. 447
It will be observed that two forms of this equation have already
been integrated ; (1) in Art. 96, where PJ = p 2 = . . . = p n = ; and
thus d n y
dae n ~
(2) the general linear equation of the first order in Art. 278, viz.
THEOREM I.* The integral of (26) depends on the integral of
the left-hand member of the equation ; that is, on the integral
of the equation when x = 0.
Let y = Uijvida?, where Ui and v\ are two undetermined func-
tions of x : then by Leibnitz's Theorem
d m
//wi 1,,.,
!=?.; (27)
and substituting the specific values of this in the several terms
of (26) we have
= x ; (28)
where QI, Q 2 , ...Q M -i are determinate functions of x and %.
Suppose now HI to be a function of x which satisfies the left-
hand member of (26), that is, suppose u\ to be a particular in-
tegral of (26) when x = 0, then the coefficient of / Vi dx in
(28) vanishes, and we have
x _
= ' (29)
an equation of the same form as (26), and of the (n l)th
order : in this equation let
Vi = u^lv^dx,
* The first of the following Theorems is due to Lagrange : the others are
the original investigations of M.G.Libri, and are taken from Crelle's Journal,
Vol. X, page 185.
448 LINEAR DIFFERENTIAL EQUATIONS. [335.
and let substitutions be made in (29) according to the process
pursued above : then if u 2 is a particular integral of (29), when
the right-hand member is equal to zero, the resulting equation
will be of the (n 2)th order, and of the form
and by a continuation of the same process we shall finally have
an equation of the first order which may be integrated by the
methods of the last Chapter; and the function which satisfies
the given equation will be determined by the successive inte-
gration of a multiple integral of the wth order. The problem
then will hereby become reduced to that of a multiple integral,
and of simple quadrature.
335.] And to indicate more clearly the form which by this
process the last integral assumes, let us consider the case of a
differential equation of the third order,
dy
^ = x - (30)
y =
d 2 y
s?
dy
and substituting in (30) we have
= , (3!)
Now if MI is, according to our supposition, a particular integral
of (30) when x = 0, the first term of (31) vanishes : also let
336.] LINEAR DIFFERENTIAL EQUATIONS. 449
then (31) becomes
2/J1. //4t. Y
= -. (32)
Again, let u z be a particular integral of this equation without
its second member : and let
then if 2-j---f Q^ = Ri2, (32) becomes
dv 2 x
h RI = - . (66)
ax
Again, let u 3 be a particular integral of (33) without its
second member; and let
v 2 = u 3 v 3 dx,
then
MI Ma
(34)
and retracing our steps we have
y = Ui u 2 dx /MS dx I - , (35)
J J J U\UiU$
where u\, u 2) u$ are particular integrals of the several equations
found as above and without their second members; and thus
the general integral is found in terms of a triple integral whose
element-function contains one variable ; and therefore by the
process of integration three arbitrary constants will be intro-
duced, and the integral will be in its most general form.
And to generalize the process : the integral of (26) will in
the same manner be
y = u\ \Uidx\Uidx ... I - . (36)
J J J UiU 2 ...U n
336.] And these quantities u\, u 2 ,...u n may be expressed in
terms of particular integrals of (26), when x = 0. To limit the
PRICE, VOL. ii. 3 M
450 LINEAR DIFFERENTIAL EQUATIONS. [336.
extent of the investigation, let us confine our attention to an
equation of the third order, viz.
dy
s- 1 -*'-" (37)
and let rji, 772, 773 be three particular integrals of this equation,
when x = 0, so that
3 2
0; (38)
then employing %, u^, u%, v\, V 2 , v 3 in the same signification as
in the last Article, let i = 771 ; and if for 771 in (38) HI lu^ dx be
substituted, it will on expansion be seen that
d* r d* r dr. r .
-j =.Wi lUvdx + PI -=5.2/1 1 u dx -f- P2T~ w i / u%dx -\- Pa MI \u z dx = : (39)
o^^ J a.z" 5 j dx J J
so that uAu^dx is a particular integral of (37) when x = 0;
suppose this integral to be 772 ; then
= rjilUzdx,
'Ja //m\
.*. W 2 = -j . v 1 *^)
dx 771
But M 2 is a particular integral of (32) without its second mem-
ber: so that 7 2 ,
OMa A ,.,.
= 0. (41)
Again, let 773 be another particular integral of (38) ; and let
u' z be another particular integral of (32) without its second
member ; then, pursuing the same reasoning as above,
* = =-: <>
ax 771
so that u z and u' z are two particular integrals of (41) : and em-
ploying w 3 as above, it will be seen that
.Wg u 3 dx + Qi-j-. 2 u s dx + Q,zU 2 u 3 dx = 0, (43)
and that therefore u 2 u 3 dx is a particular integral of (41) ; let
this be equal to u z , so that
337-1 LINEAR DIFFERENTIAL EQUATIONS. 451
r
d u'z
U 3 = -v ---
ax M 2
d_
d dx '
(44)
dx '
dx
so that now MI, u^, u$ are expressed in terms of 771, rj 2 , rj 3 , that is,
in terms of three particular integrals of the given equation,
when its right-hand member vanishes ; and these may be sub-
stituted in (35), and the final value of y thus obtained will be
, AK .
(45)
-fL..5i f A. .50 ~
I'd Vz * I'd dx rji , C d r?2 d dx -n\ [ ,
= *li -j-<tel-i-s ~dx\JL< rji-j-. -=-'-- ii > dx.
J ax r]i J dx d 772 J dx rji dx a r) 2 \
dx rji L dx r/iJ
The. same process may manifestly be extended to equations of
the order n ; the final result however is of a form too compli-
cated to be inserted : it will however involve n signs of integra-
tion, and therefore n arbitrary constants.
337.] Some examples of the above process are subjoined.
Let us first consider the linear equation of the first order, viz.
Now of this equation, when x = 0, an integral may be found
as follows: ,
+ vdx = 0,
y
y = ce~f vd * t (47)
which is r)i ; and therefore substituting this value in the gene-
ralized form of (45), we have
y e-S pdx xeS f(l *dx, (48)
and which is the general integral as before expressed in equa-
tion (69), Art. 278.
For a second example let us consider
3 M 2
452 LINEAR DIFFERENTIAL EQUATIONS. [338.
the particular integrals of the left-hand member of which are,
when the right-hand member vanishes,
_ aax . _ aZax _ f,3 ax
rji e , 772 e , 773 e ,
d 772 d
.. _ acue _ ft c>ax
, , J ~ 5 C UK
ax ri ax
da? rji
substituting which in (45) we have
y = e^jae^dx iZae^dx
e ax I e^dxfe^dxf e (m - 3a)x dx
r r ( e (m-3a)X )
_ e ax e axffa, e ax) - + d\ dx
J J (m 3a J
SjHljL p f>
__ I __ L _ L p3oar I 2. P 2ax i - p
~ (m-a)(m-2a)(m-3a) + 2a* e h a
and this is the general integral of (49) ; (49) in fact having
been deduced from it by the elimination of GI, eg, and c^.
Another example, which the reader may solve, is
d 2 y
_ <), _ /y>
dx* y
the particular integrals of which without the second member
are 771 = e x , rjz = e~ x ; and the general integral is
y = de x + c 2 e~ x x.
338.] The process which has been explained and illustrated
above also gives the following Theorems.
THEOREM II. If m particular integrals of a linear differen-
tial equation of the nth order without the second member are
known, the integration of the equation with the second member
will depend on the integration of a new linear equation of the
(n m)th order.
Let rji, 172, ... rim be m particular integrals of (26), when the
right-hand member vanishes ; and let us, in Art. 334, assume
(51)
Then substituting as in that Article, the coefficient of / v\ dx as
exhibited in (28) vanishes, and we have
339-1 LINEAR DIFFERENTIAL EQUATIONS. 453
Now of this equation, without its second member, according to
the method pursued in Art. 336, (m 1) particular integrals are
dx rji ' dx rji' " dx TJX '
let these severally be symbolized by ft, &) Cm-i ; then in (52) let
(54)
and substituting according to Art. 334, the term involving
v z dx will vanish, and we shall have
, 9 2 -f R! = 1- + ... + R n _2#2 = z- ; (55)
///v>n * /7 / y>'* 5 v> f
\AJtAj if ff. '/I al
and of this equation again without its second member, (w 2)
particular integrals are
d & d ( 3 d Cm-i ,- R .
-. . ... . . j (5o)
ff y it fi M* it d 'y it
which we may conveniently symbolize by QI, 2 , ...6 m -z ', and
by a similar process we may make the integral of (55), without
its second member, dependent on the integration of an equation
of the (n 3)th order: and in a continuance of the process
it is manifest that each of the given particular integrals of
(26) enables us to reduce by unity the order of the differential
equation ; and finally therefore the order of the equation will
be the (n m)th.
339.] THEOREM III. If TJX, 772, ... r\ n are n particular integrals
of a linear differential equation of the nth order without the
second member, and if y is a particular integral of it with the
second member, then the general integrals of the equation with
and without the second member are respectively
y = ClTli + C 2 r) 2 + ... +C n r] n + yi~\
f- (')
The truth of the proposition is evident from the form of the
equations ; because each contains n constants : these however
must be independent of each other; and the particular inte-
grals must also be independent of each other : for suppose that
T/ 3 = arji-+br]-2, then
454 LINEAR DIFFERENTIAL EQUATIONS. t34-
and which contains only n 1 arbitrary constants. ,
340.] M. Libri has in the Memoir above referred to traced
analogies between the formation and properties of algebraical
and differential equations : some of which it is good to insert.
THEOREM IV. A differential equation, linear in at least the
first two terms, may be transformed into another linear equa-
tion of the same order, and without the second term.
Let (26) be the typical equation of a linear equation of the
wth order : and let
y = uv, (o)
where u and v are two undetermined functions of x : then, cal-
d n y d n ~^y
culating -, M j[ , ...... by means of Leibnitz's theorem,
(26) after substitution will become
d n u dv d n ~ l u nn l d 2 v d n ~ 2 u
e"- 1 I 1.2
d n ~ l u ,. dv d n ~ 2 i
+ r n uv = x. (59)
Therefore if v is such that
n ^ + Vl v = 0, (60)
the second term of (59) vanishes ; and from (60) we have
v = e~ f ^ dx ; (61)
whence (theoretically at least) v may be found ; and (59) will
be a linear equation without the second term.
And more generally : A differential equation of which the
first m + 1 terms are linear may be transformed into another
linear equation of the same order, and without the (m +l)ih
term, by means of the solution of a linear equation of the mth
order. Thus, let it be required to deprive of its second term
the equation
LINEAR DIFFERENTIAL EQUATIONS. 455
substituting y = uv, we have
\ </M /</ 2 v flfo \
-Hs + (^- 3a "s +2at ') M = x; (68)
JTRH
let 2^
d#
v = e
so that (63) becomes
a
Sax
4
u = xe
2
341.] THEOREM V. If a relation be given between two par-
ticular integrals of a linear differential equation of the rath order,
the order of the equation may be diminished by unity.
Let 771 and r/ 2 be two particular integrals of (26), and suppose
them to be related by the equation 172 = <f> (rji) ', then, if in (26) we
substitute for y, first 771, and then rj 2 or (which is equivalent)
<$>(r]i), there will be two equations from which may be eli-
CLOP
minated, and the order of the resulting equation will be only
the (n l)th. Thus suppose 771 and 7? 2 to be two particular in-
tegrals Of
and suppose them to be related by the condition rjirj 2 = I ; then
we have ^ # l a *
~d^~~ "' ^- ~ =
77! =
342.] There is another property of linear differential equa-
tions which we must not omit. If n particular integrals of a
differential equation, which is without the second member, are
known, the coefficients of the several terms are functions of
these integrals, and may be found by a process analogous to
that of forming an algebraical equation whose roots are given.
456
LINEAR DIFFERENTIAL EQUATIONS.
[342.
Let the differential equation be of the wth order, and of the form
d n y d"~ l y d n ~ z y dy
dx^ '!<*-' '*dx-^ - l dx^
and let the n particular integrals be rji, 772, ... r\ n .
Substitute in (64) for y,
(64)
then we have
n(n
1.2
... +
dx r
n
+
= 0. (65)
v
Now, observing that the coefficient of lv\dx-= 0, and dividing
through by r\ l} we have
d n ~ l Vi ( n drji
dx n ~ l ( r/i dx
d n ~ 2 vi
+ ... = 0.
(66)
Let -yl + PI = Q!, and let the coefficients of the succeed-
ti ax
ing terms be Q 2 , Q 3 , ...Q n _i : so that (66) becomes
(67)
Now of this equation the (n 1) particular integrals are
d r]2 d 773 d rj n
~~3 * ' J ~~7 * ~" ~~ 9 r j ' * " y V /
dx 771 dx 771 dx rji
let us therefore repeat in (67) the same process as that to which
(64) has been subjected ; then if the successive coefficients of
the transformed equation, which will be of the (n 2)th order,
are RI, R 2 , ...R n _ 2 , we shall have
n 1 d d r)z
d 772 dx dx rji
dx 771
342-] LINEAR DIFFERENTIAL EQUATIONS. 457
n dr]i n l d z n 2
.-, PI = -- -yii -- - - .^L + R; (69)
r)i ax a 7/2 ax 1 T\\
dx rji
and by continuing a similar process in the equation which in-
volves R!, R 2 , we shall find an equation whose order is the
(n 3)th, and shall be able to express p a in terms of others of
the original particular integrals : and so on, until finally we
arrive at a value of PI expressed wholly in terms of the rfs.
By a process exactly similar, the other coefficients of (64) may
be found in terms of the particular integrals. And thus in
general, if f\(x), F 2 (^), ...F n (o?) are n functions of x y and it is
required to determine a linear differential equation of which
these are n particular integrals, we can determine the coeffi-
cients of it in terms of the particular integrals. This case is
plainly analogous to that of the formation of an algebraical
equation of which the roots are given.
In illustration of this process let it be required to form the
differential equation, of the third order, of which three particular
integrals are ^ = ^ ^_ ^ % = ^
Let the equation be
Let y = e a ^ x I v-tdx ;
then, as e a ^ x is a particular integral of (70), the coefficient of
\Vidx vanishes, and the transformed equation is, after division
by e a i x ,
P,)*! = 0, (71)
of which two particular integrals are, by reason of (53),
(aa-aOc^-"!'*, (a s ei fl s-i> ; (72)
let therefore /*
v l - (aza^e^-^^vzdx; (73)
then substituting in (71), and observing that the coefficient of
tVidx vanishes, we have
-^ + (*i + a l + 2a 2 )v 2 = 0; (74)
PRICE, VOL. II. 3 N
458 LINEAR DIFFERENTIAL EQUATIONS. [342*
of which, by reason of equation (44), e^' *** is a particular
integral ; therefore substituting we have
PI= (ai + 02 + as), (75)
substituting which in (71) we have
+ (2a 1 -o 2 -03)-^ + (ai 2 -2aia 2 -2a 1 03 + P2)fli = 0; (76)
and of this e (a *~ a i )x is a particular integral : therefore substituting,
p a = o 2 o 3 + 0301 + 0102, (77)
and substituting in (70) for PI and P 2 , and noticing that e a * is
a particular integral of (70), we have after substitution
p 3 = 010303;
therefore equation (70) finally becomes
d 3 y d^y dy
-jjs (0! + 03 + 03) Z + (0203 + 030! + 0102)^ a l a z a 3 y = 0.
And this equation might also have been found as follows :
Since e a i*, e **, e a ^ x are particular integrals, we might substi-
tute these in it, and thereby obtain these equations,
= , (78)
of which three cubic equations 01, o 2 , 03 are evidently the roots :
therefore
PI = (01 + 02 + 03),
P3 = OiO 2 O3.
Similarly let it be shewn that the equation, of which par-
ticular integrals are x~^ and a? 2 , is
SECTION 3. Integration of linear differential equations of the nth
order, whose coefficients are constants, with or without second
members.
343.3 '^ ne investigations of the last section shew that the
integration of an equation of the linear form with the second
member depends on that of the same equation without the
343-1 LINEAR DIFFERENTIAL EQUATIONS. 459
second member, and on a multiple integral the element-func-
tion of which involves the second member : in the present and
the future sections therefore we shall, if it be convenient, con-
sider properties of linear differential equations, without the
second members, and I would have the reader observe that the
generality of the investigation is not affected thereby. There
are many processes of solution, which shall be considered in
order. The general type I shall take to be
d n y d n ~ l y d n ~ z y dy ._.
5J +A 'S^ +Aa S^+- +i "-'J + A !' = x ' (79)
where AI, A 2 , ... A n are constants and x is a function of a?.
FIRST METHOD. Expressed by means of Lagrange's notation
of derived functions, (79) becomes
y() + Al2 ,(-l) + A22 ,(-2) + ... + An-ltf+Any = X, (80)
and introducing certain undetermined constants &, 0", &",...
Q(n-V f we ma y p u t (80) in the form
= x; (81)
and let us make the following substitutions ;
'Of n nfi s\n' tfl> a a?'
AI (7 = I/, A 2 v = P C7 , AS -^ P =t/C/ ......
(83)
so that (81) becomes j
(85)
and for let a be substituted : then from (83) we have
a* + A 1 a- 1 + A 2 a- 2 +...+A n _ 1 a + A n =/(a) = 0; (86)
the resemblance of which to (79) in its powers and its coeffi-
cients is evident ; and as we shall hereafter refer to this equa-
tion, it is convenient for it to bear a particular name : let it
therefore (according to a received nomenclature) be called the
characteristic equation of (79).
Now suppose the n roots of this equation to be unequal and
to be 01, a 2 , ...a n ; then there are n different values of (85), viz.
3 N 2
460 LINEAR DIFFERENTIAL EQUATIONS. [344-
e a i*| je-wxdx + Ci}, e^ x { e-^xdx + Cz \, ... (87)
which may be denoted by r/i, r]z,...t] n ; also let the values of
8', 6",...0("-v corresponding to these roots be #/, 0i",...0^ n -^,
02, 02",...02 (n - l \...0 n (n - 1 \0 n (n - 2 \...0 n (n -v> then from (82) we
have the following series :
Now on referring to Art. 150, it will be seen that this series is
similar to that marked (33) ; and that therefore
S.+ifrftfr-qftfr-q...^-!)!
" 2 . + 0S n -V 2 (-2) <? 3 (-3) . . . ^ (n _ 1} 1 '
and that the values of y ' , y",.*.y (n ~' 2 '\ y (n-1) are similar in form.
But the value of y given in (89) when expressed at length is
of the form
+^n^n, (90)
where AI, A 2 , ...A ra are constants and functions of the 0's, and
which are assigned by (89)^ but which it is easier for us to dis-
cover by the following method.
344.] Let us for the sake of a concise notation represent
(90) thus; _
y = 2,.A m ?7 TO ,
where 2 indicates the sum of a series of terms found by giving
successive values to m from 1 to n ; then
y = 2.\ m e>*e-*>**xdx + c m , (92)
2.A m x, (93)
y' = 2.a m \ m r) m + 2.\ m x; (94)
and observing the remark made in the sentence following equa-
tion (89), that y must be of the same form as y, and as this
can be the case only when 2.A m x = 0, and therefore when
S.A m = 0, we have
y = 2.a m A m i7,,,,
and therefore after differentiation
y" = 2.a w 2 A m 7 ?TO + 2.a OT A TO x ; (95)
344'] LINEAR DIFFERENTIAL EQUATIONS. 461
and as y" must also be of the same form as y, 2.a m A m x = 0,
' y" = 2.a OT 2 A m i7 m ;
and so on, until ultimately
y ln ~v = 2.a m n - l X m rj m + 2.a m - 2 A m x, (96)
whence we have 2 . a n- 2AmX = ,
and y^ = 2.a m "\ m r] m + S.a^A^x ; (97)
and as these conditions are to be accordant with equation
(79), we have after substitution
+ 2.a m "- I A m x = x; (98)
but each term of the series comprehended within the symbol of
aggregation vanishes, because 01, a 2 , ... a n are the n roots of the
characteristic equation, and therefore we have
= x,
(99)
Hence we have the following equations for the determination
of A 1? A 2 ,...A, ( ;
... + a ra A n =
+ a 2 2 A 2 + a 3 2 A 3 + . . + a 2 A n = }> . (100)
Now consider the derived function of the characteristic equa-
tion (86)
/'(a) = ( a a 2 ) (a a 3 )...(a a n ) + (a ai)(a a 3 )...(a a n )
+ ... + (a aO (a a 2 ) ... (a a n _ a ), (101)
' /'(ai) = (i 02) (ai a 3 )...(ai a n ) 1
/'(a 2 ) = (a 2 a a ) (a 2 a 3 )...(a 2 a n )
f'(a n ) = (a n ai)(a n a 2 )...(a n a n _!) J
Of these equations let us take the first to be the type : it is
plain that it is of n 1 dimensions in a\, so that
/'(oi) = a.i n - l + C l a l n - 2 + c 2 a l n -*+...+c n _ 2 a 1 + C_ l , (103)
Where c b c 2 , ... c n _ a are functions of a 2 , a 3 , ... a n ; and let us mul-
tiply equations (100) severally by c n _ a , c n _ 2 >."Cb 1 and add
462 LINEAR DIFFERENTIAL EQUATIONS. [345.
them : then the coefficient of \i is /'(ai), and the coefficients of
A 2 , A 3 , ...A n vanish, because (103) vanishes by virtue of the first
of (102) when ai is replaced by a 2 or a 3 ... or a* ; and therefore
ultimately we have
X lt f(a 1 ) = l, A =/^)5 (104)
similarly may it be shewn that
" Xn= ' (105)
and therefore the general integral of (79) is
y = 2. 7 rre x {c m +[e-**'&.dx},
J \O-rn) >
and including the constant factor in the arbitrary constant C OT
we have
y =
r go** r
\e-* x -x.dx + ... + 77? r /e-*x<fo. (106)
^ OnJ
345.] Such is the general integral of the differential equation
(79), when all the roots of the characteristic are unequal. And
if x = 0, that is, if (79) has no right-hand member, then
y = C 1 e a i* + c 2 e a 2 ii; + ... +c n e a *, (107)
an expression which is easily verified by means of substitution
in (79), and each of the terms of which is a particular inte-
gral ; and as all are different, n different arbitrary constants are
contained in it, and the integral is therefore general; and the
form of (106) indicates that the general integral is the sum of n
particular integrals, each of which involves or may involve a
different arbitrary constant.
If there are pairs of impossible roots in the characteristic of
(79) they enter as conjugates : suppose a pair to be cij, a,- : so that
= abVl,
= e a *{ (Ci + Cj) cos bx + (Cj cj) */^-\ sin bx]
ke ax cos(y + ba?), (108)
if c< + Cj k cos y, (q Cj) */ 1 = k sin y ; and where of course
345-1
LINEAR DIFFERENTIAL EQUATIONS.
463
k and y are possible quantities. In the case therefore of a pair
of imaginary roots, two terms of (107) will in combination pro-
duce a trigonometrical function of the form (108), and instead
of the arbitrary constants c t and Cj we have the new constants
(equally arbitrary) k and y. And a similar process of combina-
tion is also applicable to the latter unintegrated terms of the
general expression (106).
I may by the way observe that, if 77 = e"*, the multiple inte-
gral on the right-hand side of (45) gives after reduction a series
of the form (106). An example is subjoined :
S
whence we have
11 a 2 -Q" = Off
- 6 a 3 = 60"
~+6r) = e" 1 *,
6a*y = e mx . (109)
(110)
(111)
= 0,
(112)
e=a, = 2a, = -3a; (113)
and therefore in accordance with equation (86)
/(a) = (a-a)(a-2a)(a-3a),
/'(a) = (a 2a)(a 3a) + (a 3a)(a-a) + (a a)(a 2a),
/ 7 (3a) = 2 a 2 ,
. y =
Ex. 2.
3a)
(114)
T~
= cos nx.
464 LINEAR DIFFERENTIAL EQUATIONS [346.
Let + * = u
a- =
dri
.-. -r 1 -f 6rj = cos nx
dx
= a V 1, = a </ 1,
/(a)0-f*,
/'(a) = 2a,
COS/M7
= cos
cosnxdx, (116)
where k and y are two arbitrary constants.
346.] In the preceding investigations we have, at least tacitly,
supposed all the roots of the characteristic to be unequal : for
if two or more of them are equal, the value of y, as expressed in
(89) and found by elimination from the group of equations (88),
becomes indeterminate, and the subsequent processes of Art. 344
fail. Or, to take a particular case, let us suppose two roots to
be equal, say a 2 = ai, then the terms corresponding to these
two roots become
and thus the two particular integrals will introduce only one
arbitrary constant, and the general integral will contain only
n 1 different constants : and therefore its generality is lost.
Let us return then, and suppose m roots, a\, a 2) ...a m , of the
characteristic to be equal, that is,
<*i = 02 = = o. m ;
and, for the sake of simplicity, I will consider a differential
equation which has no second member, and observe that the
generality of the process is not lost by the restriction.
First, let us suppose
346.]
then
WITH CONSTANT COEFFICIENTS.
465
(117)
...}, (118)
if c'=
C"=
c (m) =
1 O A . 1N
J. ./... ^//t -f- 1 )
(119)
Of these equations let us take the first m to determine the
new constants c', c", ...c (m) ; and then let us suppose i = 0, so
that all the subsequent terms vanish, and the m roots of the
characteristic become equal ; and thus ultimately for the gene-
ral integral we have
y = {c / + c"a?+...+c^ wt }e a i- F + c M+1 e a -n ;p +... + c n e a - ;c ; (120)
thus if two roots of the characteristic are equal
y= {c' + c"#}e a i- r + c 3 e a ' r +...+c n e a ' r . (121)
Or let us consider the case of equal roots in the following
manner : and this is perhaps more direct.
Let the equation be
2 ^> + A 1 ^- 1 > + ...+A w _ 1 y' + A n y = 0. (122)
Let y = ue ax ; (123)
where a is a constant and M is a function of x ; and substituting
in (122), and assuming
o + A 1 a- 1 + A a o- 8 +... +A n _ia + A n = =/(o),
we have
Now this equation is satisfied if u = a constant, and f(a) = 0,
that is, if for a we substitute one of the roots of the character-
istic : let then ai be substituted for a, and Ci for u, so that (123)
becomes y _ c go,*
which is a particular integral ; and in the same way may the
PRICE, VOL. ii. 3 o
466 LINEAR DIFFERENTIAL EQUATIONS [347.
other particular integrals be found, and hereby, the general
integral. If however two roots of the characteristic are equal,
say a 2 = a b then f(ai) 0, f'(ai) = 0, and (124) is satisfied
when d z u ^
~dtf - '
u c' + c"#;
.'. y = (c f + c"x)e a i*.
And similarly if m roots of the characteristic are equal, it is
necessary that ^ m . u
~
u = c +
and thus we have the form of the general integral when m roots
of the characteristic are equal*.
347.] SECOND METHOD. I propose to apply to the solution
of linear equations with constant coefficients the process of suc-
cessive reduction which has been investigated in the last Sec-
tion. Taking (80) to be the type, let
y = e ax lu^dx, (125)
where a is an undetermined constant, and u\ is a function of x :
and let us as heretofore suppose
a n + A 1 a w - 1 + A 2 a"- 2 +...+A n _ 1 a + A n = /( a ) ; (126)
then substituting (125) in (80) we have
/""Ha) d-* Ul /"(a) d-Hn _
*" 1.2.3.. .(n-1) dx"~* + 1.2...rc dx n ~ l '
Now as a is undetermined in (125) and (127), let us suppose
it to be a root of (126), say a = ai, so that/(cti) = 0, then the
first term of the left-hand member of (127) vanishes, and there
remains a differential equation of the (n l)th order: and ob-
serving that /"(a) = 1.2.3...( l)w, it is of the form
* A more general investigation of the form which the result takes when
many roots of the characteristic are equal will be found in Moigno's Calcul
Integral, Vol. II. page 608. Paris 1844.
347-1 WITH CONSTANT COEFFICIENTS. 467
Now supposing all the roots of the characteristic to be un-
equal, there are n different equations of this form corresponding
to these roots, a\, a 2 ,...a n ; also to solve (128) let
M! = e?*ju z dx; (129)
substituting which in (128) we have
and expressing the first term of the left-hand member in the
following form, and adding /(cti), which is equal to zero by
reason of 01 being a root of (126), we have
; (181)
= (133)
by reason of the form of (126) ; and therefore a x + /3 is a root of
(126) : let this root be o 2 , then ai + /3 = a 2 , and ^3 = 02 01;
and as (132) is an algebraical equation of (n 1) dimensions,
the other roots are 03 o l5 ... a n a\ ; let these be represented
by /3i, J3 2 , ...'j3 B _ij and as a n _ 8 is evidently unity in (131),
(131) becomes
- ~
-"- < 134 >
Again, let
u z = t
and pursuing the same process y = /3 2 /3i = a 3 a 2 ; and as
the equation for determining y will be of n 2 dimensions, the
other roots will be a 4 o 2 , a 5 a 2 , ... c^ o 2 ; and the differential
equation for determining w 3 will be of the form
Cv U (M U$ dU^ _ a j; /1QP\\
302
468 LINEAR DIFFERENTIAL EQUATIONS [347-
Again, let /
% = e s *Ju 4 dz; (136)
and the equation for the determination of 8 will be of n 3
dimensions, and its roots will he a 4 a 3 , a 5 a 3 , ... c^, a 3 ; and
we shall continue the processes until we ultimately arrive at
tt,,_! = et*i>*u H (to, (137)
u n = xe~ a * x ; (138)
and thus, returning through the several steps,
y = e*i* I (*-**>* dor <*-*>* dx I .. .e^-^-i^dx xe- a x dx', (139)
and as a constant is to be introduced at each successive inte-
gration, it is manifest that in the course of the process n such
will be introduced, and therefore that the integral is general.
And the general form of it is
y = c' e a i* + c" ev x + . . . + c (n) e a * x
-^dx. (140)
.
If x = 0, that is, if the given differential equation has no
second member, then
y = c' 6*1* +c"es*+... + c <*>*. (141)
An examination of the form of the constant which will be
introduced at the several integrations of the multiple integral
in (139) shews that the result is of a form precisely the same
as that indicated in equation (106).
I may observe that this method of solution is the same as
that investigated in Art. 335, but the general form of that Ar-
ticle is too complicated to be of useful employment, and there-
fore I have chosen to give a special inquiry.
Should there be a pair of imaginary and conjugate roots in
the characteristic, the corresponding result may be reduced to
a circular function.
This process is also applicable when two or more of the roots
of the characteristic are equal ; also the general result in equa-
tion (139) holds good. Thus suppose all the roots to be equal,
then rn
y = e"*! e-^xdx";
and the several integrations will plainly introduce n arbitrary
constants.
348.] WITH CONSTANT COEFFICIENTS. 469
348.]
l*tl/ *l*/ tt <</ ltd.
The characteristic is
.-.5 1 Q 3 i OA 2 i GO ~ i 1 /"M A .
Ct ^~ J. O Ct H~ <CO Ct "7- O<v Cl -j- X V/rt ^^ v/ 1
.-. a = 1 + y^Tl =3 + 2y^I| =4.
Therefore by (141)
*" ^ & /*/"4Q / 'J* I *\t \ _j_ Z a&X r*f\G
A. i c CUo \& -p /l^ "f" 2 " i>US
Ex.2.
The characteristic is
a 3 -7aa 2 + 160 2 a-12a 3 = 0;
of which the roots are 20, 2, 3 ; and therefore by virtue of
y = (c' + c"x)e za
c? n y d n ~ l y n(n
+ a " 4 "
, i
..-I - - -a n ~ z - + -a n ~ l - + a n v = 0.
1.2 dx*^I dx^
Of which the characteristic is
j 1\
n = 0,
and of which the n roots are equal, and each is equal to a ; so
that (120) gives
y = [c r + c"x + ... +cW
The roots of the characteristic are 3 a, a, 20; therefore by (139)
y _ e -3a* e *a*dx le^dx le (m -^
[ r
_ e -3a* e^dx e
J J
/
e
- ~
m2a
Ci o(ma)x
e*+ - - - - dx
a (m a)(m 2a)
r f f 2ax i r "aax i r'"t> 3
470 LINEAR DIFFERENTIAL EQUATIONS [349.
The roots of the characteristic are equal, and each is equal to 1 .
Therefore (139) becomes
y = e* lxe~ x dxdx
EK.6. + = .
The characteristic is a 2 -f- a 2 = 0,
and therefore the roots of the characteristic are a */ 1, a -%/ 1 .
y = e -a.>J~\x\ e ia.>J^\x ( i x e~ a '^-' lx ^.dx,
whereby when x is given the general integral can be found.
Let x = cos nx =
y= -
i a) A/ 1 ( + ) A/ 1)
COS 7Z#
= A: cos (ax + y) 5 5 .
/jt,2 a 2
1
T . rt/%a /j-x 1 facucv 1 _i_ /> ax v 11
.iJC/L A. C^Oo ttt*/ ^^ T^- "S c ~f~ c j j
,-, r /-, ( x e -w>j~\\
_ e -a</-Ix e 2aV-lx ) c i (.
j 12 4 tt y_i$
^ sn aa?
349.] Let the right-hand member of the equation contain a
constant only, so that the equation is of the form
d n v d n ~ l y dy
-r- 4+-.. +A M _!-+A M y = A, (142)
-
then it may be expressed as follows :
350.] WITH CONSTANT COEFFICIENTS. 471
n n ~ lt u dy
^
Now replace y by y -\ -- ; then (143) becomes
A n
d n y d n ~ l y dy
5F + A '^ + - +A "-'S + A " ! ' = ' (144)
and is therefore of the form which has been discussed above.
^
In the final result we shall have to replace y by y -- ; and
A n
therefore if ai, a 2 , ...a n are the n roots of the characteristic of
(144), A
y = -- h Cie a i* + c 2 e a 2*-f ... + c n e a *. (145)
A-n
Also from (139) we shall derive the same result. Let x = A,
then r r r r
y = e a i x le^~ a d x dx\e( a *~ a 'i >x dx\ ...e (a *-i- a J x dx\A>e- a x dx
... +c n e a
( ) n aia 2 ...a n
In the cases of the characteristic having impossible roots,
and having equal roots, the results are similar in form to those
investigated above.
.-. y e a i x je^~ a ^ x da? ik*e~ a **
r
_ e ai* I e (a. 2 -o.\
a 2 i
k 2
dx
i ao.' v ///y
1 r M-cc-
1
X _J_
a 2
k 2
350.] THIRD METHOD. By the calculus of operating symbols.
We shall, as heretofore, assume (79) to be the typical form
of the equation, whose integral we shall investigate. Now each
term of the left-hand member which involves differentiation is
subject to the laws of repetition and commutation, and the sum
472 LINEAR DIFFERENTIAL EQUATIONS [350.
of all is subject to the distributive law* ; and therefore we may
place the subject of these operations outside of the operative
symbols, and express the differential equation as follows :
d n ( d n ~ l d
and which, for the sake of a concise notation, we may express as
/(|)y = x. (148)
Where f(-f-) expresses an operation to be performed on y, and
is such that when performed on it, it changes it into x. If
therefore we perform on both members of this equation the
operation which is inverse to/(-r-j, the left-hand member be-
comes y, and we have
and it is this operating process which I have now to investigate.
Now /(-3-j is evidently an algebraical expression of n dimen-
sions in terms of -=- ; suppose it to be resolved into factors, and
CttX/
the corresponding roots to be ai, a 2 , ... c^,, so that
< 150 >
then, as such operating symbols are subject to the law of repe-
tition,
and the right-hand member is a rational fraction in powers of
; and we can therefore decompose it into a series of simple
fractions, according to the process explained in Chap. II, Sect. 2,
of this volume.
First suppose all the roots to be unequal ; then by (27) Art. 19,
1 1 (d \~ l ltd
* See Art. 364, Vol. I.
350.] CALCULUS OF OPERATIONS. 473
Therefore, introducing the subject of the operative symbols,
d
But by equation (34) Art. 369, Vol. I,
e** e-^xdx; (154)
and substituting this in (153),
i
-f- -Z-,
r
-Z-, - 2* le~ a
f (a z ) J
+ srr^ ea ** e ~ a *** d *'> ( 155 )
/ () J
an expression which involves n signs of integration, and there-
fore n arbitrary constants ; and if these are introduced the
result becomes
1C If
^ eWe-w x (& + -& ewle-^Jidx + ...
(<*i) J (02 J
which result is identical with that marked (106).
If there is one pair, or are many pairs, of imaginary roots, we
may transform the expression according to Article 345. Thus
suppose cii and a/ to be a conjugate pair of imaginary roots
a, = a + bV 1,
then erf"** + Cje'j* =
e^kcaatfx + y), (157)
where k and y are two new undetermined constants ; and if
(158)
/'(a,-) = M N \/ 1
PRICE, VOL. II. 3 P
474 LINEAR DIFFERENTIAL EQUATIONS. [351.
g
77
f
ga-i x r e a } x r
then 7- e~ a i*xdx + 37 Ic-
(*i)J J (aj)J
2e ax (L cos bx + M sin bx)
L" + M'
) f _ aa
'-le- ax smbxdx\ (159)
and this again may be further reduced by substituting
L = r cos 0, M = r sin 0.
351.] Suppose however that m roots of /(-T-) are equal to
each other ; that is, that a x = a 2 = . . . = a m ; then, according to
Art. 21, if ^r(x) is equal to the reciprocal of 0(<), where <(#) is
the product of all the factors of f(-r-) short of the equal factors,
CLOO '
d
1.2.3. ..
- (160)
and to all these terms imagine the subject x to be affixed : then,
by reason of equation (33) Art. 369, Vol. I,
d \~ r C r
j -- a) x = e ax I e~ ax \dx r , (161)
/TO v/^'Cai) /*i-i
g-a^x^m 4. r v iy ei*^ c-
1.2.3...
+i x r e a * x r
r /rfAxfc + ... + 7=7^-, / ^x * ; (162)
+i^ J
and as the constants introduced by integration are arbitrary, in
the first m terms, wz, and only m, constants will be brought in,
and the remaining n m constants will arise in the other inte-
grations. If the roots corresponding to the sets of equal factors
are imaginary, the process of integration is the same; the result
352.] CALCULUS OF OPERATIONS. 475
however is so complicated that it is not worth while to express
it at length.
I may observe that the general result in (106) does by a
process of evaluation explained by M. Moigno, and referred to
in the foot-note of page 466, give a result the same as that just
arrived at.
352.] Ex.1. The first linear equation with constant coefficients.
dy
= e{/e :
Let x = x n ,
. y = ce^+e * e~ ax x n dx
n
nn l)... 3.2.1
'
( a a* a 6
Let x = e mjc ,
atlUf
y = C6 ax -i --
m a
Let x = e ax ,
y ce
Ex.2.
-
1 id \~ l 1 id v- 1 . 1 id
\~ , i , v-
-r ~ a ) x -- 2b-- 2a x
a# i?\dx
pax r gZax r fj3ax /"
y = ^5 g le-^x^dx -- f^-t^x^daf+'-s-s e~
litt J CL J lid, J
whereby the result is dependent on simple quadratures.
3 p 2
476
LINEAR DIFFERENTIAL EQUATIONS.
[352.
T2
= e Zx e-
Ex - 4 '
/ o \
y = [ -j 2 a ) sin W.T
W.r
,nx*/ I
/2
e -
, nxV^i
2ax
y =
Ex.5.
(4 a 2 w 2 ) sin w# + 4 a# cos
a
gaVix r g aVi.r r
and a constant must be added at each integration.
Let x = ;
ni g-i ad *v \X I f** a Q, *J \X
y v^l C *T" l^o C;
= k cos (a# + y).
Let x = cos mx ;
cos ?w
V = -= s + * cos (a.r + v).
353'] CALCULUS OF OPERATIONS. 477
Let x = cos ax ;
x sin ax cos ax
2a 4a a
x sin ax
. a
A cos (ax -f B).
353.] The preceding process, it will be observed, involves
operations represented by symbols of the general forms
ft v ~~ T / //2 \ T / riffo \
--a) x, -f- 2 - 2 *.(;!=-") x >
flb? / V//.77 2 / ' W.7?" / '
-n\ v (_ n*\ Y ^^ ^m 1
\dx
where r is unity or some other positive and integral number;
and as the operation which such a symbol represents is subject
to the laws of distribution and of repetition, we may expand
the operative symbol, and operate on x with the several and
ld_
\dx
pressed in either of the following forms :
(d \~ r
~j \- a] may be ex-
d\~
dx ^ 1.2
the former of which involves integration only, and the latter
differentiation only : and as integration introduces arbitrary
constants, and differentiation does not, it may be thought that
the latter expansion is inapplicable ; it may however always be
employed, provided that we take care to introduce the arbitrary
constants, or the supplementary function which they are in-
volved in ; and this we may do as follows :
/ d x-r /V
But \ + ) = e~ ax Odx r
= e- a *{c 1 + c 2 x + c 3 x 2 +...+c r x r - 1 }. (163)
Similarly,
478 LINEAR DIFFERENTIAL EQUATIONS [353.
+ -
Of which expression the latter part is equal to
~ "
1.2.3.4.5
^
= G! cos a# -f - sin ax. (165)
Other forms of operative symbols may also be expressed in
terms of differentiation ; and as that to which x is affixed always
admits of such an expansion, we infer that if we can integrate
a linear differential equation when the right-hand member is
equal to zero, we can by means of differentiation only find the
integral when the right-hand member is a function of x.
Ex.1. 8.
d
1.2
(/i a)
Ex. 2. -r4 4- w 2 y = cos ax.
CLOC
354-1 VARIATION OF PARAMETERS. 479
T *> J. N
( n z n* )
y = s 5- r ^ . . . > cos ax + c cos ax + c 2 sin ax :
( a* a 4 a 6 }
the last part being concluded from (165).
cos ax
The form of each example however will generally suggest the
process most convenient for this solution.
I regret that I cannot enter more deeply into the process of
solving differential equations by the calculus of operations, and
that I cannot insert a complete analysis of (1) Mr. George
Boole's paper on a General Method of Analysis, Philosophical
Transactions, 1844, (2) Mr. Hargreave's papers on Differential
Equations, Phil. Tr. 1848 and 1850; the want of space alone
hinders me ; the papers are most valuable, and exhibit in all
their breadth the comprehensiveness of the theorems of the
new calculus, and their applications to questions of the integral
calculus : other information on the same subject will be found
in the works mentioned in Art. 370, Vol. I.
354.] I must not conclude this section without a few words
on a method invented by Lagrange, and now called " The me-
thod of variation of Parameters," by which he deduced the
integral of the linear differential equation of the form (79) with
a second member, from the general integral of the same equa-
tion without the second member.
Let the two equations be
y (n) +Aiy (n ~ 1) +A 2 y (w ~ 2) + ... -f A. n -\y'-\- A. n y --- x, (166)
/ M \ /,. -i \ /._ o\ / /-\ r ~t /?fV\
z (n) -f AI z (n " l > + A 2 z (n ~ J > + . . . + A. n _i z + A. n z 0; (Io7)
and suppose Zi, z 2> ... z n to be n particular integrals of (167), so
that the general integral z is
z = c\z\ -j- -02^2 + +c n n ; (168)
then it is always possible to determine n functions of x, u\, u 2) ...
u n , so that the general integral of (166) may be
y = Wi^iH u 2 z% + ... -f u n z n , (169)
- y i/? (170}
j-~i //.** * V * /
that is, the integrals of (166) and (167) are of the same form,
but the arbitrary quantities Ci, c a , ... c n , which are constant in
the integral of (167), are functions of x in that of (166).
Suppose therefore that (170) is the integral of (166) : differ-
entiating we have
480 LINEAR DIFFERENTIAL EQUATIONS.
dy dz du
-f- = Z.u-^ + Z.z^; (171)
ax ax ax
(I*)!
and moreover suppose that -jf- is of the same form in (168) and
(169): then
2.z^=0. (172)
dz
Differentiate again (171) subject to this condition : and we
and again suppose -~ to be of the same form in (168) and
. , =
and continuing the same process, and making similar substitu-
d n ~ l y
tions up to - -- 7 , we have
dx n ~
^d^du_ d*zdu_ d-*zdu_ _
*'*~ -"~~- ">'~ *"-*~
d n ~ l y d n ~ l z
~
(176)
and substituting these values throughout in (166), we have
da?*- 1
Now of the expression on the left-hand side of this equation,
the first part vanishes by reason of z\, z<>, ... z n being particular
integrals of (167) ; and therefore
2 .^!I^^ =X . (178)
(*7 (.vuU
Hence (169) is the general integral of (166), the values of the
w's being found from the following system of equations :
\- Z U$ -p ... -(- Z n U n "^ U
Zs'Us ' + +ZnUn =
| (179)
354-3 VARIATION OP PARAMETERS. 481
These equations will of course in general give n different
values of M/, M 2 ', . . . tt n ' in terms of the z*s and of x, and each
value will have x as a factor ; suppose the other factors to be
i, v 2 , ... v n , so that
Ui = GI -f IV
= Vi x
= c*+fi
>; (180)
u n =
u n = c n + \v n
and substituting these in (169), we have the general integral of
(166).
Now in this process we have made no restriction as to the
coefficients of the given differential equation; they may be
either constants or functions of x : suppose however that they
are constants, so that if ct l5 a 2 , ... a n are the roots of the charac-
teristic of (167),
Zi e***, Zz = e a ^ x ) ...... z n = e*** ;
and these must be substituted in the series (179); and thence
may be deduced the values of v\, v^,...v n which are required
for (180).
Let us take an example of this process j
z =
y =
l2 .
dx dx ax
Let ^^i + ^^i^o;
dx dx
.. = 1 2 -
dx z dx dx
and substituting in the given example we have
dx dx
from which, combined with the supposition made above, we have
PRICE, VOL. II. 3 Q
482 LINEAR DIFFERENTIAL EQUATIONS [355.
a(m-2a)x
dx ' a a(m-2a)'
du 2 1 _
dx " a r a(m-8a)'
gWW
.-. w = Cie 2ax + c z e 3ax -\ -- .
^(w-2a)(-8a)
As another example of this process let us take the linear
differential equation of the first order with variable coefficients,
of which the general form is
+ = ,; (181)
where x and X! are functions of x. Consider
dz
z
y =
dx dx
and substituting these in the given differential equation we have
.. u = c + X
xte^ + fxieSx^dx}; (182)
.-. y = e-x
and this is the integral of (181).
SECTION 4. Integration of some particular forms of linear
differential equations with variable coefficients.
355.] The linear differential equation of the following form
admits of being reduced to one with constant coefficients by
means of a change of variable, and therefore its integral may
be completely determined.
B y = 0;(183)
355-1 WITH VARIABLE COEFFICIENTS. 483
and I may at once remark that if the equation admits of inte-
gration when the right-hand member vanishes, it may also be
integrated when the right-hand member is a function of x.
Let a + bx = z; and as x is equicrescent in (183) so will also
z be, and therefore after the substitution the equation is
so that the form of the equation is
n n ~ l
= - < 185 >
dx
Let = dt, .'. x e*;
x
dy fa
dt ' dx'
d*y _ d*y dx dy dx
~dt* " X ~dx*~dt^Tx~Tt
dx* dt '
_ ,d*y dx d z y dx
dt* ' dx 3 dt "* dx* dt "*" dt*
_
dx* dt* dt '
and so on; hereby may x-jr> x * IT*' '" ^ e ex P ressed ^ n terms
of ~, -~ ... and (185) will become a linear differential equa-
(It Lit
tion with constant coefficients.
"CV 1 7*2 " y i r ___/ u m
Ex.1. -^3+*^ y-
Let a? = e',
dy _ ^
rfa?~ rf/ J
2 d*y d*y dy
V dx* == ~di* ~ Hi '
'' ~di*~ y ~
3 Q 2
484 LINEAR DIFFERENTIAL EQUATIONS
/ d 2 \~ i
y = I 1 I p"l
\dt*
' m 2 -!
-' + -4 T
Ex.2. xz-3x + 4y = x
dx
Let x e l :
- ( m _2) 2 ^
x m
= (m-2) 2 " f "
356.] The form of linear differential equation which I shall
consider next is
. d"y ' d n ~ l v
a x} + (flo + b Q x) y = 0, (186)
where the a's and the b's are constant. Now as a priori there
is no reason why any particular circular, logarithmic, or alge-
braical function should be an integral of this equation, we must
assume an unknown function of a more general and transcen-
dental form : and such in all cases is a definite integral ; let us
suppose then - MI
y = I e ux \du, (187)
-S,
where u is a new variable independent of x, v is a function of u,
356.] WITH VARIABLE COEFFICIENTS. 485
and Ui and w are the limits of integration and are independent
of x ; and let us consider the result of the substitution of this
quantity in the given differential equation ; differentiating (187)
dv /*"'
-?- = ue ux \du.
dx -.' dx* '
~- = I u n e ux vdu\ (188)
///W f
and moreover let us substitute as follows :
[ , (189)
so that (186) becomes
Vo + Uix}e u * vdu = 0; (190)
and integrating by parts,
r i" 1 f Wi
e^uxv +/ {U O VM d.Vi\}e"' t ' = 0. (191)
L J(/ *A/
Now as v is an undetermined function of u, let us assume that
= 0,
v = e f ^ du . (192)
And in consequence of this assumption (191) becomes
[WnvT^O,
L _k
.-. [ce ll *" l " / 5*'T 1 ==.0, (193)
and therefore ,. MI Ufl
y = I e ux +J uj dM du ; (194)
but in this expression u\ and M O are undetermined ; they must
however satisfy (193) ; and as there will in general be no rela-
tion between them, each separately must satisfy it : and there-
fore we must discover the roots of the equation
e +/*te = .
suppose them to be u , MI, w 2 , ... % ; and if we take u to be the
inferior limit in all cases, and the others in turn to be the supe-
rior, then we have the following k values of y, viz.
486 LINEAR DIFFERENTIAL EQUATIONS [357-
f"' M *+/HQ dB dW
y = / de -S
u
(195)
and from the form of the equation it is plain that the sum of
these also satisfies the equations. If therefore it is possible to
find n + 1 such values of u, the resulting expression, of the form
(195), is the general integral of the given equation ; in other
cases it may be only a particular integral.
And I must observe that the definite integrals which enter
into the final result generally do not admit of further reduction ;
and hence we infer that the integral of a differential equation
of the form (186) is a transcendent of a higher order than any
of the commonly tabulated functions.
357.] Ex.1. (flz + M) + (ai + M)^ + (ao + *off)y = 0.
In the first place, for x write x ^ ; and substituting, the
Ov.
equation takes the form
= '
therefore from (189),
u =
du
i
? a) + Blog(w /3), (196)
if a and /3 are the roots of the denominators, and A and B are
determinate constants dependent on a\, OQ, b\ } b : so that from
(193) we have
e iuc (u a) A (u /3) B = 0, (197)
and this equation is satisfied by u = a, u = /3, u = oc ; and
therefore from (195)
C a C^
y = GI / e ux (u d) A ~ l (u fi)*~ l du + C 2 / e"^(M a) A-1 ('M 8) t> ~ l du, (IS
J- X /.
which is the general integral.
357-1 WITH VARIABLE COEFFICIENTS. 487
u = au, Ui = M 2 b 2 ,
/U , f au du
= / -5 F
U! J U 2 b 2
= |log(w 2 -^ );
therefore from (193)
c *( tt a_$2)f _ 0,
. . u = oo , =6, = b,
.-. y = ci/ e wc (u z -b^~ l du + c 2 \ e(v? b 2 fi~ l du; (199)
v 00 / 00
and the definite integrals do not admit of further reduction.
Ex.3.
In this example (193) becomes
M 3 OM
e^ + 36 + y = 0;
which is satisfied by u 3 = oo ; therefore u 3 + a 3 = 0, if in the
result a = oo ; and of this equation, if r is a primitive cube root
of 1, the roots are
a, ar, ar z ,
and therefore
f ar ux + ^ + rr* ux + + ^
y = C! e b 3& e?tt + c 2 / e b 3b du, (200)
J a. *'
and in the final result a = oc .
Therefore from (193) e a ( w+1 ) = 0,
.-. w +1 = -oo. (201)
i
If therefore the primitive roots of ( 1)" +1 are 1, r, r 2 ,...r n }
and a is a quantity which, in the result, is infinite, the roots of
(201) are
a, ar, ar 2 , ...ar",
488 LINEAR DIFFERENTIAL EQUATIONS. [357-
/
.'
r- r+ _
+c n e a < n+ Vdu. (202)
Ex. 5. As the last example of this method let us take equa-
tion (403) in Art. 327, which is equivalent to Riccati's equation,
and exhibit the function, which satisfies it, in the form of a defi-
nite integral. The equation may be put in the form
1 ' (2 3)
so that (193) becomes
e u*(v? b*) n =0,
.-. u = oo, = + b, = b,
? ux (u z b z ) n ~ l du. (205)
The complete consideration of this solution belongs to the
subject of definite integrals, and cannot therefore be introduced
at this part of our treatise. The reader desirous of further in-
formation may consult with advantage the Integral Calculus
of M.Moigno, Legon XXVIII, Paris 1844 : and Integration der
linearen Differentialgleichungen, von Joseph Petzval, Erster
Band, Wien, 1853, pp. 106108.
I may however remark that the integrals in (205) admit of
integration in finite terms whenever n is a positive whole num-
ber : and therefore since (see Art. 327)
m
n
2(m + 2)
4<n
whenever m
2n-I'
which is one of the conditions determined in Art. 326. And if
n is a whole negative number, then
m 4ft
.-. m = -,
~2(w + 2)'
and this is the other condition found in Art. 326. Hence arises
a reason why Riccati's equation can be integrated for these
values of m.
358.] DIFFERENTIAL EQUATIONS. 489
SECTION 5. Integration of some particular differential equations
of higher orders and degrees.
358.] As there is no general theory for the integration of
differential equations of all orders and degrees, we are obliged
to have recourse to artifices, which analysts have from time to
time discovered, for the integration of particular examples; I
propose therefore to examine these as concisely as possible and
in order; and hereby also we shall obtain a more exact knowledge
of the present state of the science. And firstly I shall take dif-
ferential equations of higher orders, where the highest derived
function is a function of either the one next, or the two next,
inferior to it.
Let/ n (#) be the highest derived function; then the problem
is, to discover the integral of the equation
(206)
Let /.-(*)=*; .-. /-!(*) = , /(*)
d 2 z idz \
' a? = '(*') < 207 >
and the equation becomes a differential equation of the second
order ; of which suppose the integral to be
z = <J(a?), (208)
*-, (209)
so that the final value of f(x) depends on the integration of a
function of x taken (n 2) times in succession. Some examples
are subjoined.
Ex.l. ? = , .-. = -.
dx z dx dx
where b is an undetermined constant.
dy dx x c
-
yb ' a '
where c is another undetermined constant.
Ex. 2. d 2 y = dx (dx 2 -f dy 2 )*, where x is equicrescent.
PRICE, VOL. II. 3 R
490 DIFFERENTIAL EQUATIONS. [358-
i = _ ipx-a e -(x-a)\
dx 2 {
y -b = ife'-a + e-^-")}.
Or we may integrate as follows : the equation is
dx* " ( dx*
It will be observed that in the former of the two methods we
have integrated first with respect to x, and in the latter first with
respect to y. The final integral also might have been found by
ftt/
eliminating -^- by means of the two first integrals.
= xc
,2 '
whence may y be found by integration.
358.] DIFFERENTIAL EQUATIONS. 491
d z y a
Ex. 4.
ax" x'
dy x x
-y- = d log i y = O, {X log
dx
-- l - z t
which is linear of the second order, and with constant coefficients.
Ex.6. r*=a + *> .
= 1, =TT = ^
dy a . 1,
~- = t&nakfxc), yo = -^ log sec ak(x c).
dx k k*
Ex.7.
Multiply both sides by 2 dy,
dx*
a
And I may by the way observe, that by this process all equa-
tions of the form
are to be integrated : viz. multiplying both sides by 2 dy,
3 R a
492 DIFFERENTIAL EQUATIONS. [359.
and of this the root must be extracted, and a subsequent inte-
gration performed.
359.] Secondly let us examine differential equations of the
// - f
cond order which involve -=-
dx
which are therefore of the form
// - ft i tJll
second order which involve -=-*fi -T-, and either x or y, and
dx 2 dx
Ex. 1. "" dx
a-\
i 2 ^- 4- ^-4-1 =
k (V. U , "1 TO 1^ A ^^ V '
where c and A; are the arbitrary constants of integration.
Ex. 3. a 2 d 2 y(a 2 + x z )* + a z dxdy
d z y dy 1 a? 2
which is a linear equation of the first order in terms of -j- , and
therefore may be integrated.
E,4. 1 +
360.] DIFFERENTIAL EQUATIONS. 493
tan- 1 ~ + tan" 1 a? = tan- 1 c,
c x
dx 1 + ca?'
whence y may be found.
Ex. 5. dx*dy-xdx*d 2 y = a
where s is the equicrescent variable.
ds z = dz 2 + dy 2 ,
= dx d 2 x + dy d*y ;
.'. dx*dyxds 2 d 2 y = ads 2 d 2 y,
dy
dy dy
dx ' dx '
dx
but as ^ becomes a transcendental function of x. the next
dx
integration cannot be performed.
360.] Thirdly, let us consider homogeneous equations of the
second order : the principle of homogeneity being as follows :
the variables x, y, and their differentials dx, dy, d z y are con-
sidered to be factors of the first degree ; and each term of the
equation is of the same degree in respect of them; thus the
equation, x 3 d 2 y (y dxx dy) 2 = 0, is homogeneous and of the
fourth degree. Now in such an equation let us make the fol-
lowing substitutions,
y = xz, ,' . dy = x dz + z dx, (210)
and it is manifest that x will enter in the same power into all
494 DIFFERENTIAL EQUATIONS. [360.
the terms, and therefore may be divided out ; this property in
fact is the characteristic of the equation ; and thus the result-
fj 1 ?/
ing equation will contain z, v and -j- ; let, for convenience of
di/
notation, -j- = p : therefore from (210)
Dm?
p dx = xdz + z dx,
dx dz
x pz
dp d 2 y v
andas "-
... = __ = (212)
x pz v
and v may be expressed in terms of * and p by means of the
given equation, and therefore by the last two members of the
equality we shall have a differential equation of the first order
in terms of p and z, whereby p may be expressed in terms of z :
and therefore from the first two members of (212) we shall have
a differential equation of the first order in terms of x and z ;
and this after resubstitution will give the required integral.
Ex.1. x 3 d z y = (ydx-xdy) 2 .
x % v = x z (yp) 2 ; .'. v = (zp) z ;
dp
therefore from (212)
, ,
p-z (p-z) z
p = z + I+ce*,
dx _ dz
x ~~ 1+ce*
e~ z dz
X Ci
log-^- = -log (c + e-*); .-. -
C i 3C
and this is the required integral.
Also differential equations which become homogeneous, if we
consider x to be of one dimension, y of n, -^- = p of n 1, and
dx
d z y
j-~ of n 2 dimensions, may be integrated by a process similar
dx*
to that above, by assuming
y = zx n , p = ux n ~ l .
dx*
361.] DIFFERENTIAL EQUATIONS. 495
It is to Euler that we are indebted for these processes ; other
examples will be found in his works, and in the ordinary collec-
tions of such problems ; and particularly in the Integral Cal-
culus of M. Moigno.
361.] And with two other examples I shall conclude this
part of the subject.
Ex. 1. Suppose that we have an equation of the form
where x and Y are functions respectively of x and y only : divide
through by -^ and integrate
log c -j- + / x dx + / Y dy = ;
' dy =. e~J
,j<-,
Ex. 2. Again suppose that there is given the differential equa-
tiou f(*,y,y',y",...y (n} } = o, (213)
and that its integral is
y = F(#,ci.c 2 , ... c n ), (214)
then y and its derived-functions depend not only on x, but also
on the values of the n undetermined constants ; but as may be
considered independent of them : suppose now that any one,
say c, of these constants varies ; then the variation of (213) is
XT , , dy dy dz dy" d z z
Now let ~ = z; then -- = -j- , - = 3-5, and so on ;
dc dc dx dc dx*
now from (214) y, y',...yW are functions of x and of Cj, c 2 ,...c n :
if then we substitute these in (213) (-}> (-/-,), become
\dy' \dy >
functions of x, Ci, c 2 , ... c w ; and therefore the coefficients of z
and of its derived-functions in (216) are variable, and the equa-
496 PARTIAL DIFFERENTIAL EQUATIONS [362.
tioii is linear ; and we know that a particular integral of it is
z = -~, because the equation was found by making -^ = z:
and as for the general value c we may substitute each of the
c's, so the general integral of (216) is
(217)
Let this process be compared with Art. 249.
SECTION 6. Integration of partial differential equations
of higher orders.
362.] The integration of partial differential equations of the
higher orders is surrounded with difficulties ; and only some few
cases have at present yielded to the powers of Analysis ; of those,
which are integrable, most arise in the more abstruse branches
of Physical Mathematics, and therefore the discussion of them
would be undertaken with inadequate means at this stage of
our treatise : we shall therefore pass them by ; and only intro-
duce in the two following Articles Monge's method of integrat-
ing those of a simple class ; and afterwards prove some proper-
ties of the most simple forms.
First then I shall consider Monge's method of solving linear
partial differential equations of the second order, which are of
the form
=
where R, s, T, v are functions of x, y, z, and the partial derived-
functions (-T-) and (-7-); and let us employ the symbols (69),
v
Art. 359, Vol. I, so that (218) becomes
R r + ss + Tt = v; (219)
where R, s, T, v are functions of x, y, z,p, q. By virtue of our
symbols it is manifest that
dp = rdx + sdy i .
dq = sdx -\-tdy )
and by means of these eliminating r and t from (219), we have
ndpdy \dydx + tdqdx s{tidy 2 sdydx + Tdx 2 } = 0. (221)
362.] OF HIGHER ORDERS. 497
Now suppose that
ndpdyvdyd.v + Tdqdx = 0, (222)
R uy" 1 s cty dx -j- T dx ^= u, (,-t^o)
then as we have also
dz = pdx + qdy, (224)
we may suppose that it is possible to satisfy these last three
equations by equations of the form
/ifoMrftf) : C O . (225)
and assuming this to be so, then
/i = F(/ 2 ), (226)
where F is the symbol of an undetermined function, will be the
general first integral of the proposed equation.
To prove this statement; let -^- = y' ; and let y' be the
general symbol of the roots of (223), so that (222) becomes
ndpy'vy'dx + Tdq = 0. (227)
Now taking the total differential of the first of (225), we have
/df\\ , ldf\\ , /dfi\ , /dfi\ , fdfi\ ,
I !=)<&? -f (~)dy + (-~)dz+ (4^}dp + (^)dq = 0; (228)
\dx' ^dy> \dz' \dp' \dq'
and substituting for dy and for dz and dq from (224) and
(227), we have
and this equation must be identical, because it satisfies each of
the three equations (222), (223), and (224) ; therefore
(&)
\d.v'
p) \dq' T
and we have similar equations in terms of/ 2 . Also from (226)
PHICE, VOL. II. 3 S
498 PARTIAL DIFFERENTIAL EQUATIONS [363.
and replacing dz by its value (224), and (-^j and (^ j by their
values from (230); and similarly replacing - and Hr
(231) becomes
which we may conveniently express
ny'dp + Tdqvy'dz = T(dy y'dx); (233)
and replacing dp and dq from (220)
(R^V + TS vy f + Ty f )dx + (Ry's + T!t T)dy = 0; (234)
and as x and y are independent variables, this equation must be
identical ' ' ' =
y = , .
-T = /'
and therefore eliminating r and t, and replacing in the proposed
equation (219), we have
{E^-sy' + T} = 0, (236)
and this equation is satisfied because y is a root of (223) : T
therefore has disappeared, and as that alone in (234) and (235)
involves P, the result is true whatever be the form of F : and
therefore (226) involves an arbitrary functional symbol and is a
general first integral of (218). Let us consider the above pro-
cess, when it is applied to the solution of some examples.
363.] Ex. 1. Let R, s, T be constant, and v = ; and sup-
pose the equation to be
In this case (222) and (223) become
dpdy + 6a 2 dqda? = 0,
dx z = 0;
.-. = 2a,
dx
from the former of which, y
also 2adp + 6a 2 dq = 0, dp + 3adq = 0,
363.] OF HIGHER ORDERS. 499
p -\-3aq = c a
similarly p + 2aq = f 2 (y3ax) :
.-. p = 3/ 2 (y-8aa?)-2/i(y
.-. z <f>i(y 2ax) + fa(y Sax),
and this is the general integral of the given equation.
In this case (222) and (223) become
dpdy atdqdx = 0, dy 2 a*dx z = 0,
dy dy _
dx ~ dx~
y = ax + c-L, y =
p aq = c/, p + aq = c/,
q=f l (y ax), p + aq = / 2 (y + ax),
z = fry
and this is the complete integral.
Ex. 3. q*r2pqs+p*t = 0.
(222) and (223) become
g 2 dp dy +p 2 dq dx 0.
q z dy 2 + %pqdxdy+p z dx z = 0;
.. qdy+pdx = 0,
pqdp+p*dq = 0,
dp_dq = Q
P 9
z = d, ^= c = (#>(ci) =
.-. z =f(y + cx)
This problem is the converse of that discussed in Art. 316,
Vol. I.
3 s z
500 PARTIAL DIFFERENTIAL EQUATIONS. [364.
364.] We can also find the integral of the following linear
partial differential equation of the nth order
=
where A, B, . . . K, L are constant. Let
d n z
n z \
- - = m n - l
n - l >
and let us substitute in (237) : then
Am n + um n - l + ... +KW + L = 0. (238)
Let the n roots of this equation be Wi, m 2 , ...m n ', and be
unequal; then
z = fi(y+m 1 x)+f 2 (y + m z x) + ... +f n (y + m n x), (239)
where fi,/2, "-f n express arbitrary functions.
If two roots of (238) are equal, say m z = /i, then
z = fi(y + m l ^+xf 2 (y+m l x)+f 3 (y+m^+ ...
...+/(y +*#); (240)
and the result is analogous, if three or more roots are equal.
Ex.1. b z r-2abs + a 2 1 = 0.
a
b 2 m 2 2abm + a 2 = 0. m = T ;
o
.' . z = fi (ty + ax) +xf z (by + ax).
365.] The calculus of operating symbols has also been
applied by English writers to the solution of linear partial dif-
ferential equations*. The process is in principle the same as
that heretofore applied to total differential equations ; and only
one peculiar extension of it is required, and that is the sym-
* See Chapter VI of Integral Calculus in Gregory's Examples on the
Differential and Integral Calculus, and edition, Cambridge, 1846; and Mr.
Hargreave's memoir in the Philosophical Transactions, 1848.
365.] CALCULUS OF OPERATIONS. 501
bolical form of Taylor's theorem ; see (27), Art. 368, Vol. I : for
we shall have such an expression as
, a.
e dy f(y),
and this may be replaced byf(y + kx) ; which, in the particular
case of f(y) being a constant or a function of x, remains the
same constant or the same function of x. As the method is
applicable to differential equations of the first order as well as
to those of higher orders, we shall consider some examples of
each.
d
5 + *
As the operations symbolized by -=- and -r- are independent
CLiV Cl'^l
of each other, when one is taking place the other must be
silent, that if -=- is variable, is constant, and vice versa ;
suppose, then, that -=- is variable, (241) becomes
a
when -T- is constant ; therefore by equation (34) Art. 369, Vol. I,
1 _ 6 d r> bx d
z -e' **/ e a d vcdx; (243)
a J b a
and remembering that the operation symbolized by e a dy when
performed on a constant produces the same constant ; and that
an arbitrary function of y must be added, because an ^-partial
integration is being performed,
1 _*! Jl
Z = 6~~a ~dy {CA
a
CX bx d
= +e
CX
cx
CL
502 PARTIAL DIFFERENTIAL EQUATIONS. [365.
If the operation expressed by -=- had been first performed,
then we should have had
cy
z = -
and which is equivalent in form to the former result.
Ex.2.
I d ud d\- id d
o--^-i(^ -- a -j-} (j- + flj-
2a dy \\dx dy' \dx dy
1
ax).
Ex.3. ^+y^=^-
dx dy
Let = du,
+ -j -- n )Z =
dv
d d
- + -j
u d
( d d \~ 1 K
1^- + ^ -- n)
\du dv '
= e un <t>(vu).
--/()
366.] GEOMETRICAL PROBLEMS. 503
CHAPTER XV.
SOLUTION OF GEOMETRICAL PROBLEMS INVOLVING
DIFFERENTIAL EQUATIONS.
366.] IN Chapter XIII, and especially in Articles 274 and
315, we investigated some geometrical problems which give rise
to differential equations ; and it was done with the view of illus-
trating the analytical inquiry: but now the subject must be
treated systematically, and it will be convenient to consider
such problems according to the order of the differential equa-
tion which they involve. We begin then with those of the first
order.
Ex. 1. Determine the equation to the curve, the subtangent
of which is equal to the excess of the ordinate over the abscissa.
doc
y = y-x, ydx + (x-y)dy = 0,
which is homogeneous and of the first degree; therefore intro-
ducing the factor discovered in Art. 288 we have
_ _
- l)U - "
where c is an arbitrary constant : and therefore the curve is an
hyperbola.
If the subtangent is equal to the excess of the abscissa over
the ordinate, then
dx x -
y = x-y, y = cev.
Ex. 2. Find the equation to a curve such that the area con-
tained between it, the axis of x, and any ordinate may be equal
to the abscissa divided by the ordinate.
x y dx xdy
-, ydx-- t --
= 0, .-. y*(c 2 a? 2
504 DIFFERENTIAL EQUATIONS.
Ex. 3. Find the equation to the curve in which the perpen-
dicular from the origin on the tangent is equal to the abscissa.
/ dy dx\
'- + ( af -7- y-j-) = *>
- V ds y ds>
(xdyydx? - x 2 ds*, (x*y*)dx + Zxy dy = 0,
where 2 a is an arbitrary constant;
.. y z = 2 ax a? 2 ;
the equation to a circle, whose radius is a.
Or, again, let us take polar coordinates
p = r cos 0,
du
- = tan dd, log 2 au = log sec 0,
Ex. 4. Find the curve, when the perpendicular from the
origin on the tangent is equal to the part of the tangent inter-
cepted between the point of contact and the foot of the perpen-
dicular. 222 202
p 2 = r z p 2 , .-. r z = 2p*,
du z du ,
log au = 0, r = ae e ;
the equation to the logarithmic spiral.
Ex. 5. Find the curve in which the tangent is equal to the
radius vector of the point of contact.
* i dy dx
y = ex, and xy = k 2 ,
that is, the required line is either straight or an equilateral
hyperbola according as the lower or the upper sign is taken.
367.] The class of geometrical problems called trajectories
involves differential equations of the first order and degree ; a
trajectory is a line or surface which cuts a series of lines or
367.] TRAJECTORIES. 505
surfaces according to a given law, the series of lines or surfaces
so cut being generally formed by the variation of a parameter
contained in their general equation. And let us first consider
the trajectory to cut the given series of plane curves at a con-
stant angle.
Let /(an, y,d) = Q be the equation to any one of the curves,
the series being formed by the variation of the arbitrary para-
meter a; and let F (<r', y'} = be the equation to the required
HYl
trajectory, and be the tangent of the constant angle con-
it/
tained between the curves at their point of intersection ; then
dy dy
m dx dx
"*" = i + ^&'
dx dx
dy' dy dy' dy
'
'u
Now let -- be found from the equation of the given family
CiX
of curves, and substituted in (1), and let a be eliminated by
means of (1) and of the given equation, then if for y' and x', y
and x be substituted, (because they refer to the same point,) the
integral of (2) will be that of the required trajectory.
If the angle between the two curves is a right-angle, the
trajectory is said to be orthogonal ; in which case n = 0, and
(2) becomes , , ,
1 + & = 0. (3)
dx dx
Ex. 1. To find the equation to the curve which cuts at a con-
stant angle all circles passing through a given point, and at
that point touching a given straight line.
Let the point be taken for the origin and the given line for
the axis of y ; then the equation to the circles is
dy _ ax
dx ~ y
therefore from (2)
\ (dy y 2 -x z l
f = < 5J L a -- h
) (dx 2xy )
PRICE, VOL. II. 3 T
506 DIFFERENTIAL EQUATIONS. [367.
= 0,
which is homogeneous, and of the second degree. Therefore by
the method of Art. 288,
(mynx) (x 2 + y 2 )
.-. u x = log (a?+y 2 ) log (mynx),
u y = log (a? 2 + y 2 ) log (mynx),
.'. x 2 +y 2 = 2c(my nx),
where 2 c is the arbitrary constant of integration. The equation
is manifestly that to a circle.
If the trajectory is orthogonal, n = ; and the equation
becomes 9.2 o
a? 2 + y 2 = 2cmy,
the equation to a circle passing through the origin, and whose
centre is on the axis of y, and radius = cm.
It will be observed that the arbitrary constant of integration
leaves the particular curve undetermined, although the general
integral determines the species of it.
Ex. 2. Find the trajectory of a series of parallel straight lines.
Let the equation to the lines be
x cos a + y sin a = p,
where a is constant, and p is the variable parameter,
dy
-f- = cot a ;
dx
therefore equation (2) becomes
dy dy
mm cot a -f- = n -~ + n cot a.
dx dx
mxmcotay = ny + ncotax + c,
(msina wcosa) x (mcosa + nsma)y = csina.
The equation to a series of parallel straight lines.
Ex. 3. Find the orthogonal trajectory of a series of parabolas
expressed by the equation
dy 2a y
dx ~ y 2x
therefore by equation (3)
368.] TRAJECTORIES. 507
1+^^ = 0, 2a?te + ydy = 0,
dx Ax
^ + ^ = <?,
where c 2 is an arbitrary constant.
_
c 2 * 2c 2 "
1\*
/ 1 \*
The equation of an ellipse, whose eccentricity is ( ^ ) .
Ex. 4. Find the orthogonal trajectory of the series of hyper-
bolas expressed by
xy
x
368.] The trajectory (orthogonal or other) of a series of
curves referred to polar coordinates may be determined in a
similar manner; thus
,m .rdd .
tan- 1 = tan- 1 -- tan" 1 . , , (4)
n dr dr
rdO r'dO'
m dr dr
drdr
, de dO' dO , dtf
.' . m + mrr , -- r-r = nr -^ -- nr -r-r, (5)
dr dr dr dr
and if the trajectory be orthogonal, n = : therefore
dO dff
Ex. 1. Find the orthogonal trajectory of a series of logarith-
mic spirals expressed by the equation r = a 9 , when a varies.
dr .
therefore (6) becomes
! ,
dr logr
3 T 2
508 DIFFERENTIAL EQUATIONS. [369.
dr
Io
logr = (c 2 2 )*, r = gC 2 -* 2 )*.
Ex. 2. Determine the orthogonal trajectory of a series of
lemniscata expressed by the equation r 2 = a 2 cos 2 0.
rde _ cos 26 rd0cos20_
dr '' ~im~2lT " ~W im~20 = '
r 2 = c 2 sin 20;
which is the equation to another lemniscata whose axis is in-
clined at 45 to that of the given one.
Ex. 3. Find the equation to the orthogonal trajectory of a
series of confocal and coaxal parabolas.
2a rdQ I + cos
r =
+ cos 6' dr sin 9
r dd 1 + cos 6 _ dr _
' ~dr sin0 ' ~~r + C 3 2
2c
= 1 - COS '
the equation to a series of the confocal and coaxal parabolas.
369.] Trajectories with reference to families of curves may
also be drawn according to other laws : and although these
may or may not involve differential equations, yet it is oppor-
tune now to consider them.
Ex. 1. A series of cycloids (see fig. 52) have a common
starting point o, and a common base ox; it is required to find
the equation to the curve which cuts off from all of them an
equal length of arc o P.
Let the length of the arc be k ; and let the equation of one
of the cycloids be
x = versin" 1 - (2ay y 2 ) ,
dx dy ds
y* (2a-y)* (20)*'
s = 2(2ay)* = k, a =
370.] TRAJECTORIES. 509
Ex. 2. Many circles touch each other at a common point :
find the curve which cuts them at an angle proportional to the
rectorial angle at the point of section, the common point being
the pole, and their common diametral line being the prime
radius - r = 2acos9
.-. ^ = -cottf = tan(90+<9).
QflT
Let k d = the angle of intersection,
dQ'
.
- 1
tan
dr dr
rdO
- 90+ 0- tan- 1
dr
k-\
If k = 3, c 2 = x z y z , the equation to an hyperbola.
k = 2, c = x.
Ex. 3. Find the trajectory of a series of concentric circles,
when the arcs intercepted between the intersections and the
axis of x are of a constant length. See fig. 53.
Let OA = a, AOP = 6 : therefore the arc AP = ad = k (say) ;
r = a
- .
" e'
the equation to a reciprocal spiral.
370.] By a similar process may the equations be found of
surfaces which are trajectories (orthogonal or other) of curved
surfaces of a given family. Suppose that the equation to the
given family is ,.
F (x, y, z) = 0, (7)
and that this equation involves an arbitrary parameter a : and
let us suppose that the equation to the trajectory is
/(ff,y,*) = 0; (8)
and let us suppose that the second surface is to cut all the
members of (7) at an angle whose cosine is m ; then
510 DIFFERENTIAL EQUATIONS. [370.
*L
) + ( ) + ()() =
and therefore if the trajectory be orthogonal
f
wherein (-j-)j (j~) (j~) must be replaced by their values
from (7); and a having been eliminated, the integral of the
partial differential equation will be the equation to the required
trajectory; and as an arbitrary function will be introduced in
the integration, it appears that a whole class of surfaces will
have the required property.
Ex. 1. To find the orthogonal trajectory of a series of
spheres touching a given plane at a given point.
Let the given point be taken as the origin, and the given
plane for the plane yz ; then the equation to the spheres is
# 2 2ax+y z +z 2 = = F(#, y, z),
where a is variable ; and therefore (10) becomes
and therefore by (84), Art. 281,
%xdx _ dy dz
o Q """ ~"~ "~~ ~"~ t
y z y z
Zxzdx = ^
2xzdxx z dz
+(1 + C! 2 )2 = C 2 ,
=
where / expresses an arbitrary function.
37I-] TRAJECTORIES. 511
Ex. 2. Find the equation of the orthogonal trajectory of
! + y! + l_ * 2
i> T T o I ft - * a 3
a 2 b 2 c-
where k is a variable parameter.
In this case (10) becomes
>-, _ n
a 2 + \dy' b 2 + \dz> c 2 "
a 2 da? _ b 2 dy c 2 dz
x y z '
a 2 log x b 2 log y = c l5
b 2 logy c 2 logz = c 2 ;
a 2 log x b 2 logy = f(b 2 logy c 2 log z)
is the equation to the trajectory, where / represents some arbi-
trary function.
371.] The following geometrical problems also involve partial
differential equations of the first order.
Ex. 1. Determine the surface whose tangent planes pass
through the same point.
x a y b z c'
the general equation to conical surfaces.
Ex. 2. To determine the surface such that the intercept of
the axis of x by the tangent plane is proportional to x.
The differential equation which expresses this property is
512 LINES OF CURVATURE [372.
dx dy _ dz
n)x~ y ' z
x
,. ...
Ex. 2. Determine the equation to the surface in which the
coordinates of the point where the normal meets the plane of
xy, are to each other as the corresponding coordinates.
The equations to the normal are
x rj y __ z
dv\ /C?F\ (dv
but = ; .-.
r? y
z = d, <r + y = c 2 ;
.-. *=/( + y),
where /represents an arbitrary function.
372-3 Next let us consider the case of the differential equation
of the first order and of the second degree, which expresses the
lines of curvature of an ellipsoid.
Let the equation to the ellipsoid be
then by the general equation (7) Art. 346, Vol. I, the equation
to the lines of curvature is
' dy ' ' dz
/y2 .j/2 ~2
-m- , W/ '/ X>
Let = f, ^ = TJ, =
so that (11) and (12) become
(14)
= 0. (15)
37 2 -] ON THE ELLIPSOID. 513
Now an integral of this form may be found by a method due
to Mr. A. Cayley of Trinity College, Cambridge, and inserted in
Vol. III. p. 264 of the Cambridge Mathematical Journal, Cam-
bridge, 1843.
Suppose that there is a primitive equation of the linear form
ax + by + cz = (16)
containing three independent variables x, y, z, and three con-
stants a, b, c; and suppose that H symbolizes a homogeneous
function, and that the constants are related by the equation
H (a, b, c) = 0, (17)
and suppose also that there is another equation of the same
form as (16)
= 0, (18)
where x\,y\,z\ are simultaneous values of x, y, z: then from
(16) and (18) we have
= * = ; (19)
yzi zyi zx\ xz\ xy\yx\
and substituting in (17), there results, by reason of its homo-
geneity, x n /om
v * II / ijiW 2I7/i 2'/*i ^2,1 Vtli 7/ I ~~~" IJ
and hence conversely we infer that (20), which contains three
arbitrary constants, is equivalent to (16) and (17) taken simul-
taneously.
Or again, suppose that (18) is deduced from (16) by differen-
tiation, so that we have
adx + bdy + cdz = 0, (21)
then (20) becomes
H(y dz zdy, zdx x dz, xdy y dx) = 0, (22)
and hence we infer that (22) is equivalent to the two following
equations taken simultaneously, viz.
H(M,c) =
where a, b, c are arbitrary constants. The integral therefore of
(22) is (23), in which however the three arbitrary constants are
equivalent to only one.
It is manifest also, that if x\, y\, z\ are simultaneous values of
x, y, z, (23) may be written in the form
H(yi zyi, zxixz\, xy^ yxi) 0. (24)
PRICE, VOL. ii. 3 u
514 LINES OF CURVATURE [372.
A similar process is applicable when more than three variables
are contained in the differential equation.
Now let us apply the process to the equation (15), since we
can transform it so that it may be of the form (22) ; for dividing
lt by (1? rfC-f dr,) d-t dO (f drj-v d},
we have
but from (14)
similarly cfy = drir)d(+dr) r)d, (26)
so that (25) becomes
A2 _ x.2 r 2 _ .,2 2 _ A2
0, (27)
which is homogeneous and of the form (22). Therefore the
integral is, by reason of (24),
where, by reason of (14),
or, if/, g, h are undetermined constants
=
(29)
1
and as this last equation is to be satisfied identically, we may put
,_ 6 2 -c 2 c 2 -fl 2 fc _ a 2 -6 2
/ == ^2l^2' 5 ' = 'cTZl2-' -l2Hpr ;
and replacing f, T/, from (13), (29) becomes
6 2 -c 2 x* c*-a* y z a 2 -b 2 z 2 _
B 2 -C 2 ^ + C 2 -A 2 fi 2 " + A 2 -B 2 ^ = '
which is the equation of a cone of the second degree. Hence
we conclude that the lines of curvature are determined by the
lines of intersection of this cone with the ellipsoid.
37 2 -] ON" THE ELLIPSOID. 515
We may also prove the proposition relating to lines of curva-
ture of a surface of the second order, enuntiated in Art. 165 ;
viz. that they are formed on the ellipsoid by the intersection
with it of two confocal hyperboloids.
For suppose that we identically satisfy /, g, h by the equations
(32)
h = k(c z -0)- 1
so that the last of (29) is satisfied, and the first becomes
** >-
f 7? C
a 2 b z c z
r 2 W 2 ~2
or, 2 \ + p-1 + * - = 0. (34)
Now multiplying this equation by and adding it to (11) we have
nr>~- 1 1 ~ -> 2
__ + ^_ + __ = i, (35)
which is the equation to a confocal surface of the second order.
Suppose now that the lines of curvature are drawn through a
given point (a? a , y^ Zi) on the ellipsoid ; then (34) becomes
which is a quadratic in : and therefore gives two values of 6,
which may easily be shewn to be real, and of which, if a, b, c are
in descending order of magnitude, one will be between a and b,
and the other between b and c : and therefore (35) will for one
represent an hyperboloid of one sheet, and for the other an
hyperboloid of two sheets ; both of these surfaces being confocal
with the given ellipsoid.
I may by the way remark, that (34) is the equation to a cone
of the second order, and therefore the lines of curvature on the
ellipsoid are formed by its intersection with the ellipsoid. And
let us suppose that the lines of curvature are to be drawn
through a given point (#1, yi, Zi) on the ellipsoid, then from
(36) there will be two values of 0, both of which will be real,
and therefore there will be two cones passing through the point,
and these will by their orthogonal intersections with the ellipsoid
trace its two lines of curvature passing through the point.
If the given point, through which the line of curvature passes,
is an umbilic, = b 2 , in which case y = 0, and
3 u 2
516 GEOMETRICAL PROBLEMS. [373-
~ 1 = *i ' < 37 >
a(a 2 -6 2 )* C (a 2 -c 2 )*
that is, the cones degenerate into two planes, one of which is
that of xz. and the other passes through the axis of y, and is
expressed by the equation (37).
373.] I proceed now to the solution of some geometrical pro-
blems which involve differential equations of the second order :
and these for the most part arise from certain relations being
given between the radius of curvature of a plane curve and the
coordinates of the point at which it is drawn.
Ex. 1. Determine the curve whose curvature is constant.
Let the radius of curvature = c,
, rfy'\*
+ = c,
' dx* 2 dy 1
= + , . = +
\ d^
(y-b)dy _ {cz_ (y _ b} 2it = + (aj-a),
O i 7 x O " * ~~ * *^ * * ~
Or thus by polar coordinates,
rdr
-
if r and p simultaneously vanish ;
2cr*dO -dr
r 2 =
(dr*
T
cos- 1 g- = 6, r = 2 c cos 0.
Ex. 2. Determine the curve of which the radius of curvature
is proportional to the normal.
dy^
~dx* 2dy
^~ : ky >
+ dx*
373-] GEOMETRICAL PROBLEMS. 517
2
where k may be either positive or negative ;
(1) Let = 1 ; that is, the radius of curvature is equal to
the normal.
dy dx
the equation of the catenary.
(2) Let k= 1;
i = dx, (c 2 y 2 ) 5 = + (x a),
y z + (xa) 2 = c 2 ;
the equation of a circle, M'hose centre is on the axis of a?.
(3) Let k = 2 ; that is, the radius of curvature is equal to
twice the normal.
(x a) = 4<c(y c).
(4) Let k = 2 ;
dx = -
c , y ,,i
x - versm- 1 (cy -y 2 ) 1 ;
the equation of a cycloid, whose starting point is the origin, and
whose base is the axis of x.
Ex. 3. Determine the curve whose radius of curvature varies
inversely as the abscissa.
k ' dx xdx
> 3 = i ~i~ >
X i //2\5 AC
dx*
dy
dx
= +
2k
dx*>
.-. dy =
an equation which does not admit of further integration, but
which represents the elastic curve. Also see Art. 220.
518 GEOMETRICAL PROBLEMS. [374.
Ex. 4. Determine the equation of the curve of which the
radius of curvature varies as the cube of the normal.
dx 2 ' y^/ dy*\* d 2 y #*
- + ' ~~ ~ '
y 2 a 2 ' ( a 2_^2)i a
ydy
)i
the equation of an ellipse.
Ex. 5. Another form of condition which reduces itself to a
differential equation of the second order is
s -
-
d^_ f/ ldy\d*y_ / d
dao~ J \dxl dx*' \ + dx* ' ~ \dx> dx*
Thus suppose that
s = a an--.-,
dx
dy_
dx
^ dx 2 '
(LOO S /y2 ^^ / /y> /\2 \ a
i M- ^^^cC^^Cy j
In the Notes appended to Liouville's edition of Monge's
" Application d' Analyse &c." will be found the solution of the
problem " To determine the curve of double curvature of which
the radius of absolute curvature and the radius of torsion are
both constant."
374.] The means of integrating partial differential equations
of the second order and of the higher degree are so limited, that
only some few geometrical problems dependent on them can be
solved. The following however was solved by Monge :
To determine the equation to the surface, every point of
which is an umbilic.
375-1 GEOMETRICAL PROBLEMS. 519
By (78) Art. 359, Vol. I, the condition is
r s t
p dp 1 dq q dq 1 dp
1 +p 2 dx q doe' 1 + q 2 dy p dy '
1+p 2 = Yg- 2 , 1 + 9 2 = x/? 2 ,
where Y and x are undetermined functions of y and x intro-
duced in the x- and y-partial integrations ; hence we have
d
but since
. _ , _
we have (1 + x) J -r-=(l-fY) ff -r-;
dx dy
now this equation shews that x and Y are of the same form,
and as there is no relation between them, this identity can sub-
sist only when each side is a constant : let therefore
_ a dx 2 _ fl?Y
(1 + x) 5 5=-*= (1 + Y) V
/t
(27 ~~~ tf . , w t/ .
whence ,
dz\ x a
idz\
= U)
zc = --x-a--
which is the equation of a sphere. Whence we conclude that
a sphere is a surface, every point of which is an umbilic. See
Art. 356, Vol. I.
375.] To determine the surface of revolution at every point
of which the principal radii of curvature are equal and of oppo-
site signs.
The differential equation which expresses the stated property
is, see (77) Art. 359, Vol. I,
p*)t = 0. (38)
520 GEOMETRICAL PROBLEMS. 1.375-
Let the surface required have the axis of z for its axis of revo-
lution, so that its equation is, see (99) Art. 317, Vol. I,
z = f(x z + y 2 ), (39)
where / expresses the arbitrary function which is to be deter-
mined ;
and (38) becomes
let xP + y 2 = t, .'. z =/((") ; and we have
r\ 3 dz
and making z to be equicrescent instead of (,
4.1.
that is -y-
dz
-
the constant being determined so that f = c 2 when z =
2 2
z t
which is the equation of the surface required : and the equation
to the generating curve is
c - -*
x = ^{e c + e <},
which is that of the catenary, the axis of revolution being the
directrix.
376.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 521
CHAPTER XVI.
INTEGRATION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS.
376.] SUPPOSE that there is a system of n different equations
between the independent variable t, n dependent variables x, y,
z, ... and their derived functions with respect to t, viz. x',y',z , ...
x", y", z", . . . ; such is called a system of simultaneous differen-
tial equations, the order of which depends on that of the highest
derived functions, and the problem of integration consists in
deducing from them integral functions of x, y, z, . . . t, which also
contain a sufficient number of arbitrary constants. I may by
the way observe that some of the most important problems in
mathematical physics depend on the integration of a system of
simultaneous equations.
Let us consider first a system of simultaneous differential
equations of the first order : and let us suppose that they are n
in number, and of the form
dx
dy
-jr = Mt,x,y,z,...)
ul ? > \ i )
^=/ 8 (/,*,y, *,...)
we have to eliminate, by means of these n equations, n 1
variables y, z, ... and hereby to obtain an equation in terms of t
and of the other variable x : for this purpose we differentiate
the first of the above equations n times, and substitute each time
for -J-, -j- , ... their values given in the other n 1 equations:
by this process we obtain n 1 equations of the forms
(2)
d n x
PRICE, VOL. II. 3 X
522 INTEGRATION OF
which added to /i give us n equations, from which y, z, . . . may
be eliminated, and there will result an equation of the wth order
in terms of x and /; and this when integrated will contain n
arbitrary constants ; and from it we shall be able to derive the
several equations (2), by means of which and the remaining
equations of (1) we shall obtain the other integrals.
E *->- -=. I -
d 2 x _ dy
*" 'dl 2 = dt
= x,
x = de' + Cze-', .: y c\e { c z e-'.
doc
Ex. 2. a - + (c-b)yz = 0,
dy
-jj-
dz
Multiply the first by x, the second by y, and the third by z,
and let
..
dx
a = ''
then
at at
similarly by 2 = 2(c
The integral of which will give us t in terms of <. Also multi-
plying the three equations severally through by (1) x, y, z } (2)
#, by, cz, and adding in each case, we have
ax dx + by dy -f cz dz = 0,
a?x dx + b 2 y dy + c 2 z dz =
377'] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 523
= h 2
- k 2
which are two relations between x, y, z, which the differential
equations by their forms import.
377.] In general the final equation will be of the nth order,
and therefore its complete integral will contain n arbitrary con-
stants : in particular cases however, we shall arrive at an equa-
tion between (say) x and t of an order lower than n, and in this
case it would seem that a sufficient number of arbitrary con-
stants will not have been introduced; but it will in the result
be found that one or more of the equations in terms of t and of
one of the other variables will be of the wth order, and that the
full number of n arbitrary constants will be made complete.
Thus suppose the following equations to be given,
dx dy dz
dt dt dt
d 2 x dy dz d 2 y dx dz
*' dt*"dt + dt Ht 2= ~dt + ~dt
yc\,
dx
x = Ci + c 2 e, .-. z = Ci c^e-:
and as y contains three arbitrary constants, the result has the
required generality.
378.] Linear simultaneous equations however are those
which offer themselves for solution with the best hope of suc-
cess. Suppose that there are n variables x\, x 2 , x n , and that
t is another dependent variable, and that there are given n
equations of the form
dx\
-77-
at
-|-s n <r n = T n
where the P'S, Q'S, ... s's, T'S are functions of t only. These
equations are integrable in certain cases.
3x2
524 INTEGRATION OF [378.
Thus suppose that there are two equations
dx dy
dx dy r ' (4)
Multiply the second by 6 and add to the first ; put
, n, P! + p 2 _ p, i ^
Qi + 6 Q2 = q, TJ + e T 2 = T,
dx dy
ft (J
dy =
m
then (6) becomes ,
du
ft (J
Let dx -\ -- dy = du. x -f - y = u, (7)
m t?
T, (8)
which is linear of the first degree, and therefore u may be found
in terms of t. Now from (7)
dx -| dy = d (x + - y )
j 1 j pdqqdp
= dx + ^dy + y?-^-
and this is satisfied by
- = 2, and dl=Q; (10)
m p p
and if we substitute for these quantities from (5), and hereby
determine d, we may substitute for them in (8), and hereby de-
termine a relation between t and u.
If the coefficients in the left-hand members of (4) are constant,
then from (10)
which is a quadratic in 6, and therefore if its two roots are un-
equal, and are (say) 61 and 6 2 , we shall have two equations of the
form (8). viz., ,, ,.
m l du+piudt = f^dt -\
w 2 du +p z udt =. t^dt ) '
and from these we shall obtain two equations between u and t,
and therefore two arbitrary constants.
378.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 525
Ex.1. L + 2-
at at
_ dx dii
s dt + M
Multiply the second of these by and add, and we have
'. (12)
Let 2 + d 2 + d '' 6 = - 2
IT30~20^2' =-3,
and (12) becomes
(13)
and
which are two linear equations of the first degree, and are easily
integrated.
ff IT
Ex.2.
Wt
-~
= 0. (14)
I* -f- U V
Let a -l^ = 0,
ar+bQ
which is a quadratic in ; let the two roots be a and /3, so that
(14) becomes
d
which are the two integrals : and x and y may be separately
determined, and each will contain two arbitrary constants.
379.] Let us next consider the case of n linear equations with
constant coefficients ; and let us suppose them to be of the form
526
INTEGRATION OF
~dt
n x n = T 3
[379-
(15)
or, according to the principles of the calculus of operations,
= T
(16)
whence, according to the notation of Art. 150, equation (34),
; (17)
so that o?i will consist of a series of terms formed by operating,
with the several factors of which the denominator of (17) con-
sists, on the quantities which are contained in the numerator.
doc
Ex.1. ~
= 0,
Let the roots of the operating factor be a and /3, so that
379-1 SIMULTANEOUS DIFFERENTIAL EQUATIONS. 527
x Cie
Ex.2. \-j- + 4J x + 3y = t,
d
=
-
31 5 1
Ex. 3. Let there be three equations
= 0,
t z = 0,
d 2
? x
of this cubic let the roots be a, /3, y,
x =
528 INTEGRATION OF [380.
from which, or by a similar process, may the values of y and z
be found.
Suppose that the three roots of the cubic are equal, then
d
(ar-H
P
= e**J
380.] Simultaneous differential equations of higher orders
with constant coefficients may also be solved in a similar man-
ner by the calculus of operating factors : the following examples
indicate the process more clearly than general explanation.
Ex. 1.
Let the roots of the operating factor be real and unequal, and
be +ai, ai, +a 2 , a 2 : and let bci biC = k,
X =
x =
if the roots are impossible, the exponental expressions in x
will be replaced by the equivalent circular functions ; and by a
similar process may the value of y be found.
Ex 2
dt* dt dt
381.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 529
therefore eliminating y
and by a similar process we may find y.
381.] A similar method is applicable also to linear partial
differential equations. Thus
/ d % z \ idz\ _
\dxdyl ' a \~dyl ~'
(d*z\
C =
therefore integrating twice with respect to ^, and introducing
two arbitrary functions of y,
which is a linear differential equation of the first order : there-
fore z _ aov(y) + ^(y)-{-e ac y^(y),
where F, <J> and ^ are symbols of arbitrary functions.
382.] There is no general method of integrating simultaneous
equations which are not linear, and therefore we are obliged to
have recourse to such artifices as are suggested by the forms of
the equations : of these we have already had numerous examples
in the problems of Chapter XII ; for the sake of further illus-
trating the processes we insert two more ; and others must be
deferred until they arise in the course of the treatise, because
their constants of integration for the most part depend on cer-
tain conditions of the problem which at present we have not
means of determining.
=0, (18)
PRICE, VOL. II. 3 Y
530 INTEGRATION OF [382.
where r 2 = x* + y 2 ; multiplying (18) by y, and (19) by x, and
subtracting rf ,
'
,. . dx dii
therefore integrating, y %-rr h, (20)
til ( i I
h being an arbitrary constant. Again, from (18) and (20),
. d z x rdyy dr
h-r^ = n y ? ;
fifil rpt flf
n I i U/C
7 dx y O1
h> ~r~ f* T<IJ (."*)
dt T
similarly h -jj = p, - - c 2 ; (22)
therefore, multiplying (21) by y, and (22) by x, and subtracting,
Again, from (18) and (19),
2 y 2ij.(xdx + y dy) _
^ = 2k, (24)
at* r
where k is an arbitrary constant. And (20), (21), (22), (24) are
the integrals of the equations.
Ex. 2. Another example of simultaneous differential equa-
tions has been solved by M. Binet, which it is desirable to
insert*.
Let there be n + l variables t, x, y, z, ... whereof t is inde-
pendent, and the n others are dependent : also let R =/(r), where
and let there be a system of equations n in number
)//2 ** / //t> \
a z /wn\ /o\
, = \~J~ ) > (*W
'i Cut az
it is required to integrate them.
From (25) it follows that
(- ) = - (-)> (} ~ (~j~)> (27)
so that (26) become
x /C?R\ d 2 y y /C?R\ d 2 z z
dt 2 " r Vrfr/' dt 2 r \dr>' dt 2 ' ~ r
and taking these in pairs, and observing that the number of the
* Liouville's Journal, Vol. II, p. 457.
382.] SIMULTANEOUS DIFFERENTIAL EQUATIONS. 531
pars s
, we have that number of equations of the form
d 2 y d 2 x _
x ~ y ~'~ '
and therefore, integrating
dy dx
x df- y di = ^'
dz dx
~~ 2 '
(30)
dz dy
dy dx\ 2 i dz
dz
dx
dz \ 2
(31)
Again, multiplying (26) severally by 2 dx, 2 dy, . . . integrating
and adding, da}2 d z dz 2 ..
(32)
where B is an arbitrary constant ; so that (31) becomes
,'. dt =
rfr 2 _
dt 2 ~
(33)
(34)
(35)
Therefore combining this with the first of (28) we have
d 2 x
AX
d i 2 d x\ _^f
dt V r dt'r ' ~~ ~ r 2 "/ 7 '
__.
A. dt \ A dt r ' r
3 Y 2
(36)
532 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [382.
A eft A dr
Let = d$ = - T , (37)
' -- -='
d(p 2 r r
integrating which,
x r (a\ cos </>-(- b\ sin </>) ^
similarly y = r(a 2 cos<-j 6 2 sin <) |
,
z r(3 cos + 63810 <)
also from (34^ and (37)
t _ a= f
J
(41)
-A 2 }*
from the latter of which < may be expressed in terms of r, and
therefore by means of the former in terms of t ; and therefore
in (39) #, y, , . . . may be expressed in terms of /.
Now it will be observed that there are at present 2 n -\- 4 arbi-
trary constants, viz., 2n in (39), and A, B, a, /3 : but there are
relations connecting them, so that all are not independent ; for,
firstly,
r 2 = # 2 + y 2 + z 2 + ...
= r 2 {(cos <) 2 2. a 2 + 2sin $ cos <f> 2.a6 + (sin <f>) 2 2.d 2 } ;
and in order that this equation should be true for all values of
<, we must have
2. 2 = 1, 2. ab = 0, 2.6 2 = 1,
which are three equations of relation. Also again the constant
/3 in (41) will merely change the values of i, bi, ... in (39), and
therefore it is not independent of them : hence the number of
constants is finally reduced to 2n.
It may also be observed that the integrals determining t and
are not independent, but may be referred to a common origin.
Thus let
383.] INTEGRATION BY SERIES. 533
CHAPTER XVII.
INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES.
383.] WHEN all other means of integrating differential equa-
tions fail, we are obliged to have recourse to integration by
series ; see Art. 89 ; a process in which we assume the depend-
ent variable to be capable of expansion in a series of terms of
powers of the independent variable, and determine the coeffi-
cients and powers of these terms by means of the differential
equation. It is a method therefore manifestly to be employed
with great caution and reserve, because the assumption that the
dependent variable is capable of expansion in an algebraical
series may be undue ; and if it is capable of such expansion,
the dependent variable and the series can be used as equivalents
only when the series is convergent : and the difficulty of deter-
mining the necessary convergence may be insuperable.
The first method of integration by series is that alluded to in
Art. 361 for the purpose of proving that the general integral of
a differential equation of the nth order involves n arbitrary
constants : and it has been therein applied to a particular ex-
ample : I intend to make some other remarks on the method,
and to express by means of it the integrals of some equations.
Let the limits of integration be no y, and X Q y Q) so that y be-
comes 2/0 when x = XQ : and let y f , y ", . . . y (ra) be the values of
y ,y", ...y (w) when x = X Q ; then by equation (14), Art. 11 9, Vol. I,
, X Xn n\'E <^o) /\ \'^ ^Qi /-!,
y = ~ ' ~~
1.2.3. .. -
Suppose that the differential equation is
y (n] =/ntay,y'...y (M - 1) ); (2)
now if the series which expresses y in terms of x, and which is
deduced from this equation, is convergent, and if also y = yo,
when x XQ, then it must be of the form (1) : and if therefore
we deduce from (2), by successive differentiation and by elimin-
ation, the several quantities,
534 INTEGRATION BY SERIES. [384.
y (n+2) = / + i(*, y, y'
and then in these expressions replace x by X Q , we shall obtain
the several values of y Q (n \ yo (n+l \ > which we shall substitute
in (1), and shall hereby obtain the general integral; for the
series is by hypothesis convergent, and therefore adequately
represents y ; it manifestly satisfies the differential equation,
because it is deduced from it ; and it contains n arbitrary con-
stants, viz., the term independent of x, and the several coeffi-
cients of x, x 2 , ... x"- 1 .
Ex.1.
dx 2
'"
'" = a 2 y Q ', y Q "" =
y'"=a*y
y =
And replacing the arbitrary quantities y , y '. ... by other arbi-
trary constants, we have
y = ae
y'"= y + xy', y""= 2y' + xy",
y "= x y , yo'"= yo + zoyd, yo""= 2yo'
,XX Q (XX ) 2
.-> y = 12
which is the solution : suppose that X Q = 0, then
y = /
384.] The change of form which the solution of this last
example has undergone in the replacement of x by 0, is equi-
valent to the use of Maclaurin's series instead of Taylor's as
the fundamental one in (1) : in this case however we must be
careful that neither y ' nor y ", ... becomes infinite when x 0;
thus, to solve the equation,
384-] INTEGRATION BY SEE1ES. 535
#y" + y = 0, .-. if *? = (), yo = 0,
. xy'" + y" + y' > yo" = yo,
*tf'"+W+y"= o, yo'"=,
1.2.3'
j 23 133 j 2.3.4
1 # 3 1 # 4
)
" f '
but as this solution contains only one arbitrary constant, viz. y '
it is only a particular integral : but we may obtain the general
integral by the following process : Let the function of x in the
right-hand member of (4) be expressed by 77, and y by c ; so that
y = CYJ, (5)
and let us suppose that c is no longer a constant, but a variable,
say u, according to the method of Art. 354, so that
y = ur,; (6)
then differentiating
dy drj du
dx ~ dx dx'
du dr] d 2 u
~ +
and substituting these in the given differential equation, and
bearing in mind that 77 is a particular integral of it, we have
d 2 u _ du dn
^TT + ^TT ~r
(MU CvdU Hub
7 du
d.-j-
^** "
du n
dx
du Cd dx
so that (6) becomes
c\ dx
3-
*
and this contains two arbitrary constants, and is therefore the
general integral of the given differential equation.
536 INTEGRATION BY SERIES.
385.] Often, instead of the series of Taylor and Maclaurin, it
is convenient to assume a series with undetermined indices and
coefficients, and to determine these by means of the differential
equation ; it is in fact the only available process when the re-
quired series involves negative or fractional powers of the inde-
pendent variable.
*l + ^ = -
Let y = ax a + bx? + cxi -f- . . . (9)
and let us suppose a, /3, y, . . . arranged in the order of ascending
magnitude : then substituting (9) in (8) we have
. = 0, (10)
now a 2 is the lowest index , and as (10) cannot be satisfied
for all values of x unless the coefficients of the different powers
of x vanish, we have a(fl + 1) = Q (U)
a = 0, and a = 1.
First, let a = 1 ; then the next two lowest indices are a and
/3 2 ; these may be equal or unequal ; if they are unequal the
term bp(j3 + l)xP~ 2 cannot be compounded with any other, and
must therefore vanish of itself; and /3 cannot be equal to 1,
because it is, by hypothesis, greater than a, therefore ft = Q :
and thus (10) becomes
= 0, (12)
-0 c - - n a
u> L2'
8-2 = 0, 8 = 2,
S+1 > = ' rf =-!31V
( I ri*x w 4 # 4 )
~10Q~t"lO<lA~ ' (i (1^)
l.^.O 1.^.0.4
a i sm nx
= cos w,r + - ,
x nx
GI cos nx + c 2 sin nx
x
386.] INTEGRATION BY SERIES. 537
which contains two arbitrary constants, and is therefore the
general integral. Secondly, let a = 0, then (10) becomes
+ ... (15)
and as /3 must be greater than a, that is, greater than 0, /3 = 2 ;
therefore 6b + n 2 a = :
n 2 a
~
= 0,
1.2.3.4.5'
sn w#
^/ __ __
y '
nx
which is only a particular integral : and the general integral
may be determined by a process similar to that of the last
Article, by assuming
a smnx
41 ^_ _ ^ yi
- M, - //.
n x
We should also have found a particular integral, if in the
former case we had considered j3 2 a.
386.] For a second example of the process let us consider
the equation (401) Art. 327, which is deduced by a substitution
from Riccati's equation,
Let us assume
y =
then substituting in (16) from (17), we have
iai(ai l)a? a i- 2 + 2 012(02 I)^ a2 ~ 2 + 3a3(a3~l)^ a3 ~ 2 +
= ka l x m+a i + ka 2 x m+a * + ka3X m+a *+... (18)
and, to satisfy this equation for all values of x, we must have
o 1 (o 1 -l) = 0, .'. a 1= 0, 0! = 1; (19)
and if n corresponds to the general term of (17)
a n -2 = m + a w _i, (20)
a n o. n (a n 1) = ^a n _T (21)
PRICE, VOL. ii. 3 z
538 INTEGRATION BY SERIES. [387.
Now from (19) let 01 = 0; therefore from (20) and (21)
a a 2 = lT m *"5s+feri)'
a 4 =3(2 + m), 3 =
l _ i
fll + + 2) + +
Again, let from (19) <n = 1 ; therefore from (20) and (21)
3 =
( m + 3) ( m + 4) (2m + 5) (2w + 4)
Now each of the series (22) and (23) involves one undetermined
constant, viz. a\, each therefore is a particular integral of (16),
and the general is the sum of the two : it is plain also that the
undetermined constant is not necessarily the same in both : re-
placing it therefore by Ci and c 2 respectively, we have
(
- Cl
\ 1-4 1 L... 1
( (w -f- 2) (?w + 3) (m + 2) (m -f 3) (2m + 4) (2m + 5) )
and this is the general integral of (16).
387.] If m = 2, that is, if the equation is
all the denominators of both series vanish : but on returning to
(18) it will be seen that
1) = a 2 (a 2 1) = ... = k,
= a* =
= k] and k 2 , say ;
y = Ci x*i + c z ^* 2 , (26)
388.] INTEGRATION BY SERIES. 539
where GI and c 2 are arbitrary constants. Equation (25) may
however be integrated by the method of Art. 350 ; for let x = e f ,
and we have Z dy
-
where ki and k 2 are the roots of
z 2 -z-k = 0. (27)
388.] Now on referring to Art. 327, it will be seen that Ric-
cati's equation ,
~ + az* = bx m (28)
dx
is transformed into the equation (16) by putting
z = _ JL ^ and - = *, (29)
ay eta?
by the simple differentiation therefore of (24) we can find a 1 ,
and thereby obtain the integral of (28) in the form of a series.
And by a similar process and the obvious substitutions which
are given in Art. 327, we may find the integrals of
2n dy
The last equation however occurs in some future investigations,
and requires an independent discussion. Let us, for the sake of
greater convenience, express it
and assume y a 1 # a i + 2 # a2 + 03# a3 + (33)
this, when substituted in (32), gives
= n(n
- {b 2 aix a i + b 2 a 2 x a i + b 2 a 3 x a *+ ...}, (34)
and if m is a general value of the index,
322
540 INTEGRATION BY SERIES. [388.
i 1) = n(n 1) -|
a m = cii + Zm L (35)
w {a H ,(a m -l)-w(w-l)}+6 2 a, n _i = J
the last of which becomes, by means of the first two,
2m(2ai + 2m l)a ra + 6 2 m _ 1 = 0; (36)
and this expresses the relation between two successive coeffi-
cients. Thus for the complete integral we have
y =
2.4(3 u5
Now this series admits of being expressed in the following
form. Integrating by parts, and taking the definite integral
between the assigned limits,
Jo
Let j = 0; then
/ 2w -i-l C*
I (sin 0) 2 "- 1 c?a = ^ / (sin 0) 2 "- 1 (cos
Jo 1 .'o
1.3 Jb J> (41)
Jn + 3
1.8.6
+ 3) (2^5) r-
'
Also in these several terms, replacing n by 1 n, we have
3 2w f"'
m0) l - 2n dO = 7 (ffln0) l -*"(coB0)*d
_ (42)
1.3
388.] INTEGRATION BY SERIES. 541
Now let the arbitrary constants in (37) be replaced as follows :
d = c' Asm 0) 2 "- 1 d0, c a = c" /""(sin 6) l ~ Zn dO, (43)
Jo Jo
then, after substitution from (41) and (42), (37) becomes
/*ir ( f)2 T Z A 4 / 4 )
= c'*j[ jl- -j3( c <> s W + i33A (cos6 ^-~-l (m^) 1 - 1 *
2 r 2 A 4 r 4 )
(cos tf )*-... (sin0)i-d0, (44)
or
f*
y = c'# n ; cos(&rcos0) (sin^) 2 "- 1 ^
Jo
+ c"^ a - n /cos (&p cos ^) (sin 0) l ~ Zn de, (45)
Jo
and this is the general integral of (32).
If n = 0, the differential equation becomes
b 2 y = 0, (46)
Uti&'~
and (37) becomes
y = c' cos bx + c" sin &F,
and (45) becomes
r-ir dQ PIT
y = c' I cos (bx cos 6) - + c"a? cos (bx cos 0) sin d0, (47)
Jo sm Jo
(r ^J/3 O ^-/'
/T x^ "" <* I
cos (0,2? cos 0) - - H i sin o#.
sm b
The evaluation of the definite integral requires artifices which
are beside our present subject.
Similarly, if n = 1, the differential equation becomes of the
form (46), and (37) becomes
y = -^- sin bx + c 2 cos bx,
and (45) becomes
["it r-rr dQ
y = c' x I cos (bx cos 0) sin d0 + c" I cos (bx cos 0) . 5 , (48)
Jo Jo (sun a)
O p' ftr rJf)
= r- sin 6,r + c" / cos (fo? cos 0) - - . (49)
b Jo sm
A particular form of the differential equation (31) occurs in
applied mathematics, viz., when n = 3,
! + 2 y = o, (50)
542 INTEGRATION BY SERIES. [388.
so that (45) becomes
["" C v dd
= c'x 3 cos (bxcosd) (sin6) 5 dO + c"x~ z J cos(&rcos0) . ^,
of which the integral is
y = csin(^ + /3)l- + cos(6,r + /3), (52)
where c and /3 are arbitrary constants ; but the deduction of
this from (51) requires a greater knowledge of definite integrals
than has been arrived at in this treatise.
[ 543 ]
CORRECTIONS.
Page 12, line 20, for r read
27, line 25, for ^xzdx read (5# 2)dx
28, line 7, insert /
35, line 1 6, for log(an-i) read log(# i)
46, line 33, for (aJ 2 + a 2 )-' read (a? 2 + a 2 ) n
47, line 23, for (a 2 -a? 2 )"-' read (a 2 -a? 2 )
55, line 27, for (j:^) read (^j-)
, .. c nj , ni
oi, line 20, for read
n n
70, line 13, for -^ read -
75, line 1 8, for - read
2 2
80, line 23, for (sin a?) 3 (cos a?) 2 read (sin a;) 2 (cos a;) 3
93, line 18, for (7) read (12)
94, line 16, for cos a? I read cos a; I n
119, line i, for read
Ja Ja
fo
, 00 , ., read o*
149, line 25, for "differentiation" read "integration"
243, lines 24 and 25, insert brackets
248, line 2, for \8x read vdx
352, line 32, for dz read dx
QA
300
P713
185M
V.2
C.I
PASC
BB
