DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
75 
2. Theorem . — Retaining the suppositions made at the beginning of the last article, 
let X be a given function of x x , x 2 , .... x n ; and let us further suppose that the equations 
by which y x ,y 2 , ... y n are determined as functions of x Xi &c., are 
dX 
dX 
Vl ’ 3/2 dx/ 
so that 
dx j 
dy. dyj m 
y n — 
dX 
dXfi 
(5.) 
dxj dxi ’ 
and if we transform the equations (1.), (2.), art. 1, by this condition, we obtain the 
n equations 
dy i doCy . dy<£ dx 2 dy n dx n ^ 
dx x diji ' dx x dyi''"'dx x dy t 
djh dxj dy^ dx ? 
dx 2 dyi dx 2 dyi''"' 
dyn d%n 
dx 2 dyi 
= 0 
dy x dxi dys d_x § ^ 
dxi dyi dxi dyi' ' " "• J*- 1 
dyn dx n 
dxi dyC 
If these equations be added, after multiplying them respectively by 
dx x dx 2 dx n 
dy/ dy/ "" dyj 
dx ‘ 
the sum of the first members reduces itself by virtue of the equations (1.), (2.), to 
whilst the second side consists of the single term We have then 
dyj 
dxi dxj 
dy~ dyi 
or, in other words, if x 15 x 2 , ... x n he found from the system of equations (5.) in terms of 
y„ y 2 , ... y n , the resulting expressions are the partial differential coefficients of a certain 
function of y„ y 2 , ..., y n , so that the system inverse to (5.) is of the form 
( 6 .) 
dY _dY dY 
'Ey ~j 5 '%2 5 •••} j — • 
dy x dy 2 dy n 
The relation between X and Y is easily found as follows. The equations (5.) and (6.) 
give 
d'K=y 1 dx 1 +y 2 dx 2 + . . . +y n dx n 
dY =x x dy } +x 2 dy 2 + . . . +x n dy n ; 
whence, by addition, d(X+Y)=d{x 1 y 1 +x s y 2 + ...+x n y n ), 
and therefore X+Y =x 1 y l +x 2 y 2 + ...+x n y n (7.) 
(omitting the arbitrary constant, which might of course be added). 
The actual value of Y will then be 
Y = — (X) + (x 1 )y l + (x 2 )y 2 +... + (x n )y n , (8.) 
l 2 
