43] 
ON THE DIFFERENTIAL EQUATIONS &C. 
277 
§ 1. Let the variables x, y, z, ... &c. be connected with the variables u, v, w, ... 
by the same number of equations, so that the variables of each set may be considered 
as functions of those of the other set. And assume 
dx dy ... = V du dv ... ; 
if from the functions which equated to zero express the relations between the two 
sets of variables we form two determinants, the former with the differential coefficients 
of these functions with respect to u, v, ... and the latter with the differential coeffi 
cients of the same functions with respect to x, y, ... the quotient with its sign changed 
obtained by dividing the first of these determinants by the second is, as is well known, 
the value of the function V. 
Putting for shortness 
dx 
du a> 
d V-P . dX - n ' d V -R' 
and 
du _ . du _ „ dv _ , dv _ 
dx dy ’ ''’ ’ dx ’ dy ’ '" 
V is the reciprocal of the determinant formed with A, B, ... ; A', B', ... , &c.; or it is the 
determinant formed with a, /3,... a.', /3',..., &c. 
From the first of these forms, i.e. considering V as a function of A, B, ... 
dV 
dA 
= -V. 
dV 
ciB 
= -v/3,... 
dV 
dA' 
W. 
d y 
dB 
where the quantities a, ¡3, ... a', ¡3',... and A, B,...A', B',... may be interchauged pro 
vided - V be substituted for V. (Demonstrations of these formulse or of some equivalent 
to them will be found in Jacobi’s memoir “De determinantibus functionalibus,” Crelle, 
t. xxii. [(1841) pp. 319—359].) 
Hence 
^ dV + ad A + /3dB + ... + a'dA' + /3'dB' + ... = 0, 
or reducing by 
dA = dB dA/ _ d& 
dy dx ’ " ’ dy dx ’ " 
this becomes 
^ dV + a 
(dA , dB 7 > 
u* dx+ dx dy + -/ 
\ LLiAj / 
+ 13 
(dA 7 dB 7 
{dt dx + Ty dy+ -J 
> 
+ oc 
/dA' . dff , \ 
Kdd dx+ dS dy+ -) 
+ /3' 
(dA' , dJ5 7 > 
_ /dx+1 - dyJr ... ) 
-