PARTIAL DIFFERENTIAL EQUATIONS. 185 
The original system may also be taken to consist of four equations connecting five 
variables x, y, z , p, q, namely : 
dzjdx = p, dz/dy — q, dp/dy — dq/dx 
f(x, y, z, p, q, dp/dx, dp/dy , dq/dy) = 0 , 
and so the method of variation of parameters does not lead to any simplification of 
the problem in general. 
§ 36. The interchange of variables and parameters is again possible ; it is, perhaps, 
made clearer by taking three equations of perfectly general form, 
<f>i (aq, x. 2 , x 3> aq, x~ 0 , zq, u. 2 u 3 , zq, zq) — 0 (i — 1,2, o), 
connecting two sets, each of five quantities. 
Whichever set we suppose constant and eliminated by differentiation, we are led 
to a system of four partial differential equations connecting the quantities of the 
other set, two of the five being taken as independent variables. A new solution of 
either of these systems of differential equations will in general yield a new solution of 
the other. 
Suppose, for instance, that we have a new solution of the u equations ; this gives 
zq, tq, zq, say, in terms of zq, u 2 . Then the six equations included in 
t du r = 0 (i = 1, 2, 3) 
?•=! OV- r 
give two relations among aq, ... iq, u 2 , since the four differential equations, 
which are consequences of these six, are supposed satisfied ; by the help of these 
two, tq, u. 2 , may be eliminated from the three relations <^> 1 = 0. (f>. : = 0, </> ?j = 0, and 
thus three relations are given connecting aq, x. 2 , x 3 , x±, x 5 ; these three will constitute 
a solution of the x system of differential equations. 
§ 37. In this more general case there will not seemingly, as a rule, be any more 
solutions for either system of differential equations. For the derivatives, say, of 
x 3 , aq, x- with respect to aq, x. 2 are given in terms of these five variables and two 
others, say tq, u 2 . The forms we may assign to ?q, u. : are then restricted by three 
differential equations derived from the three conditions 
d 2 X r 
ch\ dx 2 
d% 
cbc : dxj 
(r = 3. 4, 5), 
and thus, generally speaking, no forms of zq, u % will be suitable. In some cases the 
conditions are not inconsistent, and we may form an array by the method of § 11 
such that if c/(d,y) is a combination of its determinants, then 6 = a, y = b, </q = 0, 
VOL. CXCV.—A. 2 B 
