EQUATIONS OE THE FIRST ORDER. 
451 
and equate to 0 the coefficient of in the result. We find 
dz—{t-\-'^x^)dx—ydy—xdt—Q^ 
the integral of which is 
z—xt—o?—y^—G. 
2nd. The solution of the partial differential equation 
Er+Ss+T^+U(s^— r^)=V, 
as well as of the special equations 
R7’-hSs-{~T^= V , 
Kr+Ss+T^+U(s^-rO=0, 
the theory of which constitutes an exception to that of the more general form, depends 
in general upon the integration of three simultaneous differential equations between 
five variables. To this integration the method of the foregoing sections is applicable. 
The only cases for which the theory of the ultimate solution can be said to be com- 
plete, are those in which the auxiliary system of common differential equations admits 
either three integrals of the form P=(?, or two integrals of that form. 
We may apply the method of the foregoing sections, not only to the determination 
of the integrals, but also to the discovery of the a priori conditions connecting the coeffi- 
cients R, S, T, U, V in order that each of these species of integration may be possible. 
For example, the solution of the equation 
Er+Ss+TH-(s"-^^)=V 
depending upon the integration of the system 
dg= — 
dp=— ni^dy + T dx^ 
dz = pdz ■\-qdy^ 
in which and m.^ are roots of the equation 
m®— Sm+ET— V=0, 
let it be required to determine the conditions under Avhich the system admits three 
integrals. 
Eliminating dq, dp, dz between the above equations and 
-S^dx+-j^dy+ 
dq 
and equating to 0 the coefficients of dx and dy in the result, we obtain two partial 
differential equations which may be thus represented, viz. 
AP=0, A'P=0, 
. d d m d d 
in which 
