OF PARTIAL DIFFERENTIAL EQUATIONS. 
343 
It will have been noted that Charpit’s method consists in 
determining p and q as functions of x, y, z, which render the 
equation dz = pdx + qdy integrable. This determination pre 
supposes the existence of two algebraic equations between 
x, y, z, p, q; viz. 1st, the equation given, 2ndly, an equation 
obtained by integration and involving an arbitrary constant. 
Let us represent these equations by 
F(x, y, z, p, q) = 0, <I> (x, y,z, p,q)=a.... (29), 
respectively. And let us now endeavour to obtain in a general 
manner the relation between the functions F and <E>. 
Simply differentiating with respect to x, ?/, z, p, q, and re- 
dF Y d® dF fo 
presenting S by X, ^ by X, ^ by P, ^ by P, &c. we 
have Xdx + Ydy + Zdz + Pdp + Qdq = 0, 
X'dx + Y'dy + Z'dz 4- Pdp 4- Qdq = 0 ; 
or, substituting pdx + qdy for dz, 
(X+pX) dx+ (Y+ qZ) dy 4- Pdp 4- Qdq = 0... (30), 
(X' 4-pZ') dx 4- ( Y’ 4- qZ') dy 4- Pdp 4- Qdq — 0... (31). 
Substituting these values in (31) we have 
(.X' + pZ' 4- rP’ + S Q) dx + (Y + qZ' +sP + tQ) dy = 0, 
which, since dx and dy are independent, can only be satisfied 
by separately equating to 0 their coefficients. These furnish 
then the two equations 
— (X' +pZ') — rP +sQ' 
-(Y' + qZ') = sP+tQ 
(33).