PARTIAL DIFFERENTIAL EQUATIONS. 609 



(a — v) X 



«= ^^— , 



x + y — a 



, fx"- +2xy - 2aw\' x* 



dz = ^ dx + -■ rf«, 



\ X + y - a j {x + y-af "^ 



4 , «■« 



« = - ■» + h. 



3 X + y — a 



Make b = (pa, and proceed as before. 



Another form of solution is derived from v + x = a, or 



dz = {9,x - aydx + a'dy, 



2r = - (2^7 - ffl)' + a-y + (pa. 



Resolve the same equation into 



/ 2x V / 2xv\- 



giving XV 1- X ~- = - v^ (I + v). 



ax ay 



f X^ \' ( x^ \~ i 2xv 



From the solution of the original take q= and make ( 1 = ( — '- — 



\x + y - a) y-g + y - aj \1 + v 



X + y — 2a 





This ought to be a solution of the differential equation last written, and it will be found 

 to be so on trial. 



SECOND METHOD. 

 Let there be given the equation 



(p (.i?. y, «> p, q, -r-, «. = 0. 



Interchange p and x, q and y, z and px + qy - «, r and -, s and , , 



rt - s' rt - s^ 



■ '>' 

 t and 



rt - 8'' ■ 

 giving (p {p, ,1, V- + qy - ^, ^> y, ^^, , ^, , ^^.) = 0. 



If either of these equations can be integrated, say by 



Z=^{X, Y), 

 then the solution of the other is obtained by cliiiiiiiating -V and Y from 



rfZ dZ 



zm xX + yY - Z. 



