608 jir. de morgan, on methods of integrating 



Let px + qy = V, p + q = — . 



V 



Form tlie four other equations and eliminate, which gives 



dv dv 



(u + a) — + («■- +y) — = v, 

 diV dy 



or -=f[ . 



Let /« = au ; then v = \/ , 



^ \ — a 



1 



y/{\ -a)' ^(y-ax)' ^ -y/(l-a) y/{y - asc)' 



/y-ax 

 ^ I - a 



For -a (l - a)"' write a: then we deduce the general solution by eliminating a between 



^ = 2 ^/{a{x -y) + y] + <pa, 



^- y 



= — --. j + (ha. 



■s/{a{.^-y) +y\ 



Let Ap"^ + Bpq + Cq" = D 



which can be resolved into {p + K q) (p + Lq) = MN. 



Let p .^ Kq = Mv, p = f LMv \ {K - L) " ', 



p + Lq=-, q=\Mv \{K-L)~\ 



V \ V J 



Tiie two values of s thence derived, equated to each other, give the equation for determining v. 

 Accordingly, since A &c. may be any functions of .v and y, the general equation of the second 

 degree is reducible to ordinary differential equations, provided that sr do not appear in it. 



In these examples, I have chosen, merely for simplicity, cases in which p and q are explicitly 

 found, and the values of s equated. This amounts to exhibiting (p = under the form of 

 A — and B = 0, and determining v so that pdx + qdy may be a complete differential. And 

 in like manner as every particular value of v leads to a particular value of sr, so does each 

 value of sr lead to one of v. And in this way a particular solution of one partial differential 

 equation may lead to a particular solution for another and a more difficult one. Thus, if d) = 

 be derived from A = 0, B = 0, leading to the new partial equation 17=0; and if it also be 

 derived from A' = 0, B' = 0, leading to V' = 0: by means of a solution of U = 0, leading to a 

 solution of d) = 0, one solution of tT = may be found. 



Take the instance >^p + y/q = Sa-, 



or p = (.r - vf, q = (.V + vf, 



(w + w) — + (a? - «) (x + r). 



dx dy 



