XLIII. 3Ietho(ls of Integrating Partial Differential Equations. By Augustus 

 De Morgan, of Trinity College, Cambridge, Secretary of the Royal Astro- 

 nomical Society, and Professor of Mathematics in University College, London. 



[Read June 5, 1848.] 



The following methods for the treatment of certain cases of partial differential equations of two 

 independent variables will be interesting, both as having something new, and as combining and 

 bringing together some isolated instances given by different writers. 



Let the differential equation be 



FIRST METHOD. 



(*> y, P, </) = 0, 



p and o meaning — and — . Contrive that this equation, d) = 0, shall be the result of elimi- 



d!C 



nation between two others, ^ = 0, B = 0, or, at full length, 



A {x, y, p, q, v) = 0, B (.V, y, p, q, v) = 0. 



Accordingly, v is an implicit function of ,r and y. Let r, s, and t, as usual, be the second 

 differential coefficients of ss, and form the four additional equations 



A^ + Ar + As + A„ 



dv 

 d.v 



0, 



dv 



B,+ Bj,r + B^8 + B„-— = 0, 



dx 



A^ + A^s + A^t + A„-^=o, B^ + B^s + B^t + B„-^ = 0. 



From the six equations* eliminate p, q, r, s, t; there will result an equation between 



dv dv 

 X, y, V, —- , — , which will often be more tractable than d) = 0. When, after integration, v is 

 dx dy r " 



found in terms of *■ and y, p and q can be found in the same terms from A = 0, B = 0, and 



then s! from dx = pdw + qdy. 



Tiiis method was derived from the suggestions afforded by a previous treatment of the equation 



Apq + Bp + Cq + D = 0, 



A, &c. being functions of x and y; which occurs in the process of developing any surface which 

 admits it upon a plane. Reduce the preceding to the form 



(p +P)(q + Q)= R. 



* With regard to the notation, I must state that by such a 

 symbol as A^ 1 mean the partial difterential coefficient of A with 

 respect to a, as obtained from an equation in which A is explicitly 

 given in tlie form A = f\> (a, ). I have found this notation, 



however useful it may be as an abbreviation, almost as useful in 

 the way of distinction. It points out the ultimate and elementary 

 process, on one or more of whicli the implicit differential coeffi- 

 cients depend. 



