Mr. DE morgan, ON METHODS OF INTEGRATING, ETC. 60? 



N 

 Let p + P = Mv, q + Q =—. 



V 



MN being any convenient resolution of R into two factors. We have then 



dv 



dy " 



Ndv N, 

 s + Qx=-— — + — . 

 V dx V 



,, du N dv ^ N^ 



M—+---=P,^-Q, + — - M„v, 

 dy V- ax v 



which depends on ordinary differential equations. But it must be observed that the integration 

 of this subsidiary equation frequently leads to a form from which v cannot be directly exhibited as 

 a function of w and y. Where this happens, we must obtain a particular form which contains one 

 arbitary constant ; another will be introduced in the integration by which x is obtained ; and 

 Lagrange's process may then be applied to the primary form so obtained. 



For example, let pq == pw + qy, or, (p - y) (q - ''") = ^V- 



dv 

 Let p — y = XV, s — I = X — — , 



^ ^ dy 



y y dv dv y do 



^ V V dx dy v' dx 



or x'v^ - y- =/(»). Let fv = av^, and we have 



y ^y /—i — 



y/ias- — a) V ai' — a 



z = xy + y \/ X- - a + b. 

 Let 6 = (pa : then the general solution of pq = px + qy may be obtained by eliminating a from 



z = xy + y y/{x' — «) + 0a, 



= 7/5 r- + d)'ffl. 



But if we take p - y = v, we find 



and we ultimately obtain the same form. 



We may also obtain as the primary solution 



z = ^.{.i^ - a) (y' - (pa) + xy + >// a. 



If we apply the whole process to pq = (px . xf/y, we find for a primary solution 



z =2^/{(p^x(f^y -«)} +/a, 



where ^p^a = J(pxd x, x//, y = f\j, y d y. 



Next, take the instance {p + 7) {px + qy) «= 1. 



41 i 



