610 Me. DE morgan, ON METHODS OF INTEGRATING 



The root of this theorem lies in the following, that the interchanges above mentioned do 

 not alter the truth of the equations 



dss =i pda; + qdy, dp = rdx + sdy, dq=sdx+tdy, 



so that we have 



d{px + qy ~ s) = xdp + ydq, 



t s 



doc = — dp : dq, 



rt - s' rt - s' 



s r 



dy = — ; dp + dq. 



" rt - s' ^ rt- 8= ^ 



Let ,17, y, z be considered as functions of p and q, derived from the equations 

 f{x, y, z) =0, /. + /, p = 0, /y +f..q = 0; 



but remark that there is a case of exception, namely, when the second and third equations 

 give simultaneous elimination of x, y, and sr, or lead to ■'jr { p-, q) — 0. Since 



z = px +qy - J {xdp + ydq), 



xdp + ydq must be a complete differential. Let it be dv, then we have, v being a function 

 of p and q, 



dv dv 



x = — , y=^r' z = px +qy-v. 

 dp dq 



Let the second differential coefficients of v be p, a, t, we have then 



dx = pdp + cr dq, dp = , dx - r dy, 



' pT — a p T — a m 



dy=(jdp + Tdq, dq = j dx + — , dy, 



" ^ ' jOT-(7* pT - <x 



T ~ C P 



whence r = ; , s = , t = ; 



pT~a pT-a- pT-a 



Hence, in order to make p and q the independent variables instead of x and y, we must 

 assume a function v, of p and q, such that 



dv dv dv dv 



"=d^' ' = Tq' '-Pd-p^'Tq-''' 



and then we must find v by integrating 



idv dv dv dv t -a- P \ n 



^\dp dq''^dp ' dq '^ ^ p t - a^ pT-a^ pr-aJ 



The manner in which I first stated the theorem changes the meaning of the letters x and y 

 without changing the letters themselves. 



Of this method, I find one instance. Legendre (see Lacroix, Vol. ii. p. 622) has employed it 

 as a casual artifice for the reduction of 



/i (P> 9) • »• +/s (P' l)-s +f3ip,q) ■( = 0, 

 to /i {x, y) ■ r -f., («, y) -s +f^ (x, y).t = o. 



