DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 33 



Multiply the latter by H 0, and add it to the former. In the resulting equa- 

 tion, the coefficient of dg/dx is 



which is 



the coefficient of dgjdy is 



- Hp - B 2 + (1H - 6) p + F = - f(iH + 0)p + Eg] + F 



But this coefficient, multiplied by A, 



= AF - 



and therefore the terms, involving derivatives off/, are 



apj 



(r, U + 



A 



The terms involving derivatives of a are 



A A^. 2 J" 1 - 6 d 



A ., . + 



\fb: A dy 



and those involving derivatives of h are 



i_ m _L -R /i TT _i_ m a. 



") ~T -- r & 7~ = (i-tl + ") \ ~i -- ---- A~ 



' dx dy ' \dx A 



The equation is now in its simplest form. The other pairs may be treated in the 

 same way ; and thus, corresponding to the equation 



we have a system of three subsidiary equations free from p and q in the form 



Y + A8/i + (H + 6) Bb + (|G + ^) 8/= 



Z H 



where 



3= ^ 



dx A 



VOL. CXCI. A. F 



