DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 85 



But now. operating on this equation with 8 lf we have 



8 l [28 d g + 48'.] = 8l (A8.tr). 

 The left-hand side is 



+ 4S ' S ' = 288 i S + 28/ V + 28/Su, + 2 (8'S, - 8/8) 

 = ^{288^} + 2 (S'8 t - 8/8) w. 



The first term is 2 - ; the second, on reduction, gives 



and therefore the left-hand side is 



The right-hand side, viz., 8, (AS 3 w), becomes on expansion and reduction 



2 (p'q - pq') S 3 ' = A2^ + 2 (//^ - ^ ? w. 







j 



Hence, in virtue of the above equation, we have 



dl dz 



-- = r/A = <7 -r-. 



du " thi 



Similarly we find 



dm _ , dz du dz 



du dii? (hi, du 



and therefore the equation determining ui is 



that is, 



When a solution of this equation is obtained, a solution of the equation for v can 

 be constructed, and conversely. In particular, the identification of any solution of 

 the f-equation, as being included in the general solution, can be made as in 

 former cases. 



