DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 

 Let the various operators be denoted by 8,, 8 2 , 8 3 , say 



d 2 



S I 



6, = or J. 



z 



d* d 3 



lx dy dij" 



d* 



3 = 



moreover, let 



83 



be another operator. Tlien from the relations between in and n, I and n, it follows 

 that a function exists such that 



m = S/ H ), 



and then, as 



<lx 

 (lv_ 



dy 



= 



,/; a 



e 



dv / <P (P , 



= m + nu = 2 ( p + a -- } 



ay \ ii.i' ihi (/-//- , 



we have 



= 2 p 

 - 1 



& = 280. 



From the relation between n and c, we have 



or, if = Siv, then 



28c = S 3 n = S 3 2 , 



We thus have, from the foregoing relations, the results 



v = 2 



7 O^O 



I = bb^ 

 m = 8S^ 

 n = S8 3 ? 



c = iS 



J 



The values of a, &,/ g, h, con be obtained by using these values of I, m, n, c, and 

 we find 



M 2 



