228 Mr. R. Moon on the Integration of the General Linear 



Substituting these values in (13 1 ), and equating to zero the 

 coefficient of U_ 1 in the result, we get 



O^mJ'^-f^ + iu + ^+ef^, .... (18) 



it being assumed for the present that f„ and A do not con- 

 tain U_ 2 . 



The last equation enables us to determine f p . Putting in 

 the preceding one Cc + D for f y , where C and D are functions 

 of x and y, which we are entitled to assume do not contain <p or 

 its derivatives, and equating to zero the coefficient of z in the 

 result, we have 

 /7A •/A 



0=m ^-£- (a+ ^ +e/ ft )A - 5+K - WJi)e/ ft- D ' • (19) 



+ K~»»,)*4-C, . . (20) 

 the last of which determines C. 

 Again, observing that 



- , . dG dl) 



_ , . dC dV 



substituting these values with the above values of f a ,f'Jz),f' a (y) 

 in (14/), we get the three following equations, viz. 



dC dC dm ln d , „ . . 



0=4-0-/^(2/) (23) 



Eliminating C between (20) aud the two last equations, we 

 shall have two equations of condition to be satisfied by the co- 

 efficients of the given equation. 



Hence, observing that when 8 = 0, (19) and (21) are satisfied by 

 A = D = 0, when the coefficients of the given equation satisfy the 

 conditions above indicated, the equation will be capable of being 



