DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 399 

 Now 



V) 



and the values of the remaining quantities have already been given ; when they are 

 substituted in the equation and the factor e is removed, the equation becomes 



= A {e/iT, + (n + 5)/t"P + A(n + 1) ft'} + D(n + i) M p 



which is satisfied identically, provided 



B=-2A, 



Hence, taking A to be unity, we have 



e 4 = P 4 - 2 f-' + f^-|^P 8 8 , ..... (t5) 



dx * da? * n+ 1 



which practically agrees with BRIOSCHI'S function y (I. c., p. 107) for the value n = 4, 

 the order of the equation in connexion with which the function is obtained. 



23. The invariant 5 . 



The most general form possible is 



Proceeding exactly as in the last case, it is easily found that the conditions 

 necessary for the identical satisfaction of the invariantive relation are 



B=- | A, 

 C= -VL A> 



D=- | A, 



'= 



And, therefore, taking A to be unity as before, we have 



