DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATION?. 457 



81 '"' i 

 = Tir 



. 1 *V + 3&n 



Ins ( _, 



H' 1 



85 = J 6 4 e j 6' 4 + ^ {^fy e 3i4 + ? j e'sBs.3 + T^T 



^6 = ^~ if^HS e 3,5 + TlVW ^3,3^3, 1 + ?7l 8/ 3 e 3,+ + T8 3 



Let the priminvariants of this associate sextic be denoted by 4> 3 , <J> 4 , 4> 6) * 6 ; and 

 let the Jacobian 4e 4 6' 3 8836^ a proper covariant of the quartic be denoted by 

 "V. Then for 4> 3 we have 



,1 5J i i ?a i 



" TO e, s " " e, s 



i _ QS. i i 8'sQ.i.i i B. a + SQ'gB,,! 

 * 191 0s :> (.) (_) 



that is, $ 3 is an invariant of the original quartic. Again, we have by (15) of 22 the 

 invariant * 4 given by 



for in the present case n = 6 ; and it is not difficult to prove that, when the foregoing 

 values of S are substituted, the value of * 4 is 



e3.3-e S)l 2 )-Ae 4 ...... (40). 



I give below the values of * 5 and <J> a , founded on (16) and (17) of 23, 24; the 

 analysis is long for each of them, but, as it is of a character precisely similar to that 

 for 4> 3 , it is not reproduced here. The value of <I> 6 is 



' + i .... (41), 



and the value of <!> is 



MlKVrl. XXXVIII. A. 3 N 



