DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 453 



99. In the case of the alternative normal form (28) for the quartic, the quotient- 

 equation is, as before, of the seventh order ; and, if X, <r, p be three special solutions of 

 it, we have 



= HI (6IV + /*") + u\ (12 M 'R* + V") + 6u>" + 4tt'V 



for IJL = X, a; p, and therefore 



u 1 



4X 111 + 12X%, X", X 1 

 + 12^ er", (r 1 



X"+GR 3 X", X", X 1 



p" + 



or, since these determinants are independent of R^ we have a result the same in form 

 as before 



= 0, 



X 1 ", X", X 1 

 <r', <r, cr' 



= constant. 



The results for this normal form, which correspond to those given in 95, 97, are the 

 same as are there given. 



100. For both forms it appears that, when the two primin variants vanish, four 

 solutions are given by u = 1, z, z 9 , z 3 ; hence the primitive of the equation 



the priminvariants of which vanish, is 



y=ffi[A.+ Eie-*dx + C{ie-*dx} i + {\0- 3 dx}*] 

 where 6 is determined by 



101. But, as in the case of the cubic, it was not necessary to know more than a 

 single solution of the quotient-equation in order to obtain more than one solution of 

 the original equation, so in the case of the quartic the knowledge of a single special 

 solution of the quotient-equation, not a constant, is sufficient to give two special 

 solutions. For, if /* be such as to satisfy the quotient-equation, we can from the 

 first three equations of (29) find the value of t*V w i explicitly and thence w, ; the value 

 of M 2 is then known being ILU V Similarly, from a knowledge of two solutions of the 



