DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 451 



A + B* + C* + Da" 



-" (31)- 



The generalisation to the case of the equation of order n is so obvious as to render it 

 unnecessary to give the forms of the equations explicitly. 



97. Suppose now that three solutions X, a-, p of the quotient-equation equivalent 

 to (29) are given, as in 95 ; then we have, by the first of those equations, 



= 1*! (X lT + 4Q 3 X') + 4 M ' 1 X 1 " + GttV + 4u lu l X 1 , 

 = it, (<r* + 4Q 3 <r 1 ) + 4tt l 1 a JB + Bu^o* + 4u'V, 

 = , (p* + 4<V) + 4 W ' 1 /> m + 6t*V + 4 V> 



and therefore 



X lr + 4Q S X 1 , X", X 1 

 <r"+4Q 3 cr', (T 11 , cr 1 



or, what is the same thing, 



= , 



X", X", X 1 

 or", o* cr 1 



X" 1 , X u , X' 

 o a, o-' 

 P m > P", P 1 



X m , X", X 1 

 cr 1 ", (7", tr 1 



A P", P' 



Hence, writing 



we have 



so that we may take 



A= X 1U , X", X 1 



P m P". P' 

 j* A =: constant, 



(32); 



and the primitive of the general equation is 



u = (A + BX + Co- + D/>) A-*. 



It is evident that no one of the quantities X, cr, p may be constant, nor may any two 

 of them have a constant ratio. 



98. It has already appeared, in 96, that, if the two priminvariants vanish, then 

 relations 



3 M 2 



