IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 37 



7 n(n — 1) f fdK X \ /3*i \ 



h = 2 I ap+2 ' q \~fo + K1 ) + 2a P+ 1 ««+ 1 [jty + K1K2 J 



+a P ,g+2 [q~ + K 2 2 J + (n — 1) a p+ i, g *i + a p ,g + iK 2 + a pa , 



and 72, which differs from 7i in that it replaces (- kxk 2 in the coeffi- 



dy 



cient of a p +i tQ+ i by h /q*2. Thus 



7l _ j 2 = B(B _ l )ap+lig+1 (^ - ^ 2 ). 



Here, since p + q = n —2, a p +i iq +i is an invariant of the differential 

 equation, and the other factor we already know to be such, (31). 



In (30) pi, qi are subject only to the condition pi + qi = n — 1. 

 We may, by a special choice of these numbers, considerably simplify 

 h and 1 2, or rather their sum. For putting p\ — p + 1, q\ = q, 

 T>2 = P, ?2 = q + 1, we get 



*i = — 



This would mean that k 1} k 2 had been obtained as solutions of the 

 equations 



n(a p+ 2,qKi + a p+ i >g+1 K 2 ) + a P +\,q = 0, 



n(a P +i, q +i K i + a p , a +2*2) + a P , q +i = 0; 



from which, by differentiation, we get 



dx J 



_ n ( da p+2,q 



( dn 6*2" 



n \ a P +2,q^T ~ Op+l.g+1 



dx 



dx 



<i + 



9a p +i, g +i 



dx 



«2 J 



3a P +i, ? 



dx 



n \ ap+1 ^ 1 dJ +a ^+ 2 di) 



, da p>q +2 \ 

 *1 H r k 2 — 



ty J 



da p ,q-\-\ 

 dy 



