IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 33 



1 = 



Pi9i 

 P 2 9i 

 Ps9s 



, Pi + q* = n - 1, (28) 



is an invariant of the differential equation. That it is invariant for 

 u = \p • r] is seen at once from the formulas of transformation (17), page 

 17. The adjoint invariant is 



J=(-l) n . 



c'Pi+i.ft °pi.?i+i ^ — ^— + — dy~)~ 



a P,+l , 9, fl P 2 , 9 2 +l W I q^~ + ^— ) — a 



«P 3 + 1 . 33 a Ps, 93+1 n I Q~ H ^ ) — a P393 



Pi9i 



P 2 9 2 



(29) 



And I + (— l) n ~ 1 J is an invariant of the differential equation that 

 we shall come across later. 



The remainder of the treatment is like that of the second order. 



a Pi9i a Pi.9 l +l 



dlogfl _ a Ptqi a p „g 1+ i = ^ dlog# _ 

 dx nA lf dy 



a Pi+1.9i a Pi9i 

 a P 2 +1.92 a P 2 9j 



nA 



= x 2 , (30) 



A = 



a Pi+1.9i a Pi,9i+l 

 a P 2 +1.9s a P 2 .9 2 +l 



Pi, qi being any positive integers such that Pi + qi = n — 1. The con- 

 dition for a solution is : 



d* 1 _,. d*2 __ fl 

 dy dx 



(31) 



where the expression on the left is an invariant of the differential equa- 

 tion. If k\, K2 refer to A (77) into which L(u) goes over under u = yfr-r], 



*i = *i — 



dlogij/ 

 dx ' 



"2 = «2 — 



dlogi/f 



(32) 



The invariant adjoint to (31) is — ~, where 



Xi=- 



n 



n 



f da Pi+1.9i + da Pi,Qi+l \ _ a 



\ dx dy J 



( d(hi+-i.<i* . 9q P 2 ,9 2 +A 



V dx + dy ) a 



a Pi.9i+l 



a P2- 92+1 



with a similar expression for X.2- 

 vol. xliv. — 3 



nA 



(32a) 



