54 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



(ddnO . da„_i,i\ 

 OnO Gn-1,1 2 a n _i, — re I — 1 J 



/ 5a n _i i 60n— 2 , 2 \ 

 an-n a„_ 2 , 2 2 a n _ 2il - re \-^— + _^_ j 



ai.n— 1 



flOn 



2«o,n-i-re(^- + — j 



should vanish. These three-rowed determinants are invariants, for 

 change of dependent variable, of the differential equation. For any 

 one of them may be written as I + (— l) n— * J, where I is an inva- 

 riant of the form (28), page 33, and J is the adjoint invariant (29). 



If these conditions are fulfilled, we may solve for — — , — any 



J dx dy J 



two of the equations (44) : 



dlog<£ 

 dx 



a P& — n( 



dx 



da P 2 +l,Q 2 



dx 



+ 



dy 



da p„qi+l 



dy 



) 

 ) 



a Pi.<h+l 



a 



P 2 . 9a+l 



n A 



= Li, 



dlog<j!> 

 dy 



a Pi+i.fli ^ a 



PiSi 



re 



( 



da 



dx 



+ 



da 



Pi. gi-f 1 

 dy 



«*rfi.«, ^p,<z, n { dx + dy ) 



nA 



= L 2 , 



A = 



a P!+1.3i 



a P2+1.92 



a Pl.?l+l 



a P 2 .92+l 



Pi + q% = n — 1. 



And the necessary and sufficient condition for the existence of a solu- 

 tion, log <£, is -z — = 0. Here the expression on the left is an in- 

 variant, for change of dependent variable, of the differential equation. 

 For L\ = Ax — k 1} L 2 = A 2 — K 2, the k's and Vs being given by (30) 



and (32 a), page 33 ; and — - is, therefore, the difference be- 



dy dx 



tween an invariant and the adjoint invariant. 



We shall carry the solution of the problem no further. To com- 

 plete that solution we should next have to go on and write down the 



