32 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



over by any transformation u = y]r-T] into an expression with coefficients 

 a, and H i be the same function of the a's that k^ is of the a's, we see 

 without difficulty from (19), page 18, that 



K{ — Ki ™~ 



dlogi/' 

 dxi ' 



(26) 



so that 



9k{ 8kj di<i dxj 

 dxj dxi dxj dxi 



The invariant of the differential equation adjoint (proposition 11, 

 page 25) to the invariant just found is r- 2 - — — , if \ be the same 

 function of the 6's that ic i is of the a's, that is, by (8), page 9, if 



\i = - 



2A 



(27) 



The difference, ( ^ — T^ ) ~ V a - ^ ~~ H - / > °f tnese t wo invariants 



of the differential equation is an invariant that we shall come across 

 later. 



Consider next a differential expression of the wth order. If u =■ 6 • t] 

 carry it over into a canonical form, we must have 



n 



( aiogtf , aiog0\ , 



[Op+h 3 q x + a V, 3+1 q J + a PQ ~ 0, 



dx 

 p ■+ q = n — 1, p = 0, 1, . . . (n — 1). 



Conditions necessary and, in general, sufficient for these equations 

 being algebraically solvable are that all # three-rowed determinants of 

 the matrix 



CH.n— 1 dQn O-O.n—l 



a P+l. 8 °P> 3+1 a P1 



a n Q On— 1, 1 dn—1, 



should vanish. Any one of these three-rowed determinants 



