60 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



A (0 — 2 ^kh 



I,] 



are absolute invariants. 



For one independent variable, these invariants reduce to 

 1 an f(2ai — a v . 



/\2 



(2 ai - a u y 

 16 an 



and — , ( ) respectively. The first of these is 



8 2 oi — a n ' \ «n / 



the square of the invariant J nl , page 46, for w = 2, divided by 16an ; 



while J n2 , for n = 2, is 8 A(Z) — 4 A 2 (/). Thus we have found, for the 



second order, invariants of a partial differential expression analogous 



to the invariants J n \, J n 2 of an ordinary differential expression. 



We shall accept from Cotton the fact that A 2 (/) is an absolute in- 



—j- dxidxj is an absolute co- 



i,i 

 variant, — cf. (38), page 44, — any invariant of this quadratic differen- 

 tial form of weight w will be an invariant of L{u) of weight — w. Now 

 since, page 57, the I's are contragredient to the dx's, 



lii 



Ai, 



h 



Iml 



A 

 h 



A 



I 



m 







that is, 2 ~~a~ hljt 1S an invariant of weight two of the differential 



»>; 



form, ^aaklj or A(Z) is, then, an absolute invariant of L{u). 



i,i 



Cambridge, Mass., 

 April, 1908. 







