46 PROCEEDINGS OF THE AMERICAN ACADEMY. 



which, since / is isobaric, is equal to 



h'Y-if In 1 dIn oo In ~ k - - dlIn ' k \ 



x) \(x') n ' (x'r- 1 dr" ' ""(x') n ' k, '"(x') n - k ' 1 dp '-■■) 



— (x') W I (Jn, Jnl, . . • 0, 0, . . . Jn—k, • • • Jn—k, I, • • • ) , 



if we put 



1 dUn-k 



(X')n-k-l di l 



— J n — , 



k, I- 



When we have proved that Jn—k, i is, like J n _fc, a rational invariant, 

 and that it is of weight n — k — l, w T e shall, then, have the proposition : 

 Proposition 21. Every invariant is a function of the rational 

 invariants 



_ A n - k _ 1 d l A n - k 



~ k ~~(x') n ~ k ' J n ~ k < l ~ ( x ')n-k-i dgi ' 



of weights n — k, re — k — I respectively. Here the A's are the co- 

 efficients of the canonical form into which L(u) goes over, if % satisfy 

 (41), under £ = x( x )- 



1 \drit &n , • • ■ Qm, — 1) &n — 1 > • • • &n — k, • • • &n— k, . . . ) 



= I(J n , Jnl, ... 0, 0, . . . Jn—k, • • • Jn—k, I, • • • )• 



In particular, if J be a polynomial, it is a polynominal in these invariants 

 as well. 



The simplest of the invariants in question are: 



J n = a n- 



Jnl = On' T On— 1. 



re— 1 



T re (re — I) an On" — 2n a m On-i' — 2 (re — 1) a n ' a n -i + 4a n _i 2 



Jn2 — ', 7T • 



re(re — 1) a n 



Jn—2 = 



6n(n—l)ana n -2+2n(n—l) (re— 2) (a n 'a n -i— OnOn-i')— (re— 2) (3n— Y)a n -i 2 



6w (re- 1) On 



We shall find later invariants of a partial differential expression of the 

 second order analogous to J n i and J n 2- 



