30 PROCEEDINGS OF THE AMERICAN ACADEMY. 



gree. This may be done by mathematical induction. For I n — k is such 

 an invariant. Suppose, then, that In— k, i is. 



In-k, 1+1 = - e A n - k =- eIx (A n - k ) = - Tx (I n - k , 1 6) 



J I -l (In — 1 t 

 n—k, I ln—k, I • 



n a n 



So that I n—k, m-i ls rational, and will be an invariant of the first degree 

 by the following proposition : 

 Proposition 16. If / be an invariant of degree k, then so also is 



j/ K ttn—l j 



n a n 

 For it is equal to 



— ~(na n ' — a n -\) I + a n V — ka n 'I 

 a n [_n J 



a n L 



k d 



- (na n ' — a n -i) I + a n k + l 

 n 



Here a n and na n ' — a n —\, which is simply (— l) n b' n —i, are invariants 

 of the first degree, while I/a n k , and therefore its derivative, too, is an 

 absolute invariant. It is apparent that the whole expression is an inva- 

 riant of degree k. 



§ 10. Partial Differential Expressions : Conditions for the 

 Possibility of Reduction to Canonical Form. 



We pass now to partial differential expressions. Here it is not in 

 general possible, as will appear, to reduce the expression, by a change 

 of dependent variable, to canonical form, where now by a canonical 

 form we mean an expression in which the coefficients of all the (n — l)st 

 derivatives are zero. Let us ask ourselves under what conditions this 

 will be possible. The problem is of interest, not only in itself, but be- 

 cause it will suggest to us certain expressions analogous to the invari- 

 ants An—jc/0, to which we were led, in the case of ordinary differential 

 expressions, by the reduction to canonical form ; and these expressions 

 will turn out to be, like their prototypes, invariants of the differential 

 equation L(u) = 0. We shall also find something analogous to the 



<') 

 invariants A n — k /9 of the differential expression L(u). 



Let us notice first that the property, the conditions for whose exist- 

 ence we are seeking, is an invariant property. It is evidently so for a 

 change of dependent variable ; and it is so also for a multiplication of 



