34 PROCEEDINGS OF THE AMERICAN ACADEMY. 



§ 11. Partial Differejitial Expressions : Invariants suggested 

 by the Reduction to Canonical Form. 



In the case of ordinary differential expressions we have seen (propo- 

 sition 15, page 29) that An— k/&, k = 0, 2, 3, . . . n, are invariants of 

 the equation L(u) = 0, the A's being the coefficients of the canonical 

 form derived from L(u) by putting u = 0-n, where is defined by 

 (21), page 28. Are there any corresponding phenomena in the case of 

 partial differential expressions? In the first place it is clear, and 

 might be proved in the same way, that for such partial differential ex- 

 pressions as can be reduced to a canonical form by u = 9 • 77, where 6 

 is defined by (30), the coefficients of that form divided by 6 are inva- 

 riants of the equation, to use the term in such a sense, for that particular 

 class of differential expressions. But for other differential expressions 

 the proof that these same functions of the a's and their derivatives were 

 invariants of the equation would no longer hold. It turns out, never- 

 theless, that they are in fact invariants of the differential equation, as 

 we now go on to show. 



Let us see just what it is that we wish to prove. Consider the for- 

 mulas for the a's in terms of the a's, 



^ ^ (n - I) I 6*-*<A n ~ 



apQ ~ £> Po (n-k)\i\(Jc-l-i)l ap+i ' q+k - l ~ i dxi dyk-i-i > U 7) 



p + q = n — 1c. 



Now suppose that we substitute in this formula for ty and its deriva- 

 tives and its derivatives, — , — being given by (30), that is, 



ox dy 



and the higher derivatives of 6 being determined from these formulas by 

 differentiation and the substitution, at each step of the process of differ- 



entiation, of k^O, k 2 6 for — - , — respectively. This rule, it will be 



dx dy l 



noticed, does not completely determine the expressions to be sub- 

 stituted ; for we may, to take an instance, in accordance with its direc- 



1 • <> d 2 */' . 6ki „ dO . / d*i \ 



tions, substitute tor — — either — (/ + q — . that is I H *i*2 I &, 



dxdy dy dy \dy J 



or else -— 6 + *2 t- , that is ( h ^1^2 J #• But this does not matter. 



ox dx \dx J 



We suppose the expressions to be substituted for any given derivative 



