36 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Now, since we have parallel with each other 



du 



it is evident that, however, from the formulas on the left, we may cal- 

 culate the value of any derivative of 6, that value, divided by 6, will 

 be the same function of k\, k 2 , and their derivatives, as is the corre- 



A I Zl f 



sponding derivative of u divided by u of t~/ w > x~ u > anc ^ their 



derivatives. And thus we reach the result that our expression (33) is 

 the same function of K\, k 2 , and their derivatives, that the co variant 



(35) is of — /w, -T-/M, and their derivatives. 

 dx/ dyl 



From this it follows at once that the former, like the latter, is inva- 

 riant of degree one. For the two sets of arguments in question are co- 

 gredient with each other, since we have seen, (32), page 33, that if k\, k 2 

 stand for the same functions of the as that k\, k 2 are of the a's, then 





and this parallelism, of course, extends to the derivatives of the quan- 

 tities in question. Thus the proof of our proposition is complete. 



We see from the formulas for K lf k 2 , (30), page 33, that our invariants, 

 if reduced to a common denominator, will be polynomials in the a's, 

 and their derivatives divided by a power of A . These polynomials will 

 then themselves be invariants of the differential equation. 



The simplest of our invariants are those derived from a pq , where 

 p + q = n — 2. Here we have two invariants, 



