18 PROCEEDINGS OF THE AMERICAN ACADEMY 



<H = 2 2t a U ^~ + °*^» 



a = 2^a^. + 2^^. + ^ = W)- 



»,; " * 



(19) 



It is in the invariants and covariants of this transformation that 

 we shall interest ourselves. These terms we define as follows: 



Definition. By an invariant of L(u) under the transformation 

 u = -v/r • 7) is meant a function, 7, of the a's and their derivatives 

 such that the same function of the coefficients of the transformed 

 differential expression is equal, by virtue of the formulas (17), to the 

 original function multiplied by a power of ty. 



I (a's and derivatives) = i/^7 (a's and derivatives), 



or, in a convenient abbreviated notation, 



7(a) = </^7(a). 



If p — 0, we have an absolute, otherwise a relative invariant. 



By a covariant is meant a function, not only of the a's and their 

 derivatives, but also of u and its derivatives, having an invariant 

 property defined in a manner similar to the above. 



We shall concern ourselves wholly with rational, and principally 

 with rational, integral invariants and covariants, and shall always be 

 speaking of the latter, wherever the contrary is not stated or evident 

 from the context. It will be noticed, however, that certain proposi- 

 tions are true for invariants in general. 



We begin with some generalities. Every rational invariant is homo- 

 geneous. For make the transformation u = c-v, c being any constant 

 other than zero. The coefficients of the transformed differential ex- 

 pression are each c times the corresponding coefficient of the original 

 expression, a pq = ca vq , and the same is true of their derivatives ; so 

 that we have: 7(ca) = c^ 7(a), which shows that 7 is homogeneous. 

 We shall, in accordance with the usage in vogue for homogeneous 

 functions in general, speak of yu as the degree of the invariant, even 

 when it is not a polynomial. The corresponding proposition for poly- 

 nomial covariants is that the degree of any term in the a's and their 

 derivatives minus its degree in u and its derivatives is constant and 

 equal to p. 



We proceed now to attach a weight to each of the a's and its de- 

 rivatives. 



