IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 41 



at. 



J H being the cof actor in J of — . For the second order, the formulas 



dXj 



of transformation run as follows : 



k i uxfc ox; 



2d £i , ^ d£i 

 k t dx k dxi ~* dx k 



a — a. 



(36) 



We next define what we mean by invariants for this transformation. 



Definition. By an invariant for a change of independent variables 

 is meant a function of the as and their derivatives with respect to the 

 x's such that the same function of the as and their corresponding de- 

 rivatives with respect to the £'s is equal, by virtue of the formulas of 

 transformation, to the original function multiplied by a power of «/, the 

 functional determinant of the £'s with respect to the x's : 



1(a) = J w I(a). 



What we shall have to say about invariants will, in general, as hitherto, 

 refer to polynomial invariants. 



As to covariants, besides such as we have already made acquaintance 

 with in the case of change of dependent variable, involving u and its 

 derivatives, we have here a second kind, involving dx\, . . . dx m . These 

 two kinds we may distinguish as covariant differential expressions and 

 covariant differential forms respectively. If we replace u in a covariant 

 differential expression by an absolute invariant, it is clear that we shall 

 get an invariant; thus this sort of covariant may be regarded as an 

 operator for deriving invariants ; from this point of view it is what is 

 known as a differential parameter. 



As to the general properties of invariants, we begin with the propo- 

 sition : 



Proposition 19. If we define the weights of the a's and their deriva- 

 tives as in the case of change of dependent variable, page 19, every inva- 

 riant is isobaric, of weight w, with respect to any one of the independent 

 variables. Its partial weight, then, with respect to any one of the 

 variables is the same as with respect to any other. 



Take the case of two independent variables, x and y. Make the 

 change of variables : £ = ex, n = y, c being any constant. Then 



£5i = *-.££« / = 



ipq L'-Upq, 



