44 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the x's are such contragredient variables; so that, if dx\, . . . dx m , 

 8xi, . • • &x m be two sets of differentials, the expressions 



^ Aij dxi dxj, (38) 



i,i 



2 Mi dxi 8xj (35) 



i,i 



are covariants of weight two. Their coefficients, the Aij a, are inva- 

 riants, for change of dependent variable, of the differential equation. 

 Similarly the (m + p)-rowed determinant formed by bordering A with 

 p rows and p columns, each of which consists of a set of differentials, 

 is a covariant of weight two. 



In the course of the work above we have proved, though we did 

 not at the moment note the fact, that 



3 "Si 55 (40) 



is an absolute covariant. The analogous covariant exists for differen- 

 tial expressions of the nth order. For take the terms of L(u) involving 

 derivatives of the wth order, and form an expression C(u) by substitut- 



in *' for a^...a^-W; ■ • • \^r) ' then c <"> ls the 



covariant in question. For it is easily established by mathematical 



d v i+ • • • + Vm u . 5&+ ■ ■ • +? m u . 



induction, that the coefficient of —r^ „> v in - — 5 — =- is 



6£i Y i . . . d£ m ym dxi*- . . . dx m P m 



., . ./awV' /auV-. /3mY /3m Na- 



me same as the coefficient of — - ] ... I —r~ ) in I - — ) ... I - — 



\d$i) \dinij \dxiJ \dx m J 



Whence it follows, just as for the second order, that C(u) is an 

 absolute covariant. C{J(x 1 , . . . x m )] is also invariant for change of 

 dependent variable, as well as for multiplication of L(u) by <f>. For 

 m = 3, if / satisfy C(f) = 0, / = constant is the equation of the char- 

 acteristic surfaces of L(u) = 0. See Sommerfeldin the Encyklopadie 

 der Mathematischen Wissenschaften, II A7c, Nr. 15. The substitu- 

 tion in C(u) for u of an absolute invariant yields an absolute invariant. 

 Since the coefficient of u in L(u), say a, is an absolute invariant, 

 so then also is C(a). For an ordinary differential expression C(a) 



( da \ n 

 reduces to a n ( j- ] ; so that C(a) is, in a certain sense, the analogue 



