I 
404 [527 
527. 
ON A THEOREM IN COVARIANTS. 
[From the Mathematische Annalen, vol. v. (1872), pp. 625—629.] 
The proof given in Clebsch “Théorie der binâren algebraischen Formen” (Leipzig, 1872) 
of the finite number of the covariants of a binary form depends upon a subsidiary 
proposition which is deserving of attention for its own sake. 
I use my own hyperdeterminant notation, which is as follows : Considering a 
function U = (a,..) (as, y) n , (viz. TJ 1 = (a,..) (x 1} y x ) n &c.), and writing Ï2 = d Xl d y2 - d Vl d Xi &c., 
then the general form of a covariant of the degree m is 
k(Ï2Ï3 23 Y ...) U,11,.. U m , 
where k is a merely numerical factor, the indices a, /3, 7,.. are positive integers, and 
after the differentiations each set of variables (x 1} yO, .., (x m , y m ) is replaced by (x, y). 
I say that the general form of a covariant is as above ; viz. a covariant is equal to 
a single term of the above form, or a sum of such terms. 
Attending to a single term : the sum of the indices of all the duads which 
contain a particular number 1, 2,.. as the case may be is called an index-sum ; each 
index-sum is at most =n\ so that, calling the index-sums <r 1} er 2 , ..., cr m respectively, we 
have n — a 1 , n — <r 2 , ..., n — a m each of them zero or positive: the term, before the 
several sets of variables are each replaced by (x, y), is of the orders n — a 1} n — a 2 , ..., n — <r m 
in the several sets of variables respectively. 
The term may be expressed somewhat differently: for writing V 1 = xd Xi + yd Vl , 
V 2 = xd Xi + yà yq &c.—then (except as to a numerical factor) it is for a function 
(*)(#!, y-^fP the same thing whether we change {x u 3^) into (x, y), or operate on this 
function with V jP, and so for the other sets : the term may therefore be written 
V^ ... Vk (12* I? 23 Y U m , 
being now in the first instance a function of the single set (x, y) of variables.