38 
ALGEBRAIC INVARIANTS 
equals K{A, B, . . . ; £, rj, . . .)• Now K is homogeneous 
in x, y, . . . , of order to; thus 
a~ u K(a p a, a p b, . . x, y, . . .)=a q *K{a, b, . . . ; x, y, . . .). 
Thus if K has a term of degree dma,b,. . . , then 
a~ u -a pd =a qX , pd—co = q\, 
so that d is the same for all terms of K. 
If f is a form of order p in q variables and if K is a covariant 
of degree d, order w and index then pd — w = q\. 
22. Weight of a Covariant of a Binary Form. In 
/ = aox p +paiX p ~ x y + . . . + (^ja i x p ~ i y i +. . .+a v y p 
the weight of a t is k. We now attribute the weight 1 to r 
and the weight 0 to y, so that every term of / is of total 
weight p. 
Apply to / the transformation x=£, y=m7. The literal 
coefficients of the resulting form are 
Ao = ao, Ai=aai, . . A p =a p a p . 
If K is a covariant of degree d, order co, and index X, then 
K{A 0 , . . . , A p ; £, rj) =a K K{a 0 , . . . , a p ; x, y). 
Any term on the left is of the form 
c^lo^T 1 . • . A p e Ptj r rj u ’~ r (eo+ei+ • • • +e p =d). 
This equals 
cao e oa! e i . . . a p e px r y‘°~ r a w ~ u (W = r+ei+2e 2 + . . . +pe p ). 
This must equal a term of the right member, so that 
W— co = X. But W is the total weight of that term. Hence 
every term of K is of the same total weight. A covariant 
of index X and order co of a binary form is isobaric and its weight 
is co-f-X. 
For a form /of order p in q variables, we attribute the weight 1 to x h x 2 , 
. . . , and the weight 0 to x q ; then (§ 17) every term of / is of total 
weight p. By a proof similar to the above, a covariant of index X and order 
to of / is isobaric and its weight is co+X.