408 
ON A THEOREM IN COVARIANTS. 
[527 
which is diminished by unity. Such change is possible so long as the index \ has 
not attained its maximum value, n — ctj or cr p as the case may be, and there is any 
other index \ 2 , which is not =0: that is, we may pass from Q to Q', from 
Q' to Q" and so on; and it will be sufficient to show that the last term of the 
series is inferior or sharp. We thus pass from Q to R, where 
(Tj ' "" ^2"—"<X.2 —OLm 
R=p 1 p2 ...pm [12... m] 
and a 3 -f a 3 .. + <x m = n — (t x - \ ; or else to 
R =pl ” [12 ... m], 
according as n — ct x is not greater or is greater than <r p . 
Now let [12...m] be inferior; suppose it to be so in regard to 1, that is, let a l 
be less than \n or n — o- 1 greater than \n. Then if a p be less than \n it is less 
than n — (T 1 , that is, we have for R the last-mentioned form which is inferior in 
regard to p, viz. R is inferior; if <x p is equal to or greater than \n, then R, whichever 
its form may be, is sharp as to pi, viz. R is sharp. Hence in either case Q is a 
sum of terms which are inferior or sharp; that is, assuming the theorem for a form 
for which \ + X 2 • • • + does not exceed a given value a p — 1, the theorem is true 
for the next succeeding value a p ; or being true for the case cr p —1 = 0, it is true 
generally. 
Cambridge, 24 April, 1872.