§27] 
LAW OF RECIPROCITY 
45 
of degree d and weight w— 1. By the Lemma there exists * 
an S for which tiS is any assigned S'. Thus the coefficients 
of our S'^tiS are arbitrary and hence are linearly independent 
functions of the (w; d, p) coefficients of 5. Hence the con 
dition tiS=0 imposes (w— 1; d, p) linearly independent linear 
relations between the coefficients of 5 and hence determines 
(w — 1 ; d, p) of the coefficients of S in terms of the remaining 
coefficients. Thus the difference gives the number of arbitrary 
constants in the general seminvariant S, and hence the number 
of linearly independent seminvariants S. 
27. Hermite’s Law of Reciprocity. Consider any partition 
W = Wi+W2 + . • .+«« 
of w into 8^d positive integers such that p't.nif.nz . . . ^ n s . 
Write n\ dots in a row; then in a second row write n-2 dots 
under the first ns dots of the first row; then in a third row 
write ns dots under the first ns dots of the second row, etc., 
until w dots have been written in 8 rows. 
Now count the dots by columns instead of by rows. The 
number m\ of dots in the first (left-hand) column is «5; the 
number m2 in the second column is S mi ; etc. The number 
of columns is ni £ p. Hence we have a partition 
w = mid-m2 + . • .+m v 
of iv into 7r Sp positive integers not exceeding d. 
Hence to every one of the (w; d, p) partitions of the first 
kind corresponds a unique one of the (w; p, d) partitions of 
the second kind. The converse is true, since we may begin 
with an arrangement in columns and read off an arrangement 
by rows. The correspondence is thus one-to-one. Hence 
(w; d, p) = (w; p, d). 
By two applications of this result, we get 
(w; d, p)-{w-1; d, p) = {w\ p, d)-(w-1; p, d). 
Hence, by the theorem of § 26, the number of linearly independent 
* Provided pd—2(w—1)>0, which holds if pd—2wfS). But if pd—2w<0, 
our theorem is true by the Lemma in § 25.