472 MR. SYLVESTER ON THE QUOTIENTS RESULTING FROM THE EXPANSION 
Let the n factors be constituted of m, factors n, factors factors j;,. Then 
where 
and 
where 
Hence 
+ &c. &c. 
me, 
.h 
'mi' 
rr^ 2 V{ \ I I 4- 
rtii 
Again, in U; the term containing ??, will be 
;ji2{ ('^1 — ??2) (^1 — ^3) • • • (^1 ^i) ^(^2 % • • • ^i) } 
=Wi^i X {m^.m^. . X (^1— 
2 
Hence 
Hence, therefore, U,-T, vanishes whenever the roots of/r contain only ^ distinct 
groups of equal roots, and it has been shown that U, and each contain on y 2/ 
of the coefficients of fx, so that U,-T, is a function only of and these 2 i - 1 letters, 
and consequently by virtue of the Lemma in Art. («.) U,-T, is universally zero, 1. e. 
U, is identical with T„ as was to be proved. In the same manner as observed in a 
preceding marginal note, the expression given in the antecedent articles or le 
numerator of the ith convergents having been verified for the case of the roots con- 
sisting of only * distinct groups, could have been at once inferred to be generally true 
by aid of the Lemma above quoted. 
Art. (/.) Since the coefficient of in is Zi_i X Z^, we deduce the unexpecte 
h,...hi_,)X^^{Jh h...hi)=Vl + Vl+...+^l, 
where F,=n{he-KKf^e-K) -A^h-heiJlif^ 
So that every simplified Sturmian quotient to^, when the (n) roots of/r are real, will 
be the sum of n squares. But the equation is otherwise remarkable, m exlnbitmg 
the product of the sum of squares by another sum of-^ ^^07;^ 
squares under the form of the sum of n squares. 
