52
DEMONSTRATION OF A THEOREM RELATING &C.
[112
where the summation extends to all the quadruplets formed each by the combination
of two duads such as ab and cd, or ac and db, or ad and be, i. e. two duads, which, com
bined with the same common letter (in the instances just mentioned e, or f or g), enter
as triplets into the system of quasi-equations—so that if v = 2 W — 1, the number of quad
ruplets is
h {I 0 - 1) • i O' - 3)} v . £, = ^ v (v - 1) (v - 3),
and the terms under the sign X will vanish identically if only
ee' = = u ’;
but the relation ee' = u is of the same form as the equation ee' = ££'; hence if all the
relations
ee' =
are satisfied, the terms under the sign X vanish, and we have
(w 3 + a 3 + b 3 + c 3 + .,.) = (w 2 + a 2 + b' 2 + c 2 + ...) (w 3 + a 3 + b 3 + c 3 + ...)
which is thus shown to be true, upon the suppositions—
1. That the system of quasi-equations is such that
e o a 0 bo, e Q c Q d 0
being any two of its triplets with a common symbol e 0 , there occur also in the system
the triplets
fo a oC 0 > fodo b o>
9o a c d o> gJ> 0 c 0 -
2. That for any two pairs of triplets, such as
eajb,, eji r d r and f r a r c n , f^db n ,
ooo> ooo t/OOO? J o o o 3
the product of the signs of the triplets of the first pair is equal to the product of the
signs of the triplets of the second pair.
In the case of fifteen things a, b, c, ... the triplets may, as appears from Mr Kirk-
man’s paper, be chosen so as to satisfy the first condition; but the second condition
involves, as Mr Kirkman has shown, a contradiction; and therefore the product of two
sums, each of them of sixteen squares, is not a sum of sixteen squares. It is proper to
remark, that this demonstration, although I think rendered clearer by the introduction of
the idea of the system of triplets furnishing the rule for the formation of the expres
sions w n , a„, b //} c ;/ , &c., is not in principle different from that contained in Prof. Young’s
paper “On an Extension of a Theorem of Euler, &c.”, Irish Transactions, vol. xxi. [1848
pp. 311—341].
