DEMONSTRATION OF A THEOREM RELATING TO THE 
PRODUCTS OF SUMS OF SQUARES. 
[From the Philosophical Magazine, vol. iv. (1852), pp. 515—519.] 
Mr Kirkman, in his paper “ On Pluquaternions and Homoid Products of Sums of 
n Squares” {Phil. Mag. vol. xxxm. [1848] pp. 447—459 and 494—509), quotes from a 
note of mine the following passage :—“ The complete test of the possibility of the pro 
duct of 2 n squares by 2 n squares reducing itself to a sum of 2 n squares is the following : 
forming the complete systems of triplets for (2 n — 1) things, if eah, ecd, fac, fdh be any 
four of them, we must have, paying attention to the signs alone, 
(± eab) (± ecd) — (± fac) (±fdb) ; 
i. e. if the first two are of the same sign, the last two must be so also, and vice versa; 
I believe that, for a system of seven, two conditions of this kind being satisfied would 
imply the satisfaction of all the others: it remains to be shown that the complete system 
of conditions cannot be satisfied for fifteen things.” I propose to explain the meaning 
of the theorem, and to establish the truth of it, without in any way assuming the exist 
ence of imaginary units. 
The identity to be established is 
(w 2 + a 2 + b 2 + ...) (w, 2 + a 2 + &/...) = wf + af + bf + ... 
where the 2 n quantities w, a, b, c,... and the 2 n quantities w,, a n b n c t ,... are given quan 
tities in terms of which the 2 n quantities w /t , a„, b u , c t/ ,... have to be determined. 
Without attaching any meaning whatever to the symbols a 0 , b 0 , c 0 ... I write down 
the expressions 
w + aa 0 + bb Q + cc Q ..., w, 4- a t a Q + b / b 0 + c,c Q ...,