ING [112 
112] TO THE PRODUCTS OF SUMS OF SQUARES. 51 
nultiply without making 
)ined in the way of mul- 
expression of the quasi- 
= +, may be written in 
' we form a system of 
>nIy, and the arbitrary 
the quasi-product we 
vill assume the form 
v,, a,, b,, c, ..., homo 
of w„ being obviously 
+- b,uo, cw, -I- c,iv,... but 
ets entering into the 
system of quasi-equations as on the values given to the signs +; the quasi-equations 
serving, in fact, to prescribe a rule for the formation of certain functions w„, a,,, b„, c„, ..., 
the properties of which functions may afterwards be investigated. 
Suppose, now, that the system of quasi-equations is such that 
e 0 a 0 b 0 , e 0 c 0 d 0 
being any two of its triplets, with a common symbol e o , there occur also in the system 
the triplets 
fo a cC n > fAK, 9o a od 0 , 9o h o c o; 
and suppose that the corresponding portion of the system is 
e 0 a 0 h o = e, e o c o d 0 = e, 
fo a o°o = £> fo d o h o = ?> 
g 0 a 0 d Q = l, g 0 b 0 c 0 = t, 
where e, t,, i, e', , t each of them denote one of the signs 4- or —; then e a ,f a ,g n will 
contain respectively the terms 
e {ah, - a jo) + e (cd t — c,d), 
% (ac, - a t c) + £ (db, - djo), 
t {ad, — ajd) + i {be, — b,c); 
and ef +fj + gf contains the terms 
(a 2 + b~ + c 2 + d~) (aj + bj + cj + df) — tfaf — Wbj — c 2 c~ — d?dj 
+ 2 [ee / {ab, — ab) {cd, — c,d) 
+ ££' {ac, — a,c) {db, — djo) 
-f a {ad, - a,d) {be, — b,c)] ; 
and by taking account of the terms evo, + e,w, fw, + fiv, gw,+g,w in e„, f„, g„ respect 
ively, we should have had besides in ef +ff 4- g,j the terms 
(e 2 +f 2 + g 2 ) w, 2 + {e, 2 +f 2 + gf) w 2 
+ 2 {ee, +ff,+ gg,) ww,. 
Also iv„ 2 contains the terms 
w 2 w, 2 -1- a 2 a, 2 + b 2 b, 2 + c 2 c, 2 + d 2 d, 2 
— 2 {ee, +ff, + gg,) ww 
whence it is easy to see that 
wf + a„ 2 + b„ 2 + c„ 2 +... = 
{w 2 + a 2 + b 2 + c 2 + ...) {w 2 + a 2 + b, 2 + c, 2 + ...) 
+ 22 [ee'{ab, — a,b) {cd, — cd) 
+ tjf {ac, — a,c) {db, — djo) 
+ ll {ad, — a,d) (be, — b,cj\. 
7—2