ON POLYZOMAL CURVES. 
573 
414] 
the numerical factor of the expression in question is in fact = — 4, that is, the 
norm is 
= — 4 {x 2 + y 2 + z 2 ) 2 (i 2 +j 2 + k 2 ) + &c.; 
so that attending only to the highest powers in (cc, y, z) we ought to have 
Norm {V(b—c) 2 + (b'—cf+(b" —c'') 2 + f(c —a) 2 +(c' — a') 2 + (c"—a") 2 + f (a—b) 2 + (a!—b') 2 4-(a"—b") 2 } 
= — 4 (i 2 +j 2 + k 2 ). 
It is easy to see that the norm is in fact composed of the terms 
2 (b' — c') 2 { (b — c) 2 — (c — a) 2 -{a - 6) 2 }, 
+ 2 (c - a') 2 {— (6 — c) 2 + (c — a) 3 — (a - b) 2 }, 
+ 2 (a — b') 2 {— (b — c) 2 — (c — a) 2 + (a — 6) 2 }, 
and of the similar terms (a, b, c), (a", b", c"), and in {a!, b', o'), (a", b", c"); the above 
written terms are = — 4 into 
(b' — c') 2 (a — b)(a — c) + (c— a') 2 (b — c)(b—a) + (a — b') 2 (c - a)(c — b), 
which is 
= ci' 2 (b — c) 2 + b' 2 (c — a) 2 + c' 2 (a — b) 2 
+ 2b'c' (a — b)(c — a) + 2c'a (b — c)(a — b) + 2 alb' (c - a)(b — c), 
= \a' (b — c) + b' (c — a) + c (a — 5)} 2 
= k 2 \ 
and the value of the norm is thus = — 4 (i 2 + j 2 + k 2 ), as it should be. 
Annex III. On the Norm of (b — c) + (c — a) Vi8° + (a — b) V(5°, when the Centres 
are in a Line. 
The norm of V77 + V V + V W is 
= (1, 1, 1, -l$tf, V, W) 2 , 
whence that of V U + U' + V V + V + f W + W is 
= (1, 1, 1, -1, -1, -1 $C7, V, W) 2 
+ (1,1,1,-1, -1, -1 W> r. wy 
+ 2(1, 1, 1, -1,-1, -1 $CT, V, W£U', V', W), 
where the last term is = 2 into 
U'(U- V- W)+V'(- U+ V- W) + W'(- U- V+ W) ; 
and the norm of V U + 17' + U" + V V + V + V" + V W + W + W" is obviously composed 
in a similar manner.