574 
ON POLYZOMAL CURVES. 
[414 
Now, applying the formula to obtain the norm of 
(b - c) Va 2 + 0 + a + (c - a) Vb 2 + 0 + /3 + (a - b) Vc 2 + 0 + y, 
the expression contains six terms, two of which are at once seen to vanish; and 
writing for shortness („) in place of (1, 1, 1, —1, —1, —1) the remaining terms are 
(„) ((b — c) 2 cc, (c-af/3, (a-bf y) 2 
+ 2 („) ((b - c) 2 ol, (c - ay /3, {a - b) 2 7$(b - c) 2 a 2 , (c - a) 2 b 2 , (a - b) 2 c 2 ) 
+ 20 („) ((b - c) 2 a, (c - a) 2 ¡3, (a - b) 2 y$(5 - c) 2 , (c - a) 2 , (a - b) 2 ) 
+ 20 („) ((b — c) 2 a 2 , (c - a) 2 b 2 , (a — b) 2 c 2 ][(b — c) 2 , (c — a) 2 , (a — b) 2 ); 
the first of these terms requires no reduction ; the second, omitting the factor 2, is 
(b — c) 2 a. [ (b — c) 2 a 2 — (c — a) 2 b 2 — {a — b) 2 c 2 ] 
+ (c — a) 2 ¡3 [— (b — c) 2 a 2 + (c — a) 2 b 2 — (a — b) 2 c 2 ] 
+ (a — b) 2 7 [— (b — c) 2 a 2 — (c — a) 2 b 2 + (a — b) 2 c 2 ]; 
which is 
= 2 (a — b)(b — c)(c— a) [be (b — c) a + ca (c — a)(3 + ab(a — b) 7]. 
Similarly the third term, omitting the factor 20, is 
(b — c) 2 a [ (b — c) 2 — (c — a) 2 — (a - 6) 2 ] 
+ (c — a) 2 (3 [— (b — c) 2 + (c — a) 2 — (a — 6) 2 ] 
+ (a — b) 2 7 [— (b — c) 2 — (c — a) 2 + (a — b) 2 ], 
which is 
= 2 (a-b)(b-c)(c- a) [(b - c) a + (c-a)/3 + (a - b) 7], 
and for the last term, omitting the factor 20, this may be deduced therefrom by writing 
(a 2 , b 2 , c 2 ) in place of (a, /3, 7), viz., it is 
= — 2 (a — b) 2 (b — c) 2 (c — a) 2 . 
Hence, restoring the omitted factors, and collecting, we find 
Norm {(6 - c) Va 2 + 0 + a + (c - a) V¥+ 6 + /3 + (a-b) V<f+~0+^y| 
= (6 - cy cc 2 + (c- ay ¡3 2 + (a- by 7 2 - 2 (c - a) 2 (a - b) 2 /3 7 - 2(a-6) 2 (6- C )^a-2(5-c) 2 (c-a) 2 a/3 
+ 4 6 {a-b) {b-c) (c-a) [ (0- c ) a+ ( c - a )/3+ (»-6)7] 
+ 4 (a — b) (b — c) (c - a) [be (b - c) a +ca(c - a) ¡3 + ab(a- b) 7] 
- 40 (a — b) 2 (b — c) 2 (c — a) 2 .