572
ON POLYZOMAL CURVES.
[414
and the result of the elimination then is
(\2l + /¿53 +1;(5 ) 2 + (V21 + /¿'53 + v'&y + (\"2l + /¿"53 + v"&Y = 0.
But substituting for 21, 53, (5 their values, and writing, for shortness,
— i = b'c" -b"c' + c'a"- c"a' + ah" - a"b,
-j = b"c -be" + c"a - ca" + a"b - a b",
— k = be' —b'c +c a' — c'a + a b' — a b,
A = a (b'c" - b"c) + a (b"c - be") + a" (be' - b'c),
-p = (b'c" - b"c') (a 2 + a' 2 + a" 2 ) + (c'a" - c"a') (b 2 + b' 2 + b" 2 ) + (ab" - a"b') (c 2 + c' 2 + c" 2 ),
— q = (b"c - be") (a 2 + a' 2 + a" 2 ) + (c"a - ca" ) (b 2 + b' 2 + b" 2 ) + (ab - ab" ) (c 2 + c' 2 + c" 2 ),
— r = (be' — b'c ) (a 2 + a! 2 + a" 2 ) + (ca' — c'a ) (b 2 + b' 2 + b" 2 ) + (ab — ab ) (c- + c - + c -),
-l — ( c -b ) (a 2 + a' 2 + a" 2 ) + ( a -c ) (b 2 + b' 2 + b" 2 ) + ( b -a ) (c 2 + c' 2 + c" 2 ),
— m = ( c' —b' )(a 2 + a' 2 + a" 2 ) + ( a - c' ) (b 2 + b' 2 + b" 2 ) + ( b' - a' ) (c 2 + c' 2 + c" 2 ),
-n =( c" -b" ) (a 2 + a' 2 + a" 2 ) + ( a" - c" ) (b 2 + b' 2 + b" 2 ) + ( b" - a" ) (c 2 + c' 2 + c" 2 ),
we find
X21+ /*$ + *(£
= — i (x 2 + y 2 + z 2 )
+ 2i (x 2 + y 1 + z 2 ) — 2x (ix +jy + kz) — 2 A« + ny — mz — p,
with similar expressions for V21 + ¿¿'55 + v%, + /¿"53 + v"(S, and the result is
[i (x 2 + y 2 4-z 2 ) — 2x (ix +jy + kz) — 2Ax + ny — mz—p} 2
+ {j (x 2 + y 2 + z 2 ) — 2 y (ix +jy + kz) — nx — 2 A y + lz — q} 2
+ [k (x 2 + y 2j r z 2 ) — 2z (ix + jy + kz) + mx — ly — 2 A2 — r } 2 = 0,
viz., this is
(x 2 + y 2 + z 2 ) 2 (i 2 +j 2 + k 2 )
+ (x 2 + y 2 + z 2 ) {4A (ix +jy + kz) + 2 (i (ny — mz) +j (lz — nx) + k (mx — ly))
+ 4 A 2 — 2 (ip+jq + kr) + (l 2 + m 2 + n 2 )]
— (lx + my + nz) 2 + 4 (ix 4-jy + kz) (px + qy + rz)
+ 4A (px + qy + rz) — 2 (p (ny — mz) + q(lz — nx) + r (mx — ly))
+ p 2 + q 2 + r 2 = 0,
viz., this is in the rational form the equation of the pair of cone-spheres. The
function on the left-hand side must, it is clear, be save to a numerical factor the
norm of
y/(b — c) 2 + (b' — c') 2 + (b" - c") 2 . V(x — a) 2 + (y — a') 2 + (z — a") 2
+ V(c - a) 2 + (c - a') 2 + (c" - a") 2 . V(> -b ) 2 + (y- b') 2 + (z- U 7 ) 2
+ ^(a—b) 2 + (a' — b') 2 + (a" — b") 2 . V(x - c j 2 + (y — c') 2 + (z — c" ) 2 ,
