PROFESSOR CAYLEY ON POLYZOMAL CURVES. 105 



These last equations give 



X-.Y-.Z- xA + ^B+oC 

 : x'A + /*' B + »' C 

 : X'A + /SB + /G , 



where 



X = b'c" - b"c' + (c' - b')z - (c" - b")y , 

 (i, = ca — c a + (a — c)z — (a — c )y , 

 » = a'b" - a"b' + (b' - a')z - (b" - a")y , 



\' = b"c - be" + (c" - b")x - (c - b)z , 

 im' = c"a — ca" + (a" — c")x — (a — c)z , 

 v = a"b - ab" + (b" - a")x - (b - a)z , 



\" = be - b'c + (c - b)y - (c - b')x , 

 ft'' = ca' — c'a -f- (a — c)y — (a' — d)x , 

 „" = ab' — a'b + (b — a)y — (V — a')x ; 



and the result of the elimination then is 



(XA + /.B + » C) 2 

 + (X A + p'B + v C) 2 

 + (X'A+^'B + v"Cf = . 



But substituting for A, B, C their values, and writing, for shortness, 



- i = b'c" - b"c' + c'a" - c"a' + a'b" - a"b , 

 -j = b"c - be" + c'a - ca" + a'b - ab" , 



- h —be' — b'c + ca' — c'a + ab' — a'b , 



A = a(b'c" - b"c) + a'(b"c - be") + a"(bc' - b'c) , 



- p = (b'c" - b"c') (a 2 + a' 2 + a" 2 ) + (c'a"-c"a') (b 2 + b' 2 + b" 2 ) + (a'b"-a"b') ( c 2 + C 2 +c" 2 ) 



- q = (b"c - bc")(a 2 + a' 2 + a" 2 ) + (c"a -ca")(b 2 + b' 2 + b" 2 ) + (a"b -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 2 + c' 2 + c" 2 ) 

 -I = (c -b )(a 2 + a' 2 +a"-) + (a -c )(b 2 + b' 2 + b" 2 ) + ( b -a ) (c 2 + c' 2 + c" 2 ) 



- m = (c' -V )(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 



xA + pB + »C 



= - i(x 2 + y 2 + z 2 ) 



+ 2i(x 2 + y 2 + z 2 ) — 2x(ix +jy + kz) — 2 Ax + ny — mz — p , 



with similar expressions for X'A + /t/B + i/C, X"A + //B + ^''C, and the re- 

 sult is 



U(x 2 + y 2 + z 2 ) — 2x(ix + jy + kz) — 2 Ax + ny — mz — pi 2 

 + {j(x 2 + y 2 + z 2 ) — 2y(ix + jy + kz) — nx — 2Ay + lz — q\ 2 

 + Sk(x 2 + y 2 + s 2 ) — 2z(ix + jy + kz) + mx — ly — 2az — r\ 2 = , 

 VOL. XXV. PART I. 2D 



