106 PROFESSOR CAYLEY ON POLYZOMAL CURVES, 



viz., this is 



O 2 + y 2 + * 2 ) 2 (i 2 + y 2 + & 2 ) 



+ (x 2 + y 2 + z 2 ) {4A(is; + /;/ + kz + 2(i(ny — mz) + j(lz — nx) + k(mx — ly)\ 



+ 4a 2 - 2{ip + /g + &r) + (7 2 + m 2 + n 2 )\ 



— (Ix + my + nz) 2 + 4(ix + jy + kz) (px + qy + rz) 



+ 4 A(px + qy +rz) — 2 (p>(ny — mz) + q{Jz — nx) + r{mx — ly)\ 

 + 2 ,2 + q 2 + r 2 _ 



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 



J(b-c)* + (b'-c)* + (b"- cy. J(.r-af + (2f- a y +(z-ay 

 + J(c-af + {c'-a'f + {c"-a"f. J{x- bf + {y - b') 2 + (z- b"f 

 + y/(a-b) 2 + {a'-b'f + {a" -b"f. Jicc-c) 2 + (y- cf + (z - c"f , 



the numerical factor of the expression in question is in fact = — 4, that is, the 

 norm is 



= - 4(,; 2 + y 2 + z 2 ) 2 (t» + j 2 + V-) + &c. ; 



so that attending only to the highest powers in (x, y, z) we ought to have 



Norm { J(b - cf + {V - c'f + (b" - c"f + */(c - af + (c' - a'f + (e* - a"f + V(a - bf + (a' - by + (a" - b"f} 



It is easy to see that the norm is in fact composed of the terms 



2{b'-c r f{ (b-c) 2 -(c-af-(a-b) 2 }, 

 + 2 ( C ' -a') 2 {-{b - cf + (c- a? - (a - b) 2 } , 

 + 2(a'-b'f{-(b-cf-(c-af+(a-b) 2 }, 



and of the similar terms (a, b, c), (a", b\ c") and in (a', b', c'), (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)(a-b), 

 which is 



— „'2 



— f 



a' 2 (b-c) 2 + b' 2 (c-a) + c' 2 (a-b) 2 



+ 2b'c (a -b)(c-a) + 2c'a (b -c)(a-b) + 2a'b'(c - a) (b - c) 

 {a'(b - c) + b'(c - a) + c'(a - 6)} 2 

 = k 2 ; 



and the value of the norm is thus = — 4(z 2 + j 2 + A 2 ), as it should be. 



