82 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



pressions of (a, b, c, d) in a form adapted to the formulae of No. 49, viz., 



a : b :c : d=(b — c)(c — d)(d — b) : — (c — d)(d — a)(a — c):(d — a)(a — b)(b — d): —(a — b)(b — cj(c-« 



so that, assuming 



I : m : % = g(b - cf : <e - a) 2 : r(a - hf , 



the equation - + -^ + - = , becomes 



g(b - c){a - d) + e'yC - a) (b - d) + r(a - b)(c - d) = , 



and the equation of the curve may be presented under any one of the four 

 forms 



( , *fad - c) , JiQ, - <l) , J § (c - b) ) ( JK JB, JC, J6 )= . 



Jr(c - d) , . , Jl(d - a) , Ja{a - c) ' 



Jt(d - b) , Jz{a - d) , . , Jryb - a) ' 



+/g(b - c) , Je(c - a) , Jr(o - b) , 



Case of the Symmetrical Circular Cubic — Art. No. 187. 



187. For a circular cubic we must have 



g(b - c)(a - d) + a <■ - a)(b - d) + r(« - b)(c - d) = 

 J${b - c) + yfe(c -a) + Jr(a - b) = . 



These equations give JJ: J&: Jt = 1:1:1 (values which obviously satisfy the 

 two equations), or else 



»/g '• J a '■ Ji = n + d — b — c :b + d — c — a -. c + d — a — b . 



In fact, these values obviously satisfy the second equation ; and to see that they 

 satisfy the first equation, we have only to write them under the form 



e : g : r = M- 4(6 + c)(a + d) : M- 4(c + a)(b + d) : M - 4(a + b)(c + d) , 



where M=(a + b + c+ d) 2 . The first set gives for the curve 



(b - c) V /A + (c - a) JB + (a-b)Jc = , 



but this contains the line z — not once only, but twice; it in fact is (y 2 = 0), 

 the axis taken twice; the only proper cubic with the foci A, B, C,D in lined is 

 therefore 



(6 - c)(a + d-b - c) V /A + (c - a)(b + d - c - a) Jb + (a - b)(c + d - a — b) Jc = , 

 the equation of which is, of course, expressible in each of the other three forms. 



Case of the General Circular Cubit — Art. Nos. 188 to 192. 



188. Returning to the general case of the circular cubic, the lines jBC, AD 



