PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



83 



meet in R, and if we denote by a v b v c v d x , the distances from B of the four 

 points respectively, so that b 1 c x = a l d 1 — rad. 2 M, then observing that a, b, c, d 

 are proportional to the triangles BCD, CD A, DAB, ABC, with signs such that 

 a + b + c + d = 0, we find 



a : b : c : d = — rf,(6j — Cj) : c x {a x — (/, ) : — 6 1 (a 1 — d x ) : a^&j — c t ) ; 



I 



m . n 



and this being so, the equations — + ^- + - = , Jl + Jm + Jn — , give 

 two systems of values of Jl: Jm : Jn , viz., these are 



Jl : Jm : Jn -\- c i- c x~ a i '■ a i ~ h > 

 and 



= ^i — c i • c i "I" a i '• ~ a i ~~ b i ■ 



(To verify this, observe that for the first set we have 



I .m ,n (b, 



a b c 



o 2 



+ ( c i - "l) 2 + («i - h) 2 _ 



d \(h - c i) c i( a i - d d - & i(% - d \) 



= b i ~ c i + Lfc + !i 2 -5 _ ft i\ 



— d x a x — rfj \ l c x * 6 a / 



= 6 i~°i + & i ~ c i fV - A 



— c?-! a t — d x V&jCj / 



= - 5 i ~ c i + & i ~ c i f ffi i _ l/\ - ; 

 d x a x — ^j \d 1 / 



and the like as regards the second set). 



189. These values of Jl : Jm : Jn give the equations of the two circular 

 cubics with the foci {A, B, C, D), the equation of each of them under a fourfold 

 form, viz., we have 



( 



d. 



e x — d 1 , . , d v — a x , «! — Cj 



c h ~ \ > a \ ~ d\ > b i ~ a \ ■ ^ — di 

 \ — c i > c i — «i '» a i — & i • 



& x -^, Ci-6, )(J*,jB,JC,jD) = 



(first curve) , 



and 



. , - Cl - ^ , ^ + &, , - 6i + e, )( ^A, VB, s/C, VD ) = f .O 



rfj + Cj , . , a x — d x , C]_ — a, 



-\- d x , d 1 -a 1 , . , a, + \ (second curve). 



6 X - c t , c, + «i , —«! — &!, 



190. Similarly C-4 and BD meet in £, and if we denote by a 2 , b 2 , c 2 , d 2 the 

 distances from S of the four points respectively, so that c 2 a 2 = b 2 d 2 — rad. 2 £ 

 (observe that if as usual A, B, C, D are taken in order on the circle 0, then A, C 

 are on opposite sides of S, and similarly B, D are on opposite sides of S, so that 



