PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



85 



192. The three systems have been obtained independently, but they may of 

 course be derived each from any other of them : to show how this is, recollecting 



that we have 



RA, BB, BG, RD = a v b v c v d x , 



SA,SB, SG, SB- = a 2 , \, - c 2 , - d 2 , 

 TA, TB, TG, TD = a 3 , b v c v d z ; 



then to compare 

 similar triangles 



and similar triangles 



(a v b v c v d x ), (a 2 , b 2 , c 2 , d 2 ) ; 



SBG give b x — c x : — c 2 : b 2 , 

 SAB = a x — d x : — d 2 : a 2 , 



RAG give « 5 



t' 2 '. C'j : <Xy, 



RBD = b 2 -d 2 : d x :b 1 ; 



using these equations to determine the ratios of « 2 , £ 2 , c 2 , d 2 we have 



Ctn ■""* C* 



that is 



and hence 



that is 

 but 



b 2 _ / =/ , or d x a 2 - d x c 2 - c x b 2 + c x d 2 = ; 



b 2 (— b x c x + c x 2 + a x d x — d x 2 ) + c 2 (— b x d x + c x d x + a x c x — e x d x ) = , 

 b 2 (c* - d x 2 ) + c 2 (a x c x - b x d x ) = , 



or the equation gives b 2 + J* c 2 = , or say b 2 : e 2 = b x : — d x , and this with 

 1 ~ c } = ^ '- = - 8 , gives all the ratios, or we have 



a 2 :b 2 :c 2 :d 2 = b x (a x - d x ) : b x (b x - c x ) : - ^(a x - rfj : - d^ - c x ) . 



We have then for example 



b 2 — c 2 : c 2 — a 2 : a 2 — b 2 = b x — c x : c x — a x : a x — x b ; &c, 



showing the identity of the forms in (a v b x , c x , d x ) and (a 2 , b 2 , c 2 , d 2 ) . 



Transformation to a New Set of Concyclic Foci. — Art. No. 1 93. 

 193. Consider the equation 



JlA + VmB + J^C = , 



which refers to the foci A,B, C, and taking D the fourth concyclic focus, let 



VOL. XXV. PART I. Y 



