PROFESSOR CAYLEY ON POLYZOMAL CURVES. 43 



giving rise to a similar set of forms 



a x = , — ad + ha, a'g + b'a, — eg — b'h , 



\ = - c'b- g'h, , - f'b - g'f, - f'h + c'f , 



C j = b'c + Kg, - f'c + h'f, . , f'g+g'f, 



clj = g'c + lib, — h'a + a'c, — a'b — g'a, 



and leading to 



sL a . _ £i b . _ 3L c . . _ It d 



«-c x eg x aj *■ ag 1 



1 TYl 7b T) 



so that the equation - L + - r ; + ^ + ^ 1 = 0,is transformable into 



*■ a x h x Cj dj 



''/' 7 C "/ A/ /A 



ac * c// x aj ' ag 1 



98. Let A,B,C, D, be, as above, points on a circle ; (A r , D x ) and (B v C x ) the 

 anti-points of (A, B), (B, C) respectively. Write 



A = (f -«:)(„- «'~) , A x =(§-«*)(> -8s), 

 B = (| - /&)(, - /3s) , B x = (| - #)(, - y'a), 

 c = (I - 7-) (n - y'z) , C x = (g - 7 z)(v - 8z) , 

 D = (g - as) (, - 8s) , Dj = (g - a 8 ) („ - b'z) ; 



then we have identically 



(a -a) (8'- a') B =(/3-8) (j3'-8') A + (jS-a) (/3'-a) D - (^ — 5; C/S' - a'j Aj — C,8 — a) (/S'— S'j D, , 



(d-a)(a'-a')C =( 7 -a)(/-b')A + (y-a)C7'-«)D-(7-a)(7-a')A 1 -(7-a)(/-a')D 1 , 



(«_«)(«•_«') B 1 = (i3-8)(7 , -8') A + 0-«)( 7 '-cOD-(/3-a)(7 , -« , )A 1 -(/3-a)(7'-8')D 1 . 



(a-a)(8-«')C t = (7-a)(/3'-a - ) A + ( 7 -a)( i 8'-«')D = (7-8)(|S'-a')A x -(7-a)(j8'-a') D t , 



or in the foregoing notation 



//' B = ///A + cc'D + gc'k x + cg'D i , 

 ff'C = hh'k + bb'D - hb'\ - &A'D X , 

 ff B x = gh'k - cb'D - gb'\ + ch'D 1 , 

 ff'C 1 = g'hk - bc'D + hc'\ - b'g'D 1 . 



Further Properties in relation to the same Sets (A, B, C,D) and (A v B v C V D- S ) — 



Art. Nos. 99 to 104. 



99. It is be shown that in virtue of these equations, and if moreover 



— + T- + — + ^ =0, then it is possible to find l x , m lt n u p u such that we have 



<v U C (1 



identically 



— Ik + mB + nC — pD + l l k i — m l B l — n y C x + ^jD, = 0. 



