42 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



and we may write 



a = . ah' — ah, og — a'g, gh! — gh , 



h = bh'-h% . , bf-h'f, hf'-h'f, 



c = eg - eg, cf - c'f, . fg - fg , 



(1 = cb' — cb, ac — a'c, ba — b'a, 



viz., the expressions in the same horizontal line are equal, and a, b, c, d are pro- 

 portional to the expressions in the four lines respectively. 

 95. I say that we have 



c 'f _ c 9 \ 

 ah ' ~ bh 



a f . fa 



«f 



ah 



d, 



viz.. this will be the case if 



bc'a = /"/ il , 

 a, \, = /i/'d, 

 a'bc = fg\\ , 



and selecting the convenient expressions for a, b, c, d, these equations become 



be' (gh' — ij'h) = (j'h (cb' — r'li ) , 

 ac'(hf - h'f)=fh(ac' - a 

 a 'b (ff - fg) = f'J (b"' - b'a) , 



viz , these equations are respectively bgc'h' = b'tjdi, cha'f = dh'af afb'g '=a'f'bg, 

 and are consequently satisfied. It thus appears that the equation 



is transformable into 



l m n i' 

 a + l7 + c + d = 



cf , eg af i <i 



ah * T bh '" T af " T ab 

 which is of course one of a system of similar forms. 



96. Take (A v B x ) the anti-points of A, D ; (B v C',) the anti-points of (71 C) ; 

 or say that the circular co-ordinates of A v 2? n C\, B x are (a, &, 1), (ft y\ 1), 

 (7, j8', 1), (<S, a\ 1) respectively; the points A lt B x , C 19 B t are, as above mentioned, 

 on a circle, the condition that this may be so being in fact 



1, a, 8, ad =0, 

 1, ft 7 ft/ 



1,7.1 s ' 7ft 



1, d, a, da 



af :bg : ch = af : b'g : r'h' 



equivalent to 



97. Let (a 1? b 15 c 19 d x ) be the corresponding quantities to (a, b, c, d), viz . 

 a x : \ : c x : d\ — B 1 C 1 D 1 : — C 1 B 1 A 1 : B 1 A 1 B l : — A 1 B 1 C 1 : we have 



a i : b i : c i •• d i = 



ft y, 1 



: - 



7, ft, 1 





*, «', 1 



: — 



a, 8, 1 



7, ft, 1 





d : a, ] 





a, a, 1 





ft /, 1 



d, a, 1 





*, f, 1 





ft 7', 1 





7- ft, 1 



