46 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



The equation in is 



viz., this is 

 giving 



or else 



(cc'm + bb'n)6 + cb'nf + be m = , 

 (c6 + c'm)(b'n6 + b) = , 



b 





bn 





Wij = 





c 





c 



cm 

 T' 



c'm c'm i'n 



m> w ' i = -t' ' h = --d~ 



Since in the present case l 1 p 1 = 0, we have either l x = 0, or else p 1 = 0, and 

 as might be anticipated, the two values of 6 correspond to these two cases re- 

 spectively, viz., proceeding to find the values of / : , p lt the completed systems are 



, b , a / , a' \ bn cm n 



*=--,h = h -^i K cc "> ~ lhn ) > ™i = --• "i = -'T' Pi ' 



. cm ,, n , c'm . b'n , a' [ , 77 , \ 



*=- — ,/, = () , m l =- 1) ,-,n 1 = -- r , Pl = — ( < ccm-bbv) , 



so that for the first system we have 



i- + _i- + -i = , Wjftj = mn , — /A + >»B + nC = — l x \ + ?«,Bj + n 1 C 1 , 



<lj D| Cj 



and for the second system 



^+^- + ^1 = 0, m\n\ = mn , - /A + mB + wC = — ])\D l + m\Bj + »\C, . 

 "i c i "i 



104. The whole of the foregoing investigation would have assumed a more 

 simple form if the circular co-ordinates had been taken with reference to the 

 centre of the circle ABCD as origin, and the radius of this circle been put = 1 ; 



we should then have <*' = -, &c, and consequently 



a' = — ■—- a ,b' — — - - b , c' =■ c. , f— -/',«'= — — q , h' = h : 



fry ya a& 1 aS J J /35 J yi 



but the symmetrical relation of the circles ABCD and A x B x C x D y would not have 

 been so clearly shown. 



I will however give the investigation in this simplified form, for the identity 

 — Ik + m& + nC = — / X A + m x B + n x C ; viz., in this case we have 



I _ _ m(/3 - y) (|8 - 8) _ ?i(j8 - 7) (7 - 5) 

 a 



/3(7-a)(a-5) y(« - |8) (a -y) ' 



