02 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



situate on the axes which lie in the other of the fixed directions; viz., each of 

 these loci is a circular cubic. 



200. Adopting the notation of No. 188, or writing 



BA = a„ RB = b u BO = c v BD = d x , 



(and therefore J 1 o 1 = a^^ we have 



a : b : c : d = — d^^ — c^) : c^a^— d x ) : —b 1 (a 1 — d 1 ) : a i (b 1 — e x ) . 



Moreover 



Jl x = a + (1 , J\ - a + d , 



Jm 1 = b + J^- . Jm a = b - J -j- , 



/— /bed ,— /bob 



V" 2 = c - n/ — ' V« 3 = c + ^ — , 



and we have 



bed , , X na 1 b,c, „. , ,„ /bed , ,. 



— = K-^i) 2 -^p- 1 = <h 2 (<h~ d i) >sl -^ = ~ «iK-^0 suppose ; 



and thence 



*A = Oi-^iH&i-O. «/£ = ("i-'^C & i~ c i) 



*/^L = («l-^l)( C l- rt l)' x/'" 2 = ("l -''])( ''l+"l) 

 J"l = («l-^l)("l-^l), J» 2 = («l-''l)(-«|-''l). 



that is 



-v/^ : ^/wjj : ^//^ = &j — Cj : c x — «j : a^ — & a , 



^/^ : ^/w 2 : x/»2 = ^i — c i : c i + «i : — rt i — \ > 



agreeing with the formulae No. 188. 

 The tetrazomal curve 



-<'i(A- c i) J* + CiK-'V) n/b-JjCoi-^) Jo + a^-^) V B = " 

 is thus decomposed into the two trizomals 



(h-h) J* + (Pi-'h) n/B + ((h-h) Jo = o, 



i. b i- c i) n/A + (% + «i) J* - ((h+h) JO = 0. 



201. Observe that the tetrazomal equation is a consequence of either of the 

 trizomal equations ; taking for instance the first trizomal equation, this gives the 

 tetrazomal equation, and consequently any combination of the trizomal equation 

 and the tetrazomal equation is satisfied if only the trizomal equation is satisfied. 



