414] 
ON POLYZOMAL CURVES. 
553 
and hence 
that is 
but 
K (— hci + Ci 2 + a x d x — df) + c 2 (— b x d! + c x d x + a x c x — c x d x ) = 0, 
bo (cj 2 — d x 2 ) + Co_ {a x c x — b x d x ) = 0, 
ttjCi b x d x = ^ (cr 
or the equation gives b 2 + w c 2 = 0, or say b 2 : c 2 = b x : — d ly and this with ——^ = 4 2 = ~ ’ 
cij ct-j — oq a 2 ft 2 
gives all the ratios, or we have 
a 2 : b 2 : c 2 : oi 2 = ¿q (cq — cb) : 6j (b x — cq) : — (cq — c? x ) : — eb (¿q — cq). 
We have then for example 
b. 2 — Co : Co — a 2 : a 2 — b 2 = b 1 — c 1 : Cj — cq : cq — ¿q; &c., 
showing the identity of the forms in (cq, ¿q, c 1} d 2 ) and (oq, b 2 , c 2 , d 2 ). 
Article No. 193. Transformation to a New Set of Goncyclic Foci. 
193. Consider the equation 
V ¿21 + V m23 + Vw(£ = 0, 
which refers to the foci A, B, G, and taking D the fourth concyclic focus, let (A u D x ) 
be the antipoints of (A, D) and (B 1 , G x ) the antipoints of (B, G)\ so that (A 1} B 1} G ly Df 
are another set of concyclic foci. We have 231. Qq = 23. (5, and it appears, ante No. 104, 
that we can find ¿ x , m ly n x , such that identically 
-m +m23 +w(5 = -? 1 2i 1 + m 1 23 1 + w 1 g 1 
and that m x n x = mn. The equation of the curve gives 
— ¿21 + m23 + w(£ + 2 Vmw23(£ = 0, 
we have therefore 
— ¿ x 2li 4- mj23i + %(£ + 2 V= 0, 
that is, 
A 2b + V m x 231 + V TqGq = 0, 
viz., this is the equation of the curve expressed in terms of the concyclic foci 
A lt B lt G x . 
Article No. 194. The Tetrazomal Curve, Decomposable or Indecomposable. 
194. I consider the tetrazomal curve 
fW + */mW+ Vw6° + \fp®° = 0, 
where the zomals are circles described about any given points A, B, G, D as centres, 
c. vi. 70