80 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



183. It is to be observed that in this case if the positions of the four foci are 



given by means of the circular co-ordinates (a, — ) &c., which refer to the 



centre of the circle A BCD as origin, and with the radius of this circle taken as 

 unity, then the values of a, b, c, d (ante, No. 90), are given in the form adapted to 

 the formulae of No. 49, viz., we have 



a : b : C : d = a (&yh) : — /3 yda) : y (8a,3; : - h (ajSy) , 



where {j3yS) = (/3 — 7 )( 7 — S)(S — /3), &c. The relation - +^+ - = 0, put- 

 ting therein l:m:n = pa( (3— y) 2 ; <rfi (y — a) 2 : ry (a — /3) 2 , (or, what is the 

 same thing, taking the equation of the curve to be given in the form 



(/3 —y) Vj^A + (7 — a) J^jSB + (a —13) JryQ = 0), becomes 



p (|8 - y) (a - i , + 6 y -a) (/3 - h + r (a - j8T ;> - 3) = , 



viz., this equation, considering p, <r, t, a, /3, y as given, determines the position of 

 the fourth focus Z>, or when A,B,C,D are given, it is the relation which must 

 exist between p, <t,t; and the four forms of the equation are 



( , J7{d- y), JVifi - S), vT(y - /3 ) V«A> V3B, ^7°. v^~D) - <>. 



Vrfy - 5 , . , J§ <3 — a), v/tf a — 7) 



Vtf (i - &)> J] (0 -8), . , x/r(j8 - a, 



\/fi O - 7). J' 7 - a ), *-' T (" - Z 3 ). 



viz., the curve is represented by means of any one of these four equations involv- 

 ing each of them three out of the four given foci A,B, C, D. 



Case of the Circular Cubic — Art. Xos. 184 and 185. 



184. In the case of a circular cubic, we must have 



g (/3 — y) (a - 6 4- <s{y — a) (£ — 3) + r (a — £) (7 - d) = 0, 



x/"K0- 7) + <J&b - «0 + vV' ( a - £) = °- 



which, when the foci .4, B, C, 1) are given, determine the values of p:<r\ t in 

 order that the curve may be a circular cubic. We see at once that there are two 

 sets of values, and consequently two circular cubics having each of them the 

 given points A, B, C, D for a set of concyclic foci. The two systems may be 

 written 



\/g : \Z<r : VV = \/ad — J (3y '■ J fid — Jya : \/yd — Vafi, 



viz., it being understood that +/a$ means *fa */S, &c, then, according as v^ 

 has one or other of its two opposite values, we have one or other of the two 



