PROFESSOR CAYLEY ON POLYZOMAL CURVES. 91 



which last equation, using J — to denote as above, but properly selecting the 

 signification of =b , may be written 



Jm 1 + s/n 1 = =k - a — Jl. Jm ■ 

 V/T=F Um t + Jn\) = I { (a +d) Jl + (b + c) M Jm } 



viz., V^i =F(V«^ + JnJ witn a properly selected signification of the sign =f 

 is = 0; and similarly Jf 2 =f (Jm, + Jn~ 2 ) with a properly selected signification 

 of the sign =p is = ; that is, each of the trizomals is a cubic. 



199. IV. J] : Jm : Jn : Jp = a : b : c : d (values which, be it observed, 



satisfy of themselves the above assumed equation - + -r- H '"I" ) 5 the 



order of the tetrazomal is = 6 ; and the order of each of the trizomals is here 

 again = 3. We in fact have Jf = a + d, Jm + Jn[ = b + c, and there- 

 fore JT X + Jm x + Jn^= 0; and similarly J\ + Jm 2 + Jn^ = 0; that is, 

 each of the trizomals is a cubic. 



I attend, in particular, to the case where the four circles reduces themselves 

 to the points A, B, C, D; these four points are then in a circle ; and the curve 

 under consideration is 



a J A + b JB + c VC d /D = ; 



in the general case where the points A, B, C, D are not on a circle, this is, as has 

 been seen, a sextic curve, the locus of the foci of the conies which pass through 

 the four given points; in the case where the points are in a circle then the 

 sextic breaks up into two cubics (viz., observing that the curve under considera- 

 tion is Jlk + J^B + JnC + JpD = 0, where Jl: Jm : Jn : Jp = a : b : c : d, 

 these values do of themselves satisfy the condition of decomposability 



- + tt + - + -^ = 0) , that is, the locus of the foci of the conies which pass through 



four points on a circle is composed of two circular cubics, each of them having 

 the four points for a set of concyclic foci. It is easy to see why the sextic, thus 

 defined as a locus of foci, must break up into two cubics ; in fact, as we have seen, 

 the conies which pass through the four concyclic points A,B, C, D have their 

 axes in two fixed directions ; there is consequently a locus of the foci situate on 

 the axes which are in one of the fixed directions, and a separate locus of the foci 



