PKOFESSOR CAYLEY ON POLYZOMAL CURVES. 93 



Multiply the trizomal equation by — a x + d x and add it to the tetrazomal equa- 

 tion ; the resulting equation contains the factor a x , and omitting this, it is 



( & i-^)(-7A + jD) + (a 1 -d 1 )(jB-JC) = 0, 



where observe that b x — c x is the distance BC, and a x —d x the distance AD. But 

 in like manner multiplying the second trizomal equation by — a x + d x , and adding 

 it to the original tetrazomal equation, the resulting equation, omitting the factor 

 a x , is 



&-«i)(- J& + jD)-(a x -d x )(jB- JC)=0; 



viz., it is in fact the same tetrazomal equation as was obtained by means of the 

 first trizomal equation. 



201. The new tetrazomal equation, say 



(h - %)( -JA + JD) + K - d x )(jB- JC) = 0, 



is thus equivalent to the original tetrazomal equation ; observe that it is an 

 equation of the form JJK + s/wE + *JnG + JpD = ®, where 



Jl = -Q> 1 — C i)> Jm = a 1 ~d 1) Jn = {a x -d x ), J p = b x -c 1 , 



and where consequently s/l + Jp = 0, Jm + Jn — 0, that is an equation of 

 the form (198) III., decomposable, as it should be, into the equations of two circular 

 cubics. Writing 



*a 



- J A + JD _ e y/B- JC _ q . 



a x — d x ' b x — c x 



where 6 is an arbitrary parameter, the curve is obtained as the locus of the inter- 

 sections of two similar conies having respectively the foci (A, D) and the foci 

 (B, C) ; (see Salmon, Higher Plane Curves, p. 174): whence we have the theorem, 

 that if A, B, C, D are any four points on a circle, the two circular cubics which are 

 the locus of the foci of the conies which pass through the four points A, B, C, I), are 

 also the locus of the intersections of the similar conies, which have for their foci 

 (A, D) and (B, C) respectively; and of the similar conies with the foci (B, D) and 

 (C,A) respectively; and of the similar conies with the foci (C, D) and (A,B) 

 respectively. 



202. V. Jl — Jm — Jn = Jp- The order of the tetrazomal is = 5, whence 

 those of the trizomals should be = 3 and = 2 respectively. To verify this observe 



that the equation ~ + -\7^ l "d = ^ gives ^ + b "* l ~d = ^' anc * comDmm 8' 



with a + b + c + d = 0, these are only satisfied by one of the systems 



VOL. XXV. PART I. 2A 



