72 PROFESSOR CAYLEY OF POLYZOMAL CURVES. 



171. Suppose that the centre of the variable circle is situate on a given conic, 

 then expressing the equation of this conic in areal co-ordinates in regard to the 

 triangle ABC, we have between (a, b, c) the equation obtained by substituting 

 these values for the co-ordinates in the equation of the conic ; that is, the equation 



of the variable circle is 



aA° + bB° + cC° = , 



where (a, b, c) are connected by an equation, 



(a, b, c, f, g, h) (a, b, c) 2 = . 



Hence (A,B ,C, F,G, H) being the inverse co-efficients, the equation of the 

 envelope of the variable circle is 



(A, B, C, F, G, H){K, B°, C7 = , 



and, in particular, if the conic be a conic passing through the points A, B, C, and 

 such that its equation in the areal co-ordinates (u, i\ w) in regard to the triangle 

 ABC is 



/fir + mwu + nvr = , 



then the equation of the envelope is 



(1\ m \ „2. _ mn> _ nl t - lm)(A°, B°, C ) 2 = ; 



that is, it is 



(1, 1, 1, - 1, - 1, - 1)(/A°, mB°, nCy = , 



or, what is the same thing, it is 



JJA* + JmW + JnC° = . 



172. It has been seen that the equations of the nodal tangents at the points 

 /, J respectively are respectively 



Jl(£ - «•-) + Jm(£-i3z) + Jn(i -yz) = , 

 J({r> - a'z) + *Jm(n - &*) + \ nyri - yz) = U , 



and that these are the equations of the tangents to the conic hw + mwu + 

 nuv — from the points /, ./ respectively. We have thus Casey's theorem for 

 the generation of the bi-circular quartic as follows : — The envelope of a variable 

 circle which cuts at right angles the orthotomic circle of three given circles 

 A° = 0, B° = 0, C° = 0, and has its centre on the conic km + mwu + nuv = 

 which passes through the centres of the three given circles is the bicircular 

 quartic, or trizomal 



JIN r + JmB° + J^O r = 0, 



which has its nodo-foci coincident with the foci of the conic. 



173. To complete the analytical theor}', it is proper to express the equation of 



