PROFESSOR CAYLEY ON POLYZOMAL CURVES. 79 



+ — — + — — , = implies that the curve JiK + VmB + JnC = is a 



b — c c — a a — b 



cartesian, and we have thus the theorem that the envelope of a variable circle 



having for diameter the double ordinate of a rectangular cubic is a cartesian. 



I remark that using a particular origin, and writing the equation of the rect - 



2/4 

 angular cubic in the form y 2 — x 2 — 2mx + a + — , the equation of the vari- 



able circle is 



(z — d) 2 + y 2 = d 2 — 2md + a + —^, 



CI 



that is 



2A 



x 2 + y 2 — a — 2d (x — m) — ^— = , 



d 



where d is the variable parameter. Forming the derived equation in regard to d, 



we have 



_A 

 x m — , 2 , 



and thence 



2 2 4^4 



xr + ir — a = — =-j 

 a 



(a* + f _ af=-£ = 16 A (x - m), 



that is, the equation of the envelope is (x 2 + y 2 — a) 2 — 16A(x — m) = 0, which 

 is a known form of the equation of a Cartesian. 



Focal Formidcefor the General Curve — Art. Nos. 182 and 183. 



182. Considering any three circles centres A, B, C, and taking A°, &c. to 

 denote as usual, let the equation of the curve be 



V/A 5 + V^B 5 + JnC° = 0; 



then considering a fourth circle, centre D, a position of the variable circle, and 

 having therefore the same orthotomic circle with the given circles, so that as before 



aA° + bB° + cC° + dD° = , 



the formulse No. 47 (changing only U, V, W, T into A°, B°, C°, D°) are at once 

 applicable to express the equation of the curve in terms of any three of the four 

 circles A,B,C, D. 



In particular, the circles may reduce themselves to the four points A, B, C, I), 

 a set of concyclic foci, and here, the equation being originally given in the form 



JIK + JmB + JnC = 0, 



the same formulse are applicable to express the equation in terms of any three 

 of the four foci. 



