76 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



that is, /, m, n are as the squares of the tangential distances (direct) ot the three 

 circles taken in pairs, and this being so, the equation of a pair of circles touch- 

 ing each of the three given circles is Jiff + */mB^ + *JnC~ = 0. It is clear 

 that, instead of taking the three direct tangential distances, we may take one 

 direct tangential distance and two inverse tangential distances, viz., the tan- 

 gential distances corresponding to any three centres of similitude which lie in a 

 line; we have thus in all the equations of four pairs of circles, viz., of the eight 

 circles which touch the three given circles. This is Casey's theorem in the 

 problem of tactions. 



The Intersections of the Conic and Ortlwtomic Circle are a set oj four Concyclic Foci — 



Art. No. 179. 



179. The conic of centres intersects the orthotomic circle in four points, and 

 for each of these the radius of the variable circle is = 0, that is, the points in 

 question are a set of four concyclic foci (A, B, C, D) of the curve. Regarding 

 the foci as given, the circle which contains them is of course the orthotomic 

 circle; and there are a singly infinite series of curves, viz., these correspond to 

 the singly infinite series of conies which can be drawn through the given foci. 

 As for a given curve there are four sets of concyclic foci, there are four different 

 constructions for the curve, viz., the orthotomic circle may be any one of the 

 four circles 0, R, S, T, which contain the four sets of concyclic foci respectively : 

 and the conic of centres is a conic through the corresponding set of four concyclic 

 foci. We have thus four conies, but the foci of each of them coincide with the 

 nodofoci of the curve, that is, the conies are confocal ; that such confocal conies 

 exist has been shown, ante, Nos. 78 to 80. 



Remark as to the Construction of tlie Symmetrical Carer — Art. Nos. ISO and 181. 



180. It is to be observed that in applying as above the theorem of the 

 variable zomal to the construction of a symmetrical curve, the orthotomic circle 

 made use of was one of the circles R, S, T, not the circle 0, which is in this case the 

 axis ; in fact, we should then have the conic and the orthotomic circle each of 

 them coinciding with the axis. And the variable circle, qua circle having its 

 centre on the axis, cuts the axis at right angles whatever the radius may be ; 

 that is, the variable circle is no longer sufficiently determined by the theorem. 

 The curve may nevertheless be constructed as the envelope of a variable circle 

 having its centre on the axis ; viz., writing A° = (x — azf + p 2 — <z"V, &c., 

 and starting with the form 



then recurring to the demonstration of the theorem (ante, No. 47), the equation of 

 the variable circle is aA° -f bB° 4- cC° = 0, where a. b, c are any quantities 



