40 UNIVERSITY OF COLOKADO STUDIES 



5. A General Theorem. 



We propose now to find the envelope of all circles which cut a 

 given circle orthogonally and whose centers are situated on a curve 

 of the ?i^^ order in the plane of the circle. Let C^ be the given 

 curve and C the given circle. Find the stereographic projection Q 



of C on the sphere S '2 os =0 , and determine the pole V of the 



L 1 k J 



plane J^ through Q. Project the curve C^ stereographically upon 

 JS, thus producing a curve 6'^' of the same degree. The polar sur- 

 face of C\'' with regard to /S" is a cone of class n having P" as a 

 vertex. Every tangent plane of this cone cuts aS" in a circle IT which 

 is orthogonal to Q and whose pole is a point of Cj^\ The stereo- 

 graphic projection of JT is a circle which cuts C orthogonally and 

 whose center lies on C^^. The system of circles A" on S envelope a 

 curve of order 2m, if m is the order of the cone of class fi. Hence 

 the theorem: 



The envelope of all circles whose centers lie on a curve of the 

 n^^ order and which cut a given circle orthogonally, is a curve of 

 order 2m if m is the class of the curve of order 7i\ 



If the given curve has 8 double points and k cusps, 



m =n {fi~l)—2S — 3L 

 The order of the envelope is, therefore, 



2 i7i=2[7i{ii~l)-2B—8k]. 

 Taking a bi-circular quartic with a real double-point, 

 2m-=2[4.3—2.3] = 12. 



In general the trichodals (lemniscates) give an envelope of order 12, 

 and also the Cartesians. The lima§on gives rise to a curve of order 

 8, and the cardioide of order 6. 



The cyclographic interpretation of the above theorem may be 

 stated in the form: 



' This theorem has also been proved analytically by Dr. Virgil Snyder, of Cornell Uni- 

 versity, who has published a number of valuable papers on the geometry of the circle. 

 See Bulletin of the Am. Math. Soc, Vol. VI., pp. 319-322. 



