32 UNIVERSITY OF COLORADO STUDIES 



envelope the curve S. The centers of these circles are situated on the 

 orthographic projection upon (^X^OY) of the curve of class m, and 

 these circles all intersect the trace of the plane containing the curve 

 of class ;// on the A'O J"-plane under a constant angle. Hence the 

 theorem : The envelope of all circles whose centers are situated on a 

 curve of class m and which cut a fixed straight line in their plane 

 at a constant angle is a cui've of class 2m. 



If m=2, i. e., in case of a conic, the class of the envelope will be 

 4 and its order generally 8, since it generally has two double- 

 tangents. In case of 3 double tangents the order reduces to 6. It is an 

 easy matter to locate the conic and the cone-director of the infinite circle 

 in such a manner, that these and many other configurations of circular 

 systems may be obtained. From the specializations of the general 

 result under articles S and 4 it is seen that they involve the whole 

 theory of Cyclography, or the representation of points in space by 

 the circles of a plane and conversely, a method inaugurated prin- 

 cipally by Fiedler (i) and rigorously treated by Lie, loc. cit. 



University of Colorado, December 24, 1900. 



(') Cyclographie, Leipzig, 1882. 



