284 UNIVERSITY OF COLORADO STUDIES 



center lies on the axis of collineation and also on the given circle is a 

 conic osculating the given circle at the center of collineation <'>. 

 Hence, considering in Fig. 6^^^^ the line s, joining A, with the coinci- 

 dent remaining points, as the common axis of an infinite number of 

 perspective collineations in which only the counter-axes^^) vary, then 

 the pictures of a fixed circle K through A,A2A3A^ clearly form a 

 pencil of osculating conies. 



On every ray g' (or the identical g^') through a fixed point B 

 (assumed infinitely distant) this pencil cuts out an involution whose 

 double-points are two points on the cuspidal cubic associated with B 

 in the Steinerian transformation. These points are also the points 

 of tangency of <7 ' (^Z) with two conies of the pencil. For the actual 

 construction the following simple method may be applied. Let g' 

 intersect 5 at S. From S draw the two tangents g and ^, to the circle 

 K. Through the center of collineation (cusp) draw a line / parallel 

 to the direction of B. Let T and T, be the points of intersection of 

 / with g and g^, and through T and Tj draw two lines r and 7\ paral- 

 lel to s. Considering r and 7\ as counter-axes of two collineations 

 with the same axis s and the same center, then, according to 

 the constructions of collineation, g' and g/ are the pictures of g and 

 g^ in these two collineations, and the rays joining C to G and G, cut 

 g' {g/) in two points G' and G/ which evidently are the points of 

 tangency with g' (^/ ) of the two osculating conies corresponding to 

 K in the two collineations (/*, r,). The line I cuts K at U; the tan- 

 gent at TJ cuts s at Y, and from the construction follows that the 

 line through V, parallel to I, is an asymptote. In a similar manner 

 the lines joining C to the points of tangency W and W, of the tan- 

 gents to K, parallel to s, are the directions of the asymptotes. 



By proper collineations it is not diflicult to transform the five 

 cubics constructed by means of the Steinerian transformation into 

 Newton's five symmetrical types. 



(1) Fiedler: Darstellende Geometrie. Vol. I (3rd ed.). pp. 188-190. 



(2) The branch of the cubic on the upper right side has only been indicated. In the construc- 



tion it fell beyond the border of the figure. 



(3) See Fiedler, loc cit.. pp. 47-49. 



