106 UNIVERSITY OF COLOEADO STUDIES 



by plotting curves and using elliptic tables. This exceedingly tedi- 

 ous work is mechanical and without geometrical interest. The prob- 

 lem of actually and ultimately exhibiting the results of mathemat- 

 ical investigations, either graphically or in some other way accessible 

 to technical purposes, is necessary. The most beautiful and effective 

 method of representing problems of closure graphically is obtained 

 by a combination of projective and descriptive geometry as it has 

 been established principally by Fiedler* and Disteli, loc. cit. For 

 the reasons mentioned above I shall give a short sketch of this 

 method. 



2. Two cones of the second order generally intersect each other 

 in a curve of the fourth order of the first kind, C^. Without loss of 

 generality we may assume that each of these cones has a circular base 

 in the plane of projection (plane of the paper) of a central projec- 

 tion. Designate these circles by L, and L, and the vertices of the 

 cones by Mj and M^, Fig. 4. Join M, and M^ by a straight line and 

 find its trace S^ in the plane of projection. To simplify the desig- 

 nation I shall mark the projections with the same letters as their 

 corresponding elements in space. Every auxiliary plane through M* 

 and Mj intersects each cone in two generatrices, and the four gener- 

 atrices thus obtained intersect each other in four points of the C^. The 

 trace h of every auxiliary plane passes through S,j and cuts L, and 

 L, each in two points which when connected with M, and M^ furnish 

 four generatrices in the auxiliary plane. If we now choose the centre 

 of projection C on the curve C^ itself then its projection will be a 

 circular curve of the third order, C3, since it passes through the four 

 points of intersection of L, and L^ of which two are the circular 

 points at infinity. The vertices M, and M^ are projected on L, and Lj 

 and the C3 touches L, and L, at M, and Mj. In analogy with the 

 theory of conies, every ray through C cutting the polar plane of C 

 with respect to either cone in a point C, cuts the same cone in two 

 points X and Y so that ( C ' PX Y ) = — 1 ; C ' , P, X, Y form a harmonic 

 group. To find the polar-planes T' and Y" of C with respect to 

 the cones, connect C with M, and Mj and produce these connecting- 



(•) Daratellende Geometrie. Vol. II, pp; 148-182. Vol. III. pp. 320-363. 



