188 Mr. Morton on the Focus 



Prop. II. 



If a right cone be cut by a plane which is not parallel to 

 a slant side; and if two spheres be inscribed, as in the last 

 proposition, touching the plane in two jjoints S and S' , and 

 the conical surface in two circles BDE and B'D'E ; the sum or 

 the difference of the distances SP and S'P' of any point P in 

 the conic section from the points S and <S", shall be always the 

 same ; the sum in the case of the ellipse, and the difference 

 in that of the hyperbola. 



Let JQP be the section, and let the spheres be inscribed 

 in the manner already shewn. Then, if f^P be joined, and 

 cut the circles BDE and B'D'E' in the points D and D' re- 

 spectively, the sections of the spheres made by the planes 

 yPS and F'PS' will be circles, the one touched by the straight 

 lines PD, and PS, the other by PD, PS'. Therefore PS is 

 equal to PD, and PS' to PD', and the sum (fig. 1.) or the 

 difference (fig. 3.) of PS and PS' is equal to DD', that is, to 

 BB' or AA'. 



Therefore, &c. Q. E. D. 



The solution of the following problem, which is easily 

 derived from the construction of Prop. I, will point out a re- 

 markabfe proj)erty by which the focus is still further distinguished 

 from every other point in the plane of the conic section. 



Prop. III. Problem. 



To find the locus of the vertices of all right cones which 

 have the same given plane section JQP. 



Take S the focus and A the principal vertex of the section, 

 and let ^ be the vertex of any right cone of which it is a 

 section (fig. 1, 2, 3.) Join f''^A, and let the plane VJS cut the 

 conical surface again in the line f^B. Then, from the construction 



