OF ARTS AND SCIENCES. 389 



The equation of the cone sought becomes then 



— SwtpV = ^— + — -|_ — jSwcjpw; 



^"t SwqcTir = ^ + |, -f ^, 



and — Svrcp-r^ = ^, -|- "^ + ^- ; 



- a+^)s+(i+^)s+(i+i)s=o,or ■ 



(!>' + c'}x' + (a' + c')/ + (a' + Fjz' = 0, (1) 



is the equation of the required cone. 



Now the equation of the supplementary cone being St(jp~^t = 0, 

 where t = gjo, its equation can be written 



In any cone, as Mx^ -\- Ny" -j- Pz- = 0, the equation of the cyclic 

 planes containing the axis of x, for example, is 



y^JV— M) -\- z\P — M) =z 0, 



so that (1) and (2) have the same cyclic planes. From this it follows 

 that the locus of the intersection of two tangents to a spherical conic 

 which meet at right angles is a conic concyclic with the conic supple- 

 mentary to the given conic. Reciprocally : — Since the poles of these 

 tangents are on the supplementary conic a quadrant apart, it results 

 that the locus of the pole of a chord of 9C is a conic concyclic with 

 the given conic, and the chord envelopes a conic confocal with the 

 conic supplementary to the given one. 



26. The locus of the intersection of tangents at the extremities of 

 conjugate diameters can also be found. The equations of the tangents 

 are 



^ 1^ »/.»/ , 



-^ + f- = 1, or, by Art. 9, -^ - ^ = 1 ; 

 then, squaring and adding, 



^\^i + jiT j + 'il (^^ + '^2 J = 2, 



