378 PROCEEDINGS OF THE AMERICAN ACADEMY 



By the principles explained in Art. 1, the central equation may be 

 written 1- -^ = 1, where a and b are the tangents of the princi- 

 pal semi-diameters. 



6. Certain properties of the spherical conies follow immediately 

 from the quaternion equation of the cone, and it may be well to intro- 

 duce the equation here. The general form of the equation, as given by 

 Tait, is 



SQ(fQ = 0. 

 A particular form of this is 



Q^ — SuqS^q = 0, 



where a and jS are perpendicular to the cyclic planes. 



7. To find the equations of tangent and polar arcs. 



The equation of a tangent to a spherical conic is found, as for the 



v" — y' 

 plane curve, by determining the value of ' ,/ ' / , when x" = x' and y" 



. . , , . _, . . xx' , yy' 



= y', and substitutmg and reducmg. Tlie equation is — -|- -jj = 1. 



Since this represents a tangent when x'l/' is on the curve, it must, from 

 the symmetry of the equation, represent the arc on which lie the points 

 of contact of tangents from x'y', if x'l/ is not on the curve ; that is, it 

 is the polar of x'l/'. (When x'y' is- a pole with respect to the conic, 

 I shall call it a conic pole, to distinguish it from the ordinary pole of 

 circles.) 



The symmetry of the equation shows that if x"y" lies on the polar 

 oi x'y', theq x'l/ lies on the polar of x"y". There are many properties 

 of polars to spherical conies similar to those of plane geometry. 



8. "We can now find the equation of the locus of the extremity of a 

 quadrant moving at right angles to the given conic ; that is, the locus of 

 the pole of the tangent. Thus 



X = — -7, y =: — "i^, or x' = — a\ y' = — b-y, 



but:-. + ii = i; 

 . • . aV + by = 1 



is the required equation. This is a conic the tangents of whose semi- 

 1 



b 



axes are - and r- : its semi-axes are therefore the complements of those 



