OF ARTS AND SCIENCES. 393 



From this can be found the equation of the polar of the origin. The 

 general equation of a spherical conic, transformed to spherical polar 

 • co-ordinates by 



X = cos d tan q, y = sin tan q, 

 becomes 



(a cos'^ d -\- 2h sin d cos -\- h sin^ d) tan" q 

 -\- 2 {g cos d -|-y sin 6) tan q -\- c = 0. 



Then using tan q as the variable, we find the equation of the polar, by 

 the same process as in plane conies, to be 



From this equation, it is evident that the conic polar of the centre 

 is the circle of which the centre is the spherical pole. This can also 

 be proved by putting oa = ob in the last article. This circle corre- 

 sponds to the line at infinity in plane conies. 



33. The condition that three arcs meet in a point, and the equation 

 of an arc through the intersection of two other arcs, are the same in 

 spherical co-ordinates as for lines in plane co-ordinates. We can then 

 prove, exactly as in plane conies, the following theorems : — Draw any 

 two arcs through a point O ; join directly and transversely the points 

 where these arcs cut the conic. Then, if the direct arcs intersect in p, 

 and the transverse in R, the arc PR is the polar of o. 



The lines joining the corresponding vertices of a spherical triangle 

 and its conjugate meet in a point. 



, If a quadrilateral be inscribed in a conic, each of the three points of 

 intersection of the diagonals will be the conic pole of the arc joining 

 the other two. 



All of these properties are also seen to be true by projections on a 

 plane. 



34. The polar of x'y' relatively to a conic is 



^ j^ .yy , 



a2 -J- b-2 — -^J 



— ^ — ^)> the polar of x"y" rela- 

 tively to the supplementary conic is 



a-xx" + %3^" = 1 ; 



— —^ — jTJi ^i- becomes 



^•«' + yy' = — 1> 



