OF ARTS AND SCIENCES. 



375 



XXVI. 



SPHERICAL CONICS. 



THE THESIS OP A CANDIDATE FOR MATHEMATICAL HONORS CONFERRED 

 WITH THE DEGREE OP A.B., AT HARVARD COLLEGE, AT COMMENCE- 

 MENT, 1877. 



By Gerrit Smith Sykes. 



Presented by Professor Benjamin Peirce, Jan. 9, 1878. 



1. It is convenient in dealing with spherical curves to have a sys- 

 tem of spherical co-ordinates similar to plane co-ordinates. Such a 

 system can be constructed as follows : Through the origin of plane 

 co-ordinates draw a sphere tangent to the plane with a radius equal to 

 unity, and project the plane axes upon the sphere by drawing lines 

 from each point to the centre. The plane axes will thus be projected 

 into semicircles having their extremities upon the circle of which the 

 oiigiii is the pole. (By circles and arcs I shall always mean great cir- 

 cles and their arcs, unless it. is otherwise specified.) Every point on 

 the plane will be represented by a point on the hemisphere, and this 

 latter point can be referred to the projections of the plane axes as spher- 

 ical axes. The plane co-ordinates of a point, measured on the axes, 

 will be projected into arcs of the spherical axes, whose tangents are 

 equal to the plane co-ordinates. The tangents of the arcs are there- 

 fore taken as spherical co-ordinates instead of the arcs themselves. 

 Moreover, since all lines parallel to the plane axes' meet them at infin- 

 ity, such lines will be projected into arcs passing through the extrem- 

 ities of the spherical axes. Therefore, to 

 find the spherical co-ordinates of a point 

 on the sphere, draw arcs through the point 

 and the extremities of the spherical axes, 

 and take the tangents of the intercepts of 

 these arcs as co-ordinates. Thus the co- 

 ordinates of P are tan OA = x, tan ob = n 

 y. The spherical axes may be inclined at 

 any angle, but I shall confine myself to 

 rectanarular axes. 



