275 



are shown in the following: Given a triangle with a fixed base, find the locus 

 of the vertex P such that the variable sides, p, p'. Fig. 2, satisfy: 



(1) sinp = k sinp', 



(2) cosp = k eosp', 



(3) sin^? = k sin^ . 



• (4) cosf = k cos |'. 



In CrehVs Journal, Vol. VI, 1830, pp. 244-254, Gudermann pubhshed an 

 article " Ueber die analytische Spharik," which contains a collection of spherical 

 loci connected with sphero-conics, for example, such as: (1) The locus of the 

 feel of perpendiculars drawn from the focus of a spherical ellipse upon tangents 

 to the spherical ellipse; (2) The locus of the intersection of perpendicular tangents 

 to a spherical ellipse; and other problems similar to those of plane analytics. 

 The notation employed by Gudermann is not fully explained, and is an 

 adaptation from that used by him in a private publication of his work 

 "Grundriss der analytischen Spharik, to which the present writer does not 

 have access. 



Thomas Stephens Davies pubhshed, 1834, in the Transactions of the 

 Royal Society of Edinburgh, Vol. XII, pp. 259-362, and pp. 379-428, two 

 papers, entitled, "The Equations of Loci Traced upon the Surface of a Sphere.'" 

 In these extensive papers the author uses a system of polar coordinates 

 upon the sphere, and derives the equations of many interesting curves, the 

 spherical conies, cycloids, spirals, as weU as many properties of these curves. 

 The polar equations of Davies may be transformed into great-circle co- 

 ordinates, giving equations of spherical loci in a form similar to the Cartesian 

 equations of corresponding loci in the plane. 



Spherical Analytics. 



A system of analytic geometry upon the sphere may be derived in direct 

 correspondence to that of the plane by a proper choice of axes of coordinates. 



1. Coordinates. Let us select as axes two great circles XX , YY per- 

 pendicular to each other at O, Fig. 3. The spherical coordinates of any 

 point P are the intercepts, OA = £ and OB = ??, cut off upon the axes by per- 

 pendiculars drawn from P. Let the length of the perpendiculars from P be 

 PB = £', and PA = r,'. 



