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 cosy’, 
(3) sing = k sin? , 
(4) cos? = k eos 8. 
In Crelle’s Journal, Vol. VI, 1830, pp. 244-254, Gudermann published an 
article ‘‘ Veber die analytische Spharik,”’ which contains a collection of spherical 
loci connected with sphero-conics, for example, such as: (1) The locus of the 
feet of perpendiculars drawn from the focus of a spherical ellipse upon tangents 
lo the spherical ellipse; (2) The locus of the intersection of perpendicular tangents 
lo 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 published, 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 coérdinates 
upon the sphere, and derives the equations of many interesting curves, the 
spherical conics, cycloids, spirals, as well as many properties of these curves. 
The polar equations of Davies may be transformed into great-circle co- 
érdinates, 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 codrdinates. 
1. Coérdinates. Let us select as axes two great circles xx’, -YY’ per- 
pendicular to each other at O, Fig. 3. The spherical codrdinates of any 
point P are the intercepts, OA = and OB = 7, cut off upon the axes by per- 
pendiculars drawn from P. Let the length of the perpendiculars from P be 
[PBS = 7eE eal 1Bs\ eee 
