traced on the Surface of the Sphere. 355 



ferences. So far, then, as the geometrical position of a point on the sphere, each 

 function determines two points, and only two. 



The sphere is divisible into eight octants, each bounded by three quadrantal arcs. 

 A point may be situated any where upon any one of these, or upon any one of their 

 bounding quadrants. To arrive at that point, we may go two ways from the first 

 meridian, and two ways from the pole ; and hence any one point may be denoted by 

 four different pairs of co-ordinates. For facilitating the description, we shall suppose 

 a common terrestrial globe, having its meridian in the plane of the terrestrial meri- 

 dian, and its equator in the horizon. Its north pole will then be P, its south pole 

 P', and its equator will be EQ ; and if London be brought to the brass meridian, 

 the longitudes on this globe will correspond with the values of on our sphere, whilst 

 <f will correspond with the co-latitude (or rather say N. P. D.) of the point under 

 consideration. Let us now conceive the globe to be permanently fixed in its present 

 position, but the brass meridian to revolve about the axis, either eastward or west- 

 ward ; and we shall be able to comprehend, with the utmost facility, the distribution 

 of the co-ordinates <p, 6 into the four pairs above mentioned. 



In analogy to the method of estimating rectilinear co-ordinates, we shall consi- 

 der (the sphere being fixed, as already described) that the values of 6 lying towards 

 the right hand, or east longitude, is +, and that to the left, or west, is . We shall 

 also, when we take the polar equation, consider the radius-vector <p, which coincides 

 with the prime meridian on the south side, to be +, and the opposite one to be . 

 These can never change, except by becoming respectively = 0. We shall designate 

 the eight spherical octants by " regions :"" and we shall describe the distances from 

 the origin of & by calling it the first, second, third, Szc. quadrant of positive revolu- 

 tion, or of negative revolution. The same method will be well adapted with re- 

 spect to positive and -negative polar distance, in polar curves, or to positive and ne- 

 gative latitude, in rectangular curves. 



If, therefore, the co-ordinates of a point on the sphere be 



M, (i.) 



they are also 



ft (2<r-rf), ' (2.) 



_(2 p), (2* *) (3.) 



_(2*_p), (4.) 



respectively. This can hence create no difficulty. 



Again, when we have resolved the equation f(<p, S) with respect to a. Junction 

 of one of the variables, we shall always find two points belonging to each value of 

 that variable on the sphere, for every individual value of the other variable. Now, as in 

 general the equations between these variables in trigonometrical functions, and as to 

 each value of any such function two arcs belong, we shall, (supposing only one function 

 of each variable appear in the final equation) obtain four points on the sphere. To 



