296 Mr DAVIES on the Equations of Loci 



Inserting the values of the functions of (p and from (2, 3, 4, 5) in (6), we 

 have, dropping the common denominator /l -\cosa cos <' + sin a sin<' cos &y, 

 the following result, as the transformed equation which we are in search of, 



(sinXcosa + cosAcos/csina)cot</>' = {(sin A sin a cos A cos a cos /c) cos & + sinKcpsAsinfl 1 } . . . (7.) 



sin K cos A 

 If we put K' = tan ' sinXsin a _ cos Acosacos/c' and 



A' = cot" 1 (cos sin A + cos A sin a cos/c), then we shall have (7.) trans- 

 formed into 



cot <J>' = -tan A' cos (& K') (8.) 



the same general form as before. 



Schol. 2. Many interesting inquiries respecting transformations of 

 co-ordinates must be delayed for the present, as the room which I can here 

 allot to the inquiry is already filled. One principle to be established is, that 

 no transformation changes the order of a curve on the sphere, more than on 

 a plane, or in space of three dimensions. Much as I wish to enter upon 

 this matter here, I am compelled to waive it, on account of the preliminary 

 discussion of what determines the order of a spherical curve, and what test 

 will decide it. 



XV. 



The great variety of cases in which we may have occasion to examine 

 the projection of a spherical curve upon a plane, renders it necessary to lay 

 down a general formula of projection for this purpose. This will be done 

 in the following simple lemmas. 



:.''.: 



] . A point D on the surface of the sphere is projected from a given point 

 in the axis of the sphere upon a plane parallel to the equator. Find the 

 distance of the projection from the centre * of the plane. 



* " Centre of the plane" is used to signify the point where the plane of projection is 



cut by the axis of the sphere. 

 2 



