SI 8 Mr DA VIES on the Equations of Loci 



i 

 / 



( . ?) 4&&W>'*> xxv, - ,-^ : 



SPHERICAL PARABOLA. 



There is yet one particular case of the general equation (3.) which claims 

 attention, on account of its being related to the spherical ellipse and hyper- 

 bola, as the parabola is to the ellipse and hyperbola in piano. I mean 



7T 



when a, /3, is one of the poles, and i = - . In this case, the distance of 



any point in the circumference, from the focus a, /3, , is equal to its distance 

 from the equator *. 



In this case /?, will be indeterminate, and a, = n IT, where n is a whole 

 number. The equation (3.) becomes, after slight and obvious deductions, 



tan0 = i _. COS \ , .... (1.) 

 1 _|_ sin a,, cos p a 



The gnomonic projection of which is an ellipse, having the pole of the 

 hemisphere for one of its foci. See Repository already referred to. But, 

 in addition to that, we remark, that, in uniformity with the general deter- 

 mination in the last article, there is also a plane on which it may give an 

 ellipse by orthographic projection, though a different plane from the one to 

 which in the Repository the projections were referred. 



The relations we have assumed, viz. a, = n IT, show that this property 

 holds, whichever of the poles be taken as one of the foci. In case of a a @ u 

 being on the hemisphere which contains a, /3, , then it is the sum of the 



arcs that are equal to - ; and when a, /5 ; and /3 X/ are on different hemi- 



m 



spheres, it is the difference of the arcs that make that sum. 



* In the Mathematical Repository, vol. v. p. 240, pt. 1, are solutions of this case, 

 which had heen proposed some years before by Professor WALLACE. How far that 

 gentleman had carried his inquiries into this subject, or whether he had systematically 

 entered upon it at all, does not appeal- from the Repository ; nor have I other means of 

 ascertaining. 



