268 Mr DAVIES on the Equations of Loci 



spherical co-ordinates, both as to origin and direction. Then 

 follow formula of projection, for ascertaining the equation of the 

 projection of any spherical locus, upon a plane parallel to the 

 tangent plane at the origin of spherical co-ordinates. They are 

 both simple and symmetrical ; yet simple as they are, they are 

 capable of universal application. Amongst these projections, I 

 particularly notice the three usual ones, the orthographic, gno- 

 monic, and stereographic, which are the only cases possessing 

 any analytical peculiarity. 



The remainder of the paper is devoted to the consideration of 

 various spherical curves ; and the results are often curious and 

 unexpected. I have, however, in many cases, from studying the 

 brevity necessary in a dissertation of this kind, been obliged to 

 omit discussions which would have repaid our attention to them, 

 had our limits permitted us to dilate. 



The first curve is the equable spherical spiral, a family which 

 contains several such loci as leave a quadrable residue on the 

 surface of the sphere. Amongst them are the spiral of PAPPUS, 

 the oval window of VIVIANI, and some others worthy of no- 

 tice. The connexion between these curves has never before been 

 noticed; nor has it before been shown that the construction 

 given by JAMES BERNOULLI for solving VIVIANI'S problem is 

 the same as that of the proposer himself. In the next place 

 follows the investigation of spherical loci, produced in the same 

 way as the conic sections are in piano referred to two foci, 

 and which I hence call the spherical-ellipse, hyperbola, and pa- 

 rabola. It is shown that the ellipse and hyperbola so gene- 

 rated (that is, the locus of the point, the sum of whose dis- 

 tances, and that the difference of whose distances, are con- 

 stant), may, as in piano, be comprised in one single equation : 

 that the loci themselves, in each case, formed a double system of 

 lines, were in all respects equal and similar, but surrounding 

 the two poles of the sphere, so that they would have been pro- 



