380 Royal Society of Edinburgh. 



in the Society's Transactions (vol. xii.), and of which an account 

 was given in this Magazine for February 1832, Instead of consi- 

 dering the curves traced upon a spherical surface, as is usually 

 done, as the line of intersection of two curve surfaces, one of which 

 is the sphere itself, and of referring each curve surface to their recti- 

 linear coordinate axes, Mr. Davies refers that curve to two coordi- 

 nate axes, situated upon the spherical surface itself. The system 

 which the author has already developed, is that in which points are 

 defined by the common geographical method of reference to lati- 

 tude and longitude, or else to polar distance and polar angle : he 

 had before given all the requisite formulae for facilitating the 

 use of these systems, so far as that could be done without the em- 

 ployment of differential coefficients, and had also applied them to 

 a number of the most celebrated problems that had engaged the 

 attention of mathematicians by other methods. In the present, or 

 complementary paper, he gives the formulae which result from the 

 use of differential coefficients, such as the tangent and normals, evo- 

 lutes, and radii of curvature. He gives the equation of the great- 

 circle tangent to a curve at any point, and likewise that of the normal, 

 in terms of the polar coordinates of the current point of the tangent 

 and normal, and of the differential coefficients to the curve at the 

 point of contact or of normal section. He is also led by this, and 

 the remarkable analogy which is found to subsist between the plane 

 and spherical equations, to give the equation of the tangent and 

 normal of plane curves, in terms of polar coordinates, and to sug- 

 gest that the properties of such curves can often be more elegantly 

 investigated by such means than by an equation between the ra- 

 dius vector and perpendicular upon the tangent, which has been 

 hitherto universally employed. He considers, indeed, that the in- 

 vestigation of spherical loci is now reduced to principles quite as 

 simple, and to operations quite as easy, as that of plane loci is, or 

 perhaps ever will be. 



Mr. Davies gives a few specimens of the use of these formulae, 

 in the case of the spherical conic sections (defined by the equality 

 of the sum or difference of arcs drawn from two given points to the 

 points of the curve), in which very slight modifications of the 

 verbal enunciation are found to express corresponding properties in 

 piano and in sphcero. A curve which, from the form of its equation 

 and the mode of its genesis, Mr. Davies calls the spherical logarith- 

 mic or the spherical equisubtangential curve, is also examined at 

 some length. It is such a one that the great-circle tangent to any 

 point shall cut off a constant arc upon the equator, reckoned from 

 the meridian of the point of contact. Amongst its properties are : — 



1. That its gnomonic projection upon the polar tangent-plane is 

 a logarithmic spiral. 



2. That its gnomonic projection on the equatorial cylinder be- 

 comes, when the cylinder is developed, a logarithmic curve. 



3. That if another equal sphere be described, having the pole of 

 the logarithmic for its centre, then the gnomonic projection of the 

 logarithmic upon this second sphere, is the loxodrome-, and the 



