262 Mr DAVIES on the Equations of Loci 



rical curves, like all others, were expressed by two equations be- 

 tween three variables, one of which equations was that of the 

 sphere itself. From that time forward, we find no traces of any 

 attempt to devise special methods adapted to the discussion of 

 curves traced upon specific surfaces, and to them only, amongst 

 the continental mathematicians. No doubt, so far as the inves- 

 tigation of properties common to all classes and all orders of 

 curves was concerned, this was an improvement of incalculable 

 value, and one which for that purpose was altogether indispen- 

 sable ; but it became at the same time a barrier to the inquiries 

 respecting the peculiar characters of that class of curves which 

 are traced on a spherical surface. The sphere, indeed, from the 

 uniform character of all the regions about any point of its sur- 

 face, would, we should at first sight expect, possess a great va- 

 riety of interesting, and possibly, of important, properties belong- 

 ing to itself alone, in the same manner as the circle does amongst 

 plane curves ; and certainly, by methods whose essential charac- 

 ter is their adaptation to the discovery of affections common to 

 all curves, or to all curve-surfaces, we can scarcely hope to dis- 

 cover those which are peculiar to one class amongst them. The 

 importance of the sphere in astronomical inquiries (especially in 

 that approximative astronomy which considers the earth a sphere, 

 and its orbit a circle), as well as to the discussion of several cu- 

 rious points connected with astronomical history, these conside- 

 rations render it an object of great importance that we should 

 be able to investigate its loci by means of a system adapted to 

 that purpose only, and therefore divested of every thing foreign 

 to that immediate object. Some direct and simple methods of in- 

 vestigation are essential to our intelligibly expressing the varie- 

 ties of the celestial motions, in a manner adapted to subsequent 

 determination of the loci which result from their combinations. 

 One instance of the advantages of such a method has been given 

 in my paper on the "Antique Hour-Lines," which the Royal 



