334 Mr DAVIES on the Equations of Loci 



Combining (7) with (4) and (2), we get the sun's distance from the tro- 

 pic of Capricorn, and hence from the beginning of Aries ; and likewise his 

 right ascension at the time. 



XXX. 



THE LOXODROME, OR RHUMB-LINE. 



There is no one curve, perhaps, in the whole compass of that infinite 

 variety which presents itself to us, which possesses so many interesting 

 claims upon the attention of the speculative and the practical mathemati- 

 cian. The first in all probability that was made the object of consideration 

 and inquiry upon the surface of the sphere, and, except the conic sections, 

 upon any curve surface whatever capable of very simple and direct investi- 

 gation, and well calculated to suggest to the mind the nature of analytical 

 geometry on the sphere, such as we have here considered a curve of the 

 very greatest importance in the practice of navigation, and, therefore, upon 

 which the commerce of the world, and the comforts and luxuries of civilised 

 life, in a great degree depended a curve possessed of a long train of curi- 

 ous geometrical properties, well worthy the attention of the accomplished 

 mathematician, yet, from their simplicity and elegance, well adapted to dis- 

 cipline the mind, and cultivate the taste of students less advanced in this 

 kind of speculation and, finally, a curve upon the investigation of which 

 many mistakes have been made, and, in some cases, even by mathematicians 

 of no humble pretensions. 



It is not my intention to enter largely into the discussion in the present 

 paper, as it would require one or two preliminary theorems concerning 

 spherical curvature, which will form part of a subsequent dissertation, to 

 give the inquiry all its effect ; nevertheless I shall discuss a few of the more 

 obvious properties of this curve, though rather with a view to exhibit a few 

 additional applications of the method of spherical co-ordinates, than to ex- 

 haust the inquiry of its interest to other mathematicians. In the course of 

 the present session of the Royal Society's sittings, it is not improbable I 

 may be able to furnish a continuation of the subject generally. 







