traced upon the Surface of the Sphere. 335 



t 



XXXI. 



"" The loxodrome cuts all the meridians under the same given angle- 



required its spherical equation. 



Let P be the pole, EQ an element of the equator corresponding to 



EP, PQ, the meridians passing through the extremities M, N of an ele- 



FIG. 24. 



ment of the curve. Let MP = $, and the longitude of M from some fixed 



meridian = ?. Then MPN = d 6 - EQ, and RN = dfr Hence we 

 have 



MR = EQcosEM =r sin$d0 ..... (1.) 



But considering the ultimate elements of the arcs which form the elemental 

 triangle MRN as straight lines, we have 



MR d-e.An^ (2) 



RN d(j) 



where a = MNR = angle of the rhumb. 



From (2.) we have 



d 6 . cot a = d (p . cosec <p, 

 or integrating, 



log tan ^ = 6 . cot a + c ..... (3.) 



The value of c will depend upon the meridian which we select as the 

 origin of #, the most natural (and the most convenient it proves to be) is 

 that where the loxodrome cuts the equator. This gives as simultaneous va^ 

 lues of the variables 



BS 0, and =-, 



m 



uu2 



