360 



Mr DA VIES on the Equations of Loci 



Let LC V = 6, and describe the meridian ALB. Then 

 LAD is also = 6. We have now to find AD. 



^ = - = tan CBV = tan ABD (2.) 



CB a 



But because the angle ABD is at the circumference, the arc AD measures twice 

 the angle ABD : that is, in symbolic language, 



AD = 2 tan- 1 -. 

 a 



..(3.) 



CF = ED = a sin AD = a sin <p. 



v 



Hence a sin <p = a sin 2 tan" 1 -. 







r k 9 



But sin 2 tan 1 - = sin 2 tan~ 1 by equation (1) 

 a a 



or, taking a = 1 = rad. of sphere, we have 



sin <p = sin 2 tan 1 k? = 2 sin tan 1 k? X cos tan 1 h s , 

 or, by reduction, 



sin <p = 



2k" 



i:*ff 



,(4.) 



which is the equation of the loxodrome, already found in (XXXI, 6), in terms of 

 the variables (J), 6. Hence, &c. 



We have yet to ascertain the spherical equation of the intersection of the ortho- 

 gonal cylinder on a log.-spiral base, with the surface of the sphere. 



