392 Mr DA VIES on the Equations of Loci 



But from (1) we get 



~ cos p cos <b cos A. 



cos 6 = 5_ 



sin (p sin A. 



n ,{ sin'X cos 2 p + Scosp cos X cos (f> cos 2 </> }* 



sin (f> sin X 



Or from (12), (13), 



,_ , sin (f) V sin 2 X cos 2 p + 2 cos p cos X cos (p cos* (f> 



d 6 cos <p cos p cos X 



And putting this in the value of ^, we find the value which we sought, 

 viz. 



cos (b cos p cos X 



sin il- =: ~r- r-* (15.) 



sin p 



If the pole be in the circumference of the circle, then X = p and there results 



(16.) 



XLII. 



To find the intersection of the perpendicular from the pole of astrono- 

 mical co-ordinates upon the tangent to a spherical curve at the point 



We might employ the general expression found in (X. 10), or rather in 

 the correction to that passage at the end of the present paper : but as that 

 method would, in reference to our present object, be unnecessarily operose, we 

 shall adopt a briefer one. It depends upon this, that when the radius- 

 vector is perpendicular to a curve (or, in other words, at an apse), cot 

 will be a maximum or a minimum, and hence that the equation 



cot (f) = tan X cos 6 K 

 differentiated on this supposition, gives 



sin 6 K = 



