7T 



traced upon the Surface of the Sphere. 389 



</>, instead of and accenting 6 1 in the same manner ; thus we have at 



n 

 once 



sin 6, 6 -f~ -f- cos 6, 6 sin 0, cos (f), + cos 2 (f), tan = (3.) 



d6, 



2. TAe normal XMV, at the point M, or $'&. 

 Let the astronomical and geographical co-ordinates be denoted as be- 

 fore. Then, in the former case, instead of the tangent adapted to e, we 



TT 



must employ that for + e ; that is, we must write 



v d(b' 



. ..,-. and 



if 

 - 



- + f i = sin f = -7= 



These, inserted in xi. 9. (corrected as above), give the following equa- 

 tion, 



, , (-(cos e sin frdn Ocos 6^ 

 cot 9 = cosec q/.4 



( (cos cos ^+ sin 5 sin ^) 



d(p | * ........ ("*) 



cos + sn sn ) j 



or which, by reduction, becomes 



sind' 6sm(f>' cos^ 6 cos (f)' + cot (p sin 0' = .................. (5.) 



Making the requisite change to adapt it to geographical co-ordinates, 

 we have 



dd 

 sin Q, 6 cos^-j-r' cos 0, 6 cos d) t + tan (p cos<f>, = .................. (6.) 



XLI. 



To find the equation of spherical curves in terms of its radius-vector 

 and the perpendicular from the pole upon the tangent. 

 Let the perpendicular PL = ^ ; then, by the table, 



sin* <hdd 



sm ^ = ,.......;, 



.......................... (1.) 



3 D2 



