388 Mr DAVIES on the Equations of Loci 



XL. 



The equations of the iangent and normal to a spherical curve re- 

 ferred to polar co-ordinates. 



1. The tangent SMK at the point M, or $' &. 



Let the polar astronomical co-ordinates be PE and EPM. Correcting 

 the sign of sin 6 in xi. 9. as directed at p. 35, putting for a /3 their value 

 in the present case $' & ; and for sin 6, cos e, their values in terms of </>' & 

 furnished by Table II. of section xxxix. : then we shall have for the equa- 

 tion of the tangent 



( cos 8 (sin 8 cos f + sin e cos 6 cos a) ) 



cot a) =- cosec a cosec e-j _ a r 



( + sin 6 (cos p cos e sin sin p cos a) ) 



I cos 6 (sin & -3- + cos & sin </>' cos <') 

 cosec^' 2 



I + sin 6 (cos 6' i sin # sin 0' cos ^>') 

 du 



and this is readily changed into the form, 



sin 6' d^tL + cosd' sin 0' cos </>' + sin* <' cot = (2.) 



We may change this at once into geographical co-ordinates, by writing 



