40 IT. S. COAST AND GEODETIC SURVEY. 



For the parallels we have 



_ a cos ip 

 ~ sin a + sin <p 



a cos a 



s = ~ ■ — : 



sin a + sm (f 



. ^ X sin X (sin a + sin (p) 



sm 5 = - = r- — ■■ -■ ; ^^-r 



p 1 + sm a sm <p + cos a cos X cos <p 



. s — y cos X + cos a cos (p + sin a cos X sin <p 



cos = — ^ = — 1— ^ — = ■■ r =^ -' 



p 1 + sm a sm <p + cos a cos X cos ^ 



s in this case is not reckoned from the Equator; but, 

 since we need only the derivative of s with respect to <p, 

 it will answer the purpose to leave it as it is. In fact, s 

 could be reckoned from any fixed point in the line of 

 centers and in this case it is reckoned from the origin 

 ivhich lies at latitude a. 



bd cos a sin X 



d(p 1 + sin a sin <p + cos a cos X cos (p 



bo sin a + sin (p 



dX ~ 1+ sin a sin <p + cos a cos X cos (p 



ds _ a cos a cos (p 

 d(p ~ (sin a + sin (p)^ 



dp _ g(l +sin a sin <p) 

 d<p ~ (sin a + sin <p)^ 



These values may now be substituted in the general dif- 

 ferential formulas and by that means we obtain the follow- 

 ing results : 



d^ ds^ . ^^ a cos a sin X cos < p 



^b(p dip (sina + sin^) (1 +sinasin^ + cosacosXcos^) 



a cos a sin X cos (p 



(sin cK + sin <p) (1 +sin a sin <^ + cos a cos X cos <p) ~ 



Therefore 



tan ^ = 

 or 



^ = 0. 



