110 TJ. S. COAST AND GEODETIC SURVEY. 



We shall need the derivatives of Tc with respect to f of the 

 first two orders; we have 



sin pblc n cos p^ 



— 7-^N— =r-i — -' f — 77 — cos p 



k op l+sin2> cosX ^ 



or 



yH t; ) = — cot p' + - (cosec p^ + cos X') cos p 



nsuip sm p Y^( I^ ) "" ^ ^ ^ ^^^ 2^ ^^^ V 

 — sin^ 2^ (1+cos X' sin pO, 



or 



Ti, sin p sin 2?' |^p gp - i^y^) J = sin^ 2? (1 + cos X' sm p') 



+ n cos p cos p' — n^. 



Let us first suppose n<l. Then at the two poles, that 

 is, for p = and lor p = r, we should have fc = 00 ; within 

 the interval Ic would pass upon each meridian through a 

 minimum. Denoting by a subscript m the value which 

 applies for Ic a minimum, we should have, by equating to 

 zero the first derivative of Ic with respect to p, 



cos p'm ^ COS pm 



1 + COS X' sin p'm n 



-, ^tan_£^ 

 "" tan pm 



, r 1 d-Z:~| cos p\ 



The corresponding point is situated in the Northern Hemi- 

 sphere. 



The values which the above expression for — n— v' as- 

 sumes for p = and for p = tt are, respectively, n — 1 and 

 1 — n, so that the first is negative and the second is positive. 



TT , \ . TT 



But for 2^' = p-» p{=Po)>i)'i hence the expression is pos- 

 itive for v' = 7y> and, in fact, \t is positive for p^-^- The 



