118 U. S. COAST AND GEODETIC SURVEY. 



If these values are substituted in the equation of condition 



p fds . . dp\ - 

 a^ cos (p\d(p a<p / 



we obtain for the determination of d the equation 



d — cos^ (p sin ^ = X sin^ (p. 

 In this case 



, 1 — cos^ (p cos 6 , 



fCjn= ^^ sec^ 



™ sin^ (p ^ 



I snr <p 



P 1 — cos^ ip COS B 



so that we have as required 



A^m^p C0S^=1, 



and both '^m and Tc^ are equal to unity for 6 = 0, 



If, On the other hand, we assume 



p = a cot (^ 



s = a cosec ^ 



dp » 



-j-= — a cosec^ <p 



ds 



T^ = — a cot (p cosec (p 



these values bemg substituted in the equation of condition 

 give as the formma for 6 



6 — cos (p sin ^ = X sin' ip 



and 



, 1 — cos (p cos d , 



^-= — s?9 — ^^'''1' 



, _ sinV 



'^ 1— COS (^ COS d 



SO that ^m Tcx, COS ^ = 1 and ^p = 1 for ^ = <^, and itm = sec yp at 

 the same point. 



