96 BULLETIN OF THE UNIVERSITY OP WISCONSIN 



which furnish, after a little reduction, 



tan (A -* 4 - . Sin7tsin ^ 



sin p sin 7t cos 2r 



We also obtain from (3) 



, L _. _ _ ~ 



cos r ~ sin T ) ~ si/i i (p + TT) si/i (p n) 



from which 



, ! cos 7t cos p ^ sin (r -f- A) 



sin p sin 2t 



In this expression we put 



COS 7t COS p 



- : - tan i ( p x) 

 sin p 



and find the rigorous equation 



tan $ x = fan 2 \ncot\p (9) 



for which there may usually be substituted 



2 sin 2 i it 



X ~- - : -4 COt i p 



sin 1 

 Introducing (8) into (4) it becomes 



cos (T - A) = tan 9 tan \(p-x) S -^~- (10) 



We now put 



X- T = M !T-A = iY 



and obtain 



T - r = T 2 + 4T- a a = Jf-f N (11) 



These equations suffice for the determination of ^T when 

 the latitude, <p, is known, and the effect upon ^T of an er- 

 ror in the assumed value of <p is readily shown to be 



^A- 4 q> = y. z/ (p = - 2 cosec 2 ?> cof N. A <? (12) 



Putting t 2r and eliminating the formulae requisite for 

 the reduction of an observation may be collected and ar- 

 ranged as follows: 



