96 BULLETIN OF THE UNIVERSITY OF WISCONSIN 



which furnish, after a little reduction, 



. ,, , sin n sin 2r 



tan /I - r) = — ■. -_ (6) 



sin p — sm Tt cos 2r ^ ' 



We also obtain from (3) 



( cos X , sin '^ I . sin p 



i cos r sin r ) sin i ( p + tt) sin i (i-* — tt) ^ ^ 



from which 



-_, cos TT — COS p sin (r 4- X) 



I : • r (o) 



Sin p Sin 2t ^ ' 



In this expression we put 



fos It — cos p . . , . 



i- == tan \\p — X) 



sm p) ~ 



and find the rigorous equation 



tan ^x = tan'^ i 7t cot ^ p (9) 



for which there may usually be substituted 



2 sin'- \n ^ , 



X -= : -^ cot ^ p 



sm 1 

 Introducing (8) into (4) it becomes 



cos (r - A) = tan <p tan \ {p - x) fl^iJX+il (iq) 



Sill/ t^T 



"We now put 



A - r = JW T -\ = N 



and obtain 



r - r = T.^ AT - a.^ = J/+ N (11) 



These equations suffice for the determination of at when 

 the latitude, ^. is known, and the effect upon at ol an er- 

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



^- A (D = ^ // <p = - 2 cosec IcpcotN.A (p (12) 



d <p a cp 



Putting t = 2r and eliminating the formulae requisite for 

 the reduction of an observation may be collected and ar- 

 ranged as follows: 



