192 Dr. J. R. Airey on Numerical Calculation of the 



Hence 



. / 1 7 83 6949 \ , 1ft . 



^-^^K^ + i + ^ + w + ---)]' (19) 



and (4p~l + 2n)7r 



P~ An 



(b) When n is large and p small, <p is a small angle and 

 from (2) to a first approximation, 



It follows from this, by putting 

 (/>= tan<f> — 

 and solving for tan </>, that 



, , tan 3 4> tan cf> 



6 = tan (f> — — H ^— t - 



3 5 



tan0 = X 1+T + I ^... 

 where f3(4p — l)7r~|i 



Xi= L — a — J • 



Therefore 



3 . ^ 3 3(4/)-1)tt 

 ?i tan 3 <f) = nk Y 6 = v L — — . 



With a very small error, from (2) and (3), e = tane 

 5 5 



24rctan 3 18(4/> — 1)tt' 



The second approximation, therefore, is 



„(tan*--0)=^^ + Ig(i ^. . (20) 



tan * =X+ T + l75- 

 and r8( (^>-l> 5 \1i 



L»i 4 + 18(4^-lj7r/J • • ^ 



Finally, , /, , tan s <£ tan 4 <f> \ 

 •" p p =n sec <p = n\ 1 + — = g — ... I 



=«(^¥+¥--)- • • - • • (22) 



