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



McMahon's formula can be derived from the above results 

 as follows : 



If fi= i \ a + € ] > e( l uation (2) becomes 



7T 



Also 



1 , a , a tan 3 6> , tan 5 6> 



tan = 7 — r and 0=tan0 ^r-+— * •••• 



tan 9 6 



Hence 



tan*=/*+|^ 5 ^ + j i ^- J5 ^ + .... (5) 



Lagrange's Theorem then gives 



, 1 2 13 146 ,_ 



tan^-^-^-^-j^-..., . (6) 



where o_ „ , ^ _ r(4p--l)7r 1 1 tt 



/*-/* + j - [ 4 +€ J n + 2 ' 



Since 



1 5__ 33 , 



8rc tan <£ + 24ti tan 3 </> 512n 3 tan 3 <£ ' "' .' U 



1 , 5 25 



€ ~8ntan^^24ntan 3 (/) 384n 3 tan 3 ^* * w 

 Therefore 



P = a+ 8n 2 tan + 24n 2 tan 3 <£ "" 384n 4 tan 3 ' * ' ^ 



and _ (4p — 1)tt 7r __ (4p — 1 + 2rc) tt 



4n + 2 ~ 4n 



Substitute for tan <j>, its value in terms of /3 and we get 



1 1 25 



/5 = a+ 8^ + 3^~384n 4 /3 3 '" ; ' * (10) 



and applying Lagrange's Theorem, 



Since 



1 128^-31 nn 



fcrra 3o4n*cr 



^="(^-^-2^-)'- • • • (12) 



