236 Dr. J. R. Airey on the Addition Theorem of the 



The expression in the bracket can be considerably simplified 

 by comparing it with the series for sin h. In fact, the 

 coefficient of Z^p) 



f/i h h 2 A 3 \ - 7 / A 5 A 6 \ 



/ A 7 3A 8 \ / 2A 9 \ -i 



+ \iip 2 2.v. P 3+ "'J \v\ P 2 •••;+•••] 



fp, /- A\ . , A 5 /, 3A 12A 2 25A 3 \ 



A 7 /., 3 A 6 9 A 2 \ 2 A 9 /, 3 A \ 1 



This is approximately equal to 



"G loge ( 1+ p) sinA 



A 2 / 3 A 5 A 7 Vi^ 12/* 2 \1 



360p 2 V 1 14 + 504 ' * 7 V 2p + 7p 2 " 7 J 



= - K K,( 1+ 5),„ i+i |,(.«-* + J..)( 1+ |)-]. 



Hence 



%o(p + h ) = -[« sin A— £<>*?] Zi(p), 



where p , A , 7*\ ^ ^ 2 /\ , 3A\" 2 



/ . 7 7 A 3 A 7 \ 



and ^ = ^- S inA + A-^ + MM() -...j. 



When the value of p is comparatively large and A not 

 greater than 1'6, the expression for Z (p + A) reduces to 

 one term, 



Z o (p + A)=-^log e (l+J).sinA.Z 1 (p), 



and ZqQ-A) _ log (p — A) — logp 



Z (p + /0 log (p+ A) -logp' 



