WITH AN APPENDIX BY PROFESSOR A. LODGE. 109 



We have 



P,, = -,-~- -r {sin (n+l)0 

 irk(2n + 1) I 



- (), 



with 





When n is great, approximate values may be used for the coefficients of the sines 

 in (a). To obtain LAPLACE'S expression it suffices to take 



1 1_._3 J_. 3 . 5 



2' 2.4' 2 .~4 . 6' 



but now we require a closer approximation. Thus 



1 . (2n + 3) 2 \ 2n + 2 



1.2. (27* + 3) . (2 + 5) == 2 . 4 \ ~ 2n + 2 " 2n + 4 > 



and so on. If we write 



r l 



2n' 



the coefficients are approximately 



1 1_._3 T_^3 . 5 o 



2 lT 2 .1 X 274 76 * 



and the series takes actually the form assumed by TODHUXTER for analyticnl 

 convenience. In his notation 



1 3 



C = t COS -\- ifi COS ?>0 -j- 5 COS 50 + . . ., 



2 . 4 



and 



S = t sin 6> + l < 3 sin 30 + * ' 3 * 6 sin 5(9 + . . ., 



P, = , /0 4 lS (0 sin n0 + S cos n6\, 

 TTK (2n + 1) 



where ultimately is to be made equal to unity. 

 By summation of the series (t < 1), 



t t 



C = r- cos (0 + j<^)) S = j~ sin (0 + 

 v/jo vp 



