274 

 And 



Mr. W. D. Niven on the Calculation 



sin cos M ~ 2 0=sin a cos M ~ 2 a + {(n— 2) sin 2 a cos' 1 " 3 a— cos M-1 a}^ 

 + \(n— 2)(?i-3) sin 2 a— (3w— 5) cos 2 a} sinacos t,-4 a 



+ etc. 



( 8 ) 



I now propose to neglect the squares and higher powers of \p. The 

 effect of this on the (c) integral will be that we have now to find X 

 from 



cos' 



s ?l-1 a C p du , (n — 1) cos"" 2 a sin a C p , du 

 Call the two integrals in this expression Q and B. Then 



and, taking account of equation (6), 



^ COS'^a r 1 / 1 1 \ 1 If 1 _ 1 \1 



w \27i-2^ 2w - 2 p 2n - 2 ) n-2p\q n ~ 2 p n ~ 2 ) J 



Let this be put equal to 



Q(«-/3)/. 



Then, since 



we see that 



/= 



2->i-2\g 2w - 2 <p 2n - 2 ) n-2p n \g n - 2 p n ~ 2 / 



Now let - — ? =X, so that p\( =^ —l\ may be regarded as a frac- 

 2 JP \ 2 J 



tion whose square may be neglected. "We get 



3?i 2 -7« 



/= 



24 



12 



•}• ' 



(9) 



= ^ l + («-2) P X ifWenegleCt( ^ ; 



