DEPARTMENT OF PURE MATHEMATICS. 497 



or, in terms of k, i.e. ^(4?i- - 1), 



k USk(k-l) 1.3.5 k( k-r){k-ti) 



This establishes the series for R-. 



Tojind the series for a, we have by (11) 



da' 



R 



t*m- 



Let 1 + - =l_--^-«_=^''_ 



d.v ~ .V- .r' .v'' 

 Then 



= 



,'. ;»4 = - ^= + |/b(A; - 1) = ^/c(A: - 3), 

 and ^jy, after reduction, = ^A(/i;- — lik + 15) 



Pg = |/v(5^^ - 190/c^ + 807k- 630) . 



Inserting these values, and integrating, we find (a being zero when k = 0) 

 _k k{k-3) k(^k-~Uk + 15) k{5k^ - 190F + 807/c - G30) ^ ,.g. 



This is not a very nice series for a, but when .v is large we may take 



k k k(k-3) 



a = - or, a = - + - ' 3 ^ 



as a sufficient approximation. 



Or we may, if we Hire, calculate P as well as R, and then obtain a from the 

 formula cos a = P -f- R. 



The interest attaching to the above series for R'^ lies in the fact that when 

 .r is fairly large R varies but slowly, and so, similarly, does P. 



Hence, if small tables of P and R, or, rather, of log P and log R, were cal- 

 culated for different values of .r, say, 10, 11, . . . 20, and then 30, 40, . . . 

 100, and then 100, 200, . . . for different values of ?;, say at intervals of I, i.e., 

 for ?i = i, 1 ,§,..., it would be quite feasible to interpolate in these tables not 

 only for intermediate yalues of x, but also for intermediate values of n. 



A remarkable relation between the values of P and Q for successive values of 

 n differing by unity follows from the formula 



— J„ =J,i_i — J„ + i I « • « (16) 



.V 



Thus, J,. -=- ^-^ = P„ cos (.r - ^ - n ^) - Q„ sin (.v - | - « ^) 

 = r„ cos X — Q„ sin X, say. 



•'. -T,, 1 1 -^ A /-- = P,. + 1 sin X + Q„ + 1 cos X, 

 V n.v 



1906. K K 



