ON THE CALCULATION OF MATHEMATICAL TABLES. 109 



From this expansion, putting aV i fora;andn=0, ber x=(27r.r)-*c' cos /3 

 and beia;=(27ra;)~Vsin/?, where 



a=-^-+— _ ^^ _ ^^ _ ^^'^^ _ ^75733 

 >/2 8N/2ic 384v'2a;3~128x^~5120v/2a;5~229376N/2a;7 



23797 



4096a;8 



^_ j; _«■ 1 1 25 1073 103 



n/2 8 8N/2a; IGic^ 384 N/2a;=» "^5120 v/2a;^"'" 192x6 

 375733 



229376 N/2a;7 



Putting Xv't for x and «=1 since ro(a;)=Ii(x'), ber'a;=(27rj;)-*e''cosf/. 

 and bei'a;=(27ra;)~Vsin(/) where 



_ ^ 3^_ 21 27 1899 _ 543483 _ 



'^~^/2 8V2a; 128v^2a;"3"'"l28a;^'^5120v/2a;'5~229376V2x^7 



32427 

 ~4096x«~ • • • 



x_ TT ^ _3 21 1899 27 



^~N/2'^8'^8N/2a;"^16a;2"*'l28v/2x3~5120N/2a;5~32a;« 

 543483 

 "22937672*7"*" ' " ' 



The corresponding series for kera;, keio;, &c., are obtained by putting 

 — xiorx and [ ^j*cosn7r for {2ttx)~^ 



§ 4. The ' Product Functions' 



In practical problems the functions usually appear in certain combina- 

 tions. For the ber and bei functions they are as follows : 



X6*(a;)=ber2a; + bei'''a; =la{x^\) ^^{xs/ \) 



Yb (x) = ber'2x + bei'^x = I'o(a; V ,) J,' {x^ .) 



Zb (a?)-=ber a;ber'a; + bei x bei'a?=jj(roJo + IoJ'o) 



W6 (a;)=ber x bei'x — beia;ber'a;= - (I'oJo— Wo) 



2( 



(In the last two the argument xVt is understood for lo and Jq.) 



* This notation is adapted from that in Russell's Alternating Currents, second 

 edition, vol. 1, chapter 7. 



