116 REPORTS ON THE STATE OF SCIENCE. ^191 6. 



§ 15. The expansions of the ratios, similar to those given above for 



^A(^ and 1^#-), may be noted here. 

 Xb{x) Xb{x) 



From e^''=2-!rxYb{x) ; taking logs and differentiating, and noticing 

 that Yb'{x) = 2hNb{x)-hb{x)^ 



WJ)(a;)_ , l^J^ , 1 . 3 63_ _ 27 _ 1899 _^ 



Ybix) ''2a; ^/2 2a; 8^/2x2 128s/2a;'' 32r^ 1024y2a;« 



From e'^'=2s/2^TrxZb{x), differentiating as before and noticing that 

 Zb'{x)=yb{x)--Zb{x) 



VM=2^'=2l ^ + ^ +^+ ^ + ^ -^^^- 

 Zb{x) " 1^2 8v/2a;2 8a;=' 128 /2a;'' 32a;'^ 1024V2a;6 



1161 



256x^ 



From e'^"—2V2KX^Nb{x), noticing that ^b'{x)=Xb{x)-^ \Nb{x). 



Xb{x)_ f_ f I _ ^ _ ^ _ 23 1 1153 



Wb{x)^^'^'~^ 1 V2~8v/ 2x-^ 8a;3 128 v/ 2a;^ "*" 32a;5 "^ 1024 V 2x« 



835 

 ■256a;7' 



835 



From 0=arc tan r — r, differentiating 



yb{x)~'^ v/2 8V2a;2 8a;-* 128 v/2a;^ 1024 v/2x« 16a;^ 



Yb{x) f Z6(a;) y j W5(x) 1 ^ 



Yb(x)_ JL 1_ 3 13 83 _ 25 771^_ _ 



X6(^~'^~N/2a;"^4a;2'^8v/2a;3'''32a;''"''l28>/2a;5 64a;6 102W'2x'' '" 



The ker, kei, &c., functions yield similar series with the sign of x 



* It will be seen that we may also obtain the coefficients in the series f and a> as 



