ON THE CALCULATION OF MATHEMATICAL TABLES. 115 



From this, by expanding the logarithm, we obtain 

 r7Z,/ ^ 1 „, , . » 8 3 27 9 



^'^"^ = 271:7/' "^''' ^=71"^7r.-16P-12871x-3-128x-^ 



. 1179 , 387 , 



5120\/2a;5 512x« 

 W5(x). We have /3=arc tan r^- Differentiating 



W6(aj)^o,__l +_!_ + JL + _M_--i^IL--^- 

 Xi(a;) ^ n/2 8N/2a;2 Sx^ I28v/2cc« 1024 V 2x5 32x^ 



Proceeding as for Zb{x), we obtain 

 "\V6(x) = — =— e^" where 



2N/27rX 



« 1 I 23 1 _ 1 153 ^35^ 



''"^v/2"^872^^i6^''^38472x"3 128x' 5T2"0s/2x-^ i536x« " ' * 



§ 14. The differential equations of the 4th order (see § 11) are all 

 unchanged by substitution of -x, or tx, for x; therefore the same co- 

 efficients furnish four independent solutions of each equation. 



The ker, l^ei, &c., forms are of course obtained by substituting —x for 



X and A /'" for a / -• The expansions of the mixed functions are: 



Xr(x)=\ 1 / 13 , \ fcosW r^_ 1 ^^_ 



Xu{x)= / 2x ^''^' \Qix' + • • • j \ sin i V'^^/ ^ 4 V2a; 192 V2x^ 



+ -i^+ 



2560 V 2a;^ 



1899 

 2560v/2a;* 



_3 27 1179^ ^ _ 



''"4v/2a; 64V 2x=' 2560^ 2a;'* 



W^x)-/272:^ ^"V 8x2 64x^^768x« /IcosJV 



1 , 23 __^ 1153 



4v'2x 192 v/ 2x3 2560v/2x* 



I 2 



