114 REPORTS ON THE STATE OP SCIENCE. — 1916. 



The new coefficients are (('*i— Ci+^-p I '■> ( ''^2 "" 2^2 + 5-92 j ' 



Zk{x)=[ia- loga;)2 + fg} Zb{x) + 2{a-\ogx) [-1+^11^^ 



3 11J2|4 J 



^fx , x^ , \ 11 (V2)V943 (V,)^ 



2^,22 (2.6)2 j2 ;^g ^j2 288)1^|2|4 



The new coefficients are (m, — ^Ci j ; In^ — jcA ; (^3— ^Csj ; 



Vfk{x) = I (a - log a;)2 + ^ j W&H + 2(a - log x) I ^(7 2) 



.17 (72)% ] 



I21I 11^13 J 



4 U 22 12 (2.6)2 14 "^ ■ ■ ■ / "^ 2 ^2/ 36'|1 |1 13 3600 "12 |2 |5 



(8 3 \ / 3 3 \ 



(^^^-6^^+8X2) •••• 



§ 13, Expansions in series of descending poivers. 



From the expansions, in descending powers, of her x, &e., we have 

 Xb(x) = -^e''' and Yb(x)=J-e^'' (for a and v, see § 3). 



^ ^ ^vx 2irX 



Zb{x). Putting Zb{x)=^e'\ we have ||M=27rCe^«-'". 



Now e^''=2TrxXb{x). Taking logs and differentiating 



Z&(^)^^,_1^J 1 _ 1 , 25 ^ 13 _^ 1073 

 Xfc(x) " 2x s/2 2a; 8N/2a;2 128 V 2a;* 32a;5 1024V2a;6 



Th» puUtag C=^-J^. 2g-a)=log(l-^-i-^-gl + 



25 

 128a;^ 



16N/2a;' 



