108 REPORTS ON THE STATE OP SCIENCE. — 1916. 



Pakt IV. 



Tables of the ber, bei, ker, kei, dc. functions — (continued). 



Introductory Note. 



§ 1. Definitions (Kelvin, ' Math, and Phys. Papers,' vol. iii. p. 491 

 Russell's 'Alternating Currents,' 2nd ed., vol. i., chap. 7). 



ber ic + i bei ic=Io(a;'^t) ('=>/— 1) 



ber x—i bei x=3q{x x^i) 

 ker a; + t kei a;=Ko(a;>/i) 

 kei x—i kei a;=Go(a;v^t) 



§ 2. Expansions 



In series of ascending powers. 



X* x^ ' " " 



ber a; =l—^r5—-, + ,-r— ————„+ . . . beirc=; 



2*. d'^ (2. 4. 6. 8)2 ' •■ 22 (2. 4. 6.)2^"" 



ker a;=(a— log a;) ber a; + - bei re — ;, — — „ + — -~ — ] 



^ ^ ^ ^4 2 22.42 24 2.4.6.8)2 | 



k9ia; = (a-log a?) beia;-^bera;+ |J-g^. (24 6)2+ ' ' * 

 general term, 



a=log, 2-y = -1159315. . . 



2(1) 



(2. 4 . . . 2s.)2 



§ 3. In semi-divergent series of descending potvers. 



These are obtained from I„(a;)=(27ra;)-' exp (x+^+—„ + ^^^!^ 



^ x 2x^ 6a;^ 



, 8m + 2to2 15m + 14w^ + m' 45m + 51m2 + Sm^ 

 ~Ix* 10^5 + 12^6 



630m + 807m 2 + 1 90m^ + 5m* 8r5m + 438m2 + 132m3 + 87ra^ 

 56a;^ 8x^ 



11340?» + 16704m2 + 5925m3 + 560m^ + 7m-^ 

 72a;9 



14175W + 21780m2 + 8655m^ + 1080m^ + 32m5 



20a;'o 



where vi—^ ^l 



8 



(To derive these coefficients put in Bessel's equation for I„a;, viz. : 

 y' + -y-^(l + -J=0; ?/=exp (^u.dx), whence u^ + tt' + ^u-(l+^-l\ 

 =0 ; from this the coefficients are readily deduced.) 



