and Inductance of a Concentric Main. 



529 



respect to x. The combinations Y/X, Z/X, W/X, Z/Y, and 

 W/Y also occur in this and allied problems, and so we shall 

 give formulae for these functions as well. 



By squaring the series for ber x and bei x and adding them 

 together, we get 



v , 1/,1'V, 1 /x\ s 1 Ai'V 2 1 Ai'Y 6 __ 



approximately. 



When a? is not greater than 4 this formula may be used. 

 For instance, when x is 4, (7) gives X(4) = 11-8275. From 

 the tables given in Gray and Matthews' ' JBessel's Functions ' 

 where the values of ber x and bei x are tabulated for values 

 of x up to 6, the interval of the argument being 0'2, we 

 find that ber 4= -2-56342 and bei 4 = 2*29269, and thus 

 X(4) = ber 2 4 + bei 2 4 = ll-8275. 



Similarly we get the approximate formulae : — 



*-?{^(?)'+fKf)'+ #©"}•■ • 



z =B{'4(l)'4(i)' + nb(3'}' 



(«) 

 (9) 



and 



w-f{i +( >(f)V<g.(!)V$(i)"}. ■ m 



When .?' is not greater than 2 the following formulae can 

 be employed : — 



Y _ .r f _5 /#\ 4 143 f #\ 8 7661 /arV 2 1 

 X" 4 l. "12V2/ + 720 U? 4?\1\2J J' ' 



Z x* r ll/a?V 473 /*Y 304107 /aA 12 l 

 X ~ 16 i 1 " 2iU) + 3~I0 \2 J " 4 . 12* |7_U ^ }' 



W x f\ 1/*V 19 /*Y 687 /*\ u l 



x=n 1 "3(T) + iLb)-7-TFb)>- • 



Y 4\_ 24V 2/ + 4320 W 12 2 . 360 . 56 ^2/ J '^ ; 



(11) 

 (12) 

 (13) 



.n d 



w 



Y 



f 2 f i 1 /a?Y 1 /^\ s 11 /.r\ 12 "l „_ 



- = ^1 1+ 12(2) -180(2) + 12. 28,30 (2-) J' ( 15 > 



Phil. Mag. Ser. 6. Yol. 17. No. 100. April 1909. 2 O 



