530 Dr. A. Russell on the Effective Resistance 



Formula (15) agrees with that found by Lord Rayleigh * by 

 another method. Heaviside f gives 73/(12 2 . 28 . 80). that is 

 657/ (12 2 . 360 . 56) as the coefficient of 0/2) 12 in (14). 



Formulae (7) to (10) should not be used if x is greater 

 than 4, and formulae (11) to (15) should not be used if x is 

 greater than 2. In practice, therefore, they have only a 

 limited use. 



We shall now find approximate formula? for ber^, bei.r, 

 X, Y, Z, and W, which can be used with sufficient accuracy 

 for practical purposes when x is not less than 5, and can 

 always be used for computing with a maximum inaccuracy 

 of less than 1 in 10,000, when x is not less than 10, that is in 

 those cases where the labour involved in the direct compu- 

 tation of the series becomes practically prohibitive. 



If y denote ber«r + t bei# we see from the definition we 

 gave of these functions that 



1^ + -^=^ (16) 



d.ir x ox J 



Putting «/=A'e7 */x, where A' is a constant, we get 



3»0 . /Bay , i n m . 



When x is large, 



9~x \Si + a + b t, 



is obviously an approximate solution of (17), where a and b 

 are constants. Let us assume therefore that 



= x \/i + a + b L + ai/x + a 2 lx 2 + ... 



is a solution of (17). Substituting this value for 0, equating 

 the coefficients of x~ 2 , x~ 3 , ... to zero, and noticing that 

 \/~l = 1/ \/2 + (1/ \/2>, and 1/ */7= 1/ \/f - (1/ V2>, we find 

 that 



1 



i 



ai ~8v/2~SV^' a% ~ I 6 ' 



25 25 1 _ JL3 



a% ~ 384 •! 38471' ai ~ 128' 

 &c. 



Hence we may write 



y\/x = A'e a+ao+Pt 



= Ae a cos ft + *Ae a sin ft 

 * L. e. ante. 



t ' Electrical Papers/ vol. ii. p. 64. [Dr. Heaviside lias written me 

 that he discovered this slip in 1894.] 



