430 Prof. A. W. Riicker on the Magnetic] 



values of fia/2 are required the approximation must be carried 

 to the second term in the integral, the other terms having 

 their true values ; but that an approximation neglecting all 

 terms containing powers of yua/2 above the second, will give 

 accurate values of the mean and minimum currents if the 

 value of /ia/2 used is determined to the same degree of 

 approximation. 



The evaluation of the expression 



may be proceeded with as follows : — 



Writing y=wa/2, x—a[2 = \a/2 7 



b + a[2=/3a[2 7 



the expression may be put in the form 



2IU f 1 € M«A/2_|_ 6 - M aA,2 



tt | € M«/2 + r M«/2 





Pf+l^l 



d\. 



If we expand the integral in powers of /na/2 the general 

 term is 



- 2 

 n 



Then 



(fxa^C 1 X 2n d\ ■ 2 /fiia\ 2 " 



F \ 2 "(\ + /3j 2nC \ 2 *- l (\ + l3)d\ 



2n u 2 {(\ + @y 2 + v?}i u 2 J \{(> + /3) 2 -r« 2 }i* 



Multiplying and dividing the quantity under the sign of 

 integration by (A. + /3) 2 4- u 2 , and writing A 2 » for the first term, 



we get 



F, ft = A 2n - % { F 2B+2 + 3/3F 2 „ +1 + (S/F+t^F, 



+ iS09 9 + « 2 )F 2j ._ 1 



or Fo n+2 



?M- 



Z?i 



30F 2 „ +1 - ^3^ 2 -h^(l+ i)] F, 

 -/SQ^+OF^, 



and when the limits are inserted 



,2 



!L-A - 1 f 1+S 1-/3 1 



2n 2n ~ 2rc I {(I +£)«.+ u*}» + {(l-£) a + i?}* J ' 



