424 DR. T. R. LYLE ON THE SELF-INDUCTANCE OF 



Substituting for d in terms of r and </> in the above expression for L we obtain 



aT, 8a , d 2 \ Sa 1 d 4 \ 8a ]~| 



L = 47rn 2 log-- -2 + -5 ^ 2 log - - +g,[ + J jo.log-- + ? 4 , 



L " it l ? j ct (. ) JJ 



where 



= 

 P * 2 5 .3 



4 -2085^V 2 - 11442c 4 



2-15.7 

 4. If 



A = a( l+m l ^ 7 .+m, 2 



\ a a 



d* 



:! 

 a" a a / 



!, ( 2 , v( 1; ?t a > Jl s can t* e determined so that 



shall differ very little from the value for L given in 3. 



After substituting for A and R in the above and expanding in a series in d*/a 3 , the 

 first three terms of the expansion are identified with the corresponding terms of L in 

 the usual way, and in addition, the coefficient of the fourth term of the expansion, that 

 is the coefficient of d e f(t*, is equated to zero. The n 3 term in R enables this to be done, 

 with the result that a closer agreement is obtained between the proposed formula and 

 that in 3. 



Thus 



=p 4 , n 2 = - 



Hence, when A and R have been so determined the formula 



will give the self-inductance of the coil correct to the fourth order. 



