14 Allï. G. — H. NAGAOKA : THE INDUCTANCE 



the first and second kinds, while the third term corresponds to 

 to an integral of the third kind and forms the chief dilliculty 

 in the numerical calculation. 



In the refined experiments of the present day, a formula 

 which admits of exact calculation is found essential. The usual 

 formula for M expressed in terms of elliptic integrals of three 

 different kinds is not easy to evaluate. The calculation of the 

 integral of the third kind, which is usually expressed by in- 

 complete integrals of the first and second kinds requires a good 

 deal of labour, even when Legendre's table is accessible, as it 

 is of double entry. When the integral is expanded in powers of 

 k, as in Russell's formula, the convergence is rather slow and 

 the calculation of successive coefficients by the formula of recur- 

 sion is not easy. In place of the usual method of reduction, the 

 evaluation by means of the ç'-series, which is rapidly convergent, 

 may be used with great advantage. 



§ 9. The expression for / in the above form (11) leads to 

 cumbrously large values of i/-2 and p'u, so that for exact calculation, 

 it is convenient to reduce the above to a simpler form. It can 

 be thrown into various forms suitable for numerical calculation. 



= —ecu,— 



1 ä,"{o) ' 



