2 ART. 15. — H. NAGAOKA: NOTE ON THE POTENTIAL 



little lise has been made of Jacobi's ^/-series. Mathy^^ uses 

 Weierstrass's If? -function in evaluating M, but he seems to be in- 

 clined to the use of a hypergeometric series, rather than to the 

 reduction of these integrals to a rapidly converging q-series, to which 

 the expression can be easily transformed. 



The ])roblem can, however, be attacked from another point of 

 view. In the following, I proceed by finding the Newtonian poten- 

 tial of an uniform circular disc, and derive the expression for 

 the potential and the lines of force for a circular current by simple 

 differentiation. Finally il/ is expressed by means of a simple (/-series, 

 of which a single term will generally suffice to secure a practically 

 accurate value ; the force between two coaxial coils can also be 

 expressed in a similar manner. 



§ 2. The whole investigation rests on the following lemma. 



The potential U of an homogeneous body of rotation (about 

 2-axis) satisfies Laplace's equation outside the body, which in this case 

 is given by 



s z'^ 3 x'- X S X 



X being the radial coordinate. Thus 



öiC "hz X "hz ^ "bx f 



If the potential ip of a certain distribution symmetrical about 2;-axis 

 be derivable from U by differentiation with respect to z, so that 



lU ,-r\ 



^=^r— (I.) 



o z 

 1) Mathy, Journal de Physique, torn. 10, p. 33. 1901. 



