20 Prof. H. Naoraoka on the Potential and 



'O 



of elliptic integrals or in terms of zonal harmonics. The 

 lines of force of a circular current (or stream-lines about a- 

 vortex ring) are usually obtained from the expression for the 

 mutual potential energy (denoted by M) of two coaxial cir- 

 cular currents. By using F. Neumann''s formula, M may be 

 expressed by elliptic integrals, or developed in terms of zonal 

 harmonics, which is sometimes advantageous in calculating 

 the action between thick coils. Maxwell has also given a 

 table of the coefficients of mutual induction when the coils 

 are near each other. In these calculations we are always 

 in need of Legendre^s tables. It is A'ery curious that so 

 little use has been made of Jacobins ^-series. Mathy"^ uses 

 Weierstrass' y-function in eA^aluating M, but he seems to 

 incline to the use of a hypergeometric series rather than 

 to the reduction of these integrals to a rapidly converging 

 ^-series, to which the expression can be easily transformed. 



The problem can, however, be attacked from another point 

 of Adew. In the following I proceed by finding the New- 

 tonian potential of a uniform circular disk, and derive the 

 expression for the potential and the lines of force by simple 

 differentiation. Finally, M 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 potential U of a homogeneous body of rotation 

 (about x:-axis) satisfies Laplace's equation outside the body, 

 which in this case is given by 



B^ B^IJ 1BU_. 



X being radial coordinate. Thus 



^s^ X b^x bx /' 



and 



bx'bz X B^\ "bx ) 



If the potential <^ of a certain distribution symmetrical about 

 the ^-axis be derivable from U by differentiation with respect 

 to z, so that 



*=-f , (I.) 



* Mathy, Journal dt Physiq^ie, x. p. 33 (1901). 



