378 Prof. H. Nagaoka on Magnetic 



It is found that deviations of different orders are positive 

 in one sector starting from the mean point, followed by 

 that of negative sign in alternate steps, and the curves 

 of constant deviation resemble hyperbolas. 



"Recent advance in accurate measurement of current 

 calls forth the necessity of deducing some formulae, which 

 will easily give the magnetic field accurate to about a 

 millionth part. For this purpose, the ordinary method of 

 expansion in spherical harmonics is not suitable, as the 

 convergence is not sufficiently rapid. The following cal- 

 culation was made with the view of meeting such needs 

 in practical problems. 



Let U be the Newtonian potential of a homogeneous 

 body of rotation about the y-axis ; referre d to the axial co- 

 ordinate y and the radial coordinate p — ^/x 2 -f ^ 2 , Laplace's 

 equation can be written 



^u yv i bu (1) 



or 



and 

 If we put 



^u_ _i _a/ au\ 



•df - p-'dpV-dp) 



b 2 u _ i a /; ^\ 



( au 



'dp'dy p' "by \ "dp J 



*=^y and * = P W • " " " ( } 



they satisfy the equation 



"dp * "dp 3y " By 



showing that the families of surfaces (/> = const, and 

 ty = const, are orthogonal, and the sections through the 

 ?/-axis of these surfaces represent diagrams of equipotentials 

 and lines of force. 



Referred to polar coordinates r and 0, the potential of a 

 uniform circular disk of radius a is given by 



dr d6 /0 x 



. . {5) 



fa nz 



U= J J 



Jo */o 



V p 2 — 2rp cos # + r 2 -f y l 



rp cos 6 -fr 2 , we have 1 

 unctions 



J (\R) = J {\p) J Q (\r) + 22 J n {\p)J n {Xr) cos 0, 



Putting R 2 = p 2 — 2rp cos 6 -f r 2 , we have by the addition 

 theorem of cylinder functions 



