384 Prof. H. Nagaoka on Magnetic 



Thus the values of the magnetic force, as we proceed 

 outward from the plane of the circle by f, rj, are given 

 by the formulae (A) and (B). The process seems at first 

 rather tedious, but when once the value of q is known 

 in terms of the coordinates £, rj, the expansion in ascending- 

 powers of the variables is but a simple process of addition 

 and multiplication. The usual process of expansion in zonal 

 harmonics of different orders is looked upon as a practical 

 guide to the solution of such problems. If we try to treat 

 the problem such that the magnetic force at the point f, v is 

 expanded in different powers of f , 77, the solution in spherical 

 harmonics becomes very tedious as each harmonic involves 

 different powers of f , rj. A practical calculation will convince 

 the reader of the facility of the g-series. Of course the 

 knowledge of the expansions of the integrals in ^-series 

 must be presupposed. 



The above calculation was checked by expanding the 

 component forces X and Y in zonal harmonics. The 

 advantage of expansion in power series of the coordinates 

 referred to the centre and the plane of the circle lies in 

 finding the deviations of the magnetic force from the value 

 at the mean point as we recede from it. We can group 

 terms of different powers and discuss their influence in 

 disturbing the field, just as we discuss the aberrations of a 

 lens, by expanding them according to powers of the so-called 

 Seidel's coordinates, and class the deviations according to the 

 orders of the coordinates involved in the aberrations. 



Radial Component X for a Single Coil. 



Reverting to the expression of magnetic force X given 

 in (A), we find, by assuming the radius of the coil a and the 

 axial component of force at the centre to be both unity, that 

 the deviation of the second order is given by. 



(PL VI. 

 **l - c, (15) Fig. 1.) 



which is represented by rectangular hyperbolas. By con- 

 sidering c as a parameter, and giving small fractional values 

 to it; we obtain loci of points at which the radial component 

 is c times the force at the centre. Thus the distribution of 

 the radial deviations of second order is given in fig. 1; they 

 are mapped out for equal difference of c except for curves 

 near the centre, in a square of sides equal to the radius, in a 

 plane through the centre perpendicular to the plane of the 

 circle. 



