228 
PROFESSOR J. A. EWING AND MR. W. LOW ON THE 
which gives 
X = ; tan 6 = ^/2 ; 0 = 54° 44'. 
Hence, a series of co-axial rings will be most advantageously disposed for producing 
force at a point on the axis if they lie on a cone having its vertex at that point, with 
a semi-vertical angle of 54° 44', This conclusion is independent of the distribution of 
density from ring to ring. 
§ 13, The greatest force will be produced when the pole-pieees are themselves 
saturated, so that 3 reaches its limiting value in all j^arts of the metal. In that case 
the distribution of density from ring to ring is uniform. The surface density of free 
magnetism at any point of a sloping pole-face is 3 sin 0, where 0 is the slope of the 
face to the axis of magnetisation. The whole quantity in each ring is 3 multiplied 
by the area of the ring projected upon a plane normal to the axis—a quantity which 
is independent of the slojDe of the cone. We have, therefore, the same series of 
attracting rings to deal with, whatever be the slope of the convergent forces, and 
whether that slope be uniform or not. Given, then, a certain diameter for the neck 
of the bobbin to be magnetised, the greatest magnetic force wull be produced at the 
middle of the axis of the neck if we make the expanding ends and pole faces in the 
form of cones, with a semi-angle of 54° 44',' and with their vertices at the middle of 
the neck.* 
§ 14. This cone of maximum concentrative power is not, however, the form best 
suited for producing a uniform field. At any point in the axis clF/dy and dFjclz are 
zero, axes of y and 2 being taken in a plane parallel to the rings, and the condition 
for a uniform field (uniform, namely, in the neighbourhood of the axis, over a trans¬ 
verse plane) is that d^Fjdy'^ and d^Fjdz^ shall also be zero. Consider again the 
attraction of a ring at any point O in the axis. Taking Laplace’s equation— 
rfW ^ _ 
If- dy^ ~dF ~ 
and differentiating with respect to x, \ve have 
d- ^ d^ (1^ d- dN _ 
dx- dx dy- dx dz^ dx 
By symmetry of the held about the axis of x the second and third terms are equal; 
hence, writing F for dYjdx, 
d^ 2<FF 
dx- ^ d.if 
0 . 
* The correspoutling proi^osition for trancated cones, with an air space between them, has lately been 
stated by Professor Stefan (‘Wien, Akad. Sitzber.,’ Feb. 9, 1888 ; or ‘ Phil. Mag.,’ vol. ‘25, p. 32‘2). 
