MAGNETISATION OF IRON AND OTHER MAGNETIC METALS. 
229 
The condition for a uniform field will therefore be satisfied when d^F/c/x" = 0 ; tliat 
is, when 
9x , 15a;^ 
which gives x = ; tan 6= 6 = 39° 14'. Thus, the condition is satisfied 
when the pole-faces are cones converging as before upon the middle of the neck, but 
with a semi-vertical angle of 39° 14'."^ 
§ 15. Witli a cone of any semi-angle 0, the surface density of free magnetism being 
3 sin 6, the force at the vertex due to a ring at an axial distance x, of radius r, and 
of length (U, measured along the slope, is 
27rr dl. 3 sin 9. x/P, or 27r3 siiF 6 cos 9 di'lr. 
The whole force at the vertex is 
277 siiF 9 cos 9 — j 
r 
a being the radius of the neck on which the cone converges, and h the radius of the 
base to which it spreads. 
Hence (treating 3 as uniform), with a pair of truncated cones, joined by a neck at 
the middle of which they have their common vertex, the whole force there is 
F = 4773 9 cos 9 log^ ^ , 
which, for convenience of calculation, we shall write 
F = 28’935 3 sin^ ^ cos 9 log^g “ • 
§ 16. Applying this to the cones of maximum concentrative power (§ 13), in which 
sin 9 = cos 9 = ' 
F,„„,= 11-187 3 login*, 
and the greatest value of the force will be obtained when 3 the saturation value 
(of say 1700 c.g.s. units for soft wrought iron), in which case 
= 18930 logio^ , 
an expression which measures the greatest possible force which the “ isthmus ” 
method of magnetisation can apply at a point in the axis of the bobbin (over and 
above the small force which is directly produced by the magnet coils). It is 
* We are indebted to Mr. A. Tanakadate for suggesting this calculation of the form of poles suited 
to give a uniform field. 
