MR. J. S. TOWNSEND ON MAGNETIZATION OF LIQUIDS. 
537 
Let $ be the potential of the magnetic force ; dv the element of the normal to the 
boundary of the liquid. There is a distribution cr of magnetism on the surface of the 
liquid, where cr = — k d^jdv, which gives rise to a small force f, at C, in the direction 
opposite to F. So that, if B is equal to the mean area of the secondary coil and N' 
its number of turns, there is a diminution in the total flux of induction through the 
secondary circuit equal to A/’N'p -f- (B — A)/N', due to the magnetism induced on 
the surface of the liquid. 
Since p = 1 + ink, and /‘involves k as a factor, we may put p, = 1 , and the cor¬ 
rection reduces to B/N'. 
Now, since cr = — k dfyldv, we may, in determining the normal component — d(f>/dv 
of the magnetic force, omit k and other small corrections and consider the case of a 
long solenoid of length 2 / and find the force at the ends parallel to the axis, and the 
normal force at the cylindrical boundary, when the space inside is occupied by air. 
It is easy to see that <f> = — 47 mz + <f> y , z being measured from A to B along the 
axis, and the potential of the distribution — n at A, and + ?q at B, so that at the 
ends of the solenoid the component L of the magnetic force is 
— d(f)fdz = 4 7 m — 2-rrn = 2nn at A or B. 
The following simple consideration also shows that the force parallel to the axis, at 
the ends is approximately half its value at the centre. The force at B due to the 
long solenoid AB' is 47m. This must be contributed to equally by the parts AB and 
BB', hence, when the latter is removed, the force parallel to the axis is 27m. 
The double integral jjyy ds vanishes over any closed surface within which A 2 <£ = 0 
at every point, so that by considering the closed surface made up of the two planes 
perpendicular to the axis at C and B, and the intermediate part of a cylinder of any 
section whose generators are parallel to the axis, and integrate over it. We get 
— |j ds over the plane end at C 
= jjyy ds over the length I of the cylindrical surface and the plane end at B. 
But d(f>/dv = inn at C, and — 2nn at B. 
Hence 
ds over the length l of the cylinder = — 47m (A) + 2nn (A) = — 27m (A), 
where (A) = area of the section of the cylinder. 
Now let this cylinder be a body whose coefficient of magnetization is small, and let 
it extend from A to B, since the surface distribution is — kd(f>/dv, we see that : 
MDCCCXCVI.—A, 3 Z 
