PROF. W. THOMSON ON THE MATHEMATICAL THEORY OP MAGNETISM. 285 
which produces the same potential at any point, internal or external, as the given 
solenoidal magnet. 
84. The same general conclusion may be arrived at synthetically in a very obvious 
manner, by taking into account the property of a solenoid stated in § 71, aecording 
to which it appears that any two solenoids of equal strength, with the same ends, 
produce the same force at any point whether in the magnetized substance of either, 
or not. For it follows from this, that when a magnet is divisible into solenoids with 
their ends on its surface, by joining the two ends of each solenoid by any arbitrary 
curve on this surface and laying a solenoid of equal strength along this curve, we 
obtain a series of solenoids, constituting by their superposition, a tangential distribu- 
tion of magnetism in an infinitely thin sheet coineiding with the bounding surface, 
which produces the same resultant force at any internal or external point as the given 
magnet. It is not, however, easy to deduce from, this synthesis, a formula involving 
the requisite arbitrary functions to express a superficial distribution satisfying the 
existing conditions in the most general manner. The analytical investigation given 
above, supplies, in reality, a complete solution of this problem. 
It may be remarked that the sole condition which F, G and H, considered as 
funetions of the coordinates, x, y, z, of some point in the surface of the magnet, and 
therefore functions of two independent variables, must satisfy in order that (XVII,) 
may express correctly the potential at any point — 
(m dG 
\ dy dz 
dz 
, (dG 
dxJ~^\.dx 
dy) 
(XVIIL), 
x,y and a of course being supposed to satisfy the equation to the surface; and it 
may be proved, by a demonstration independent of the investigation which has been 
given, that the second member of (XVII.) has the same value for any functions F, G, 
H whatever, which are subject to this relation. 
END OF CHAPTER V. 
