PROF. W. THOMSON ON THE MATHEMATICAL THEORY OF MAGNETISM. 283 
tions (XII.) show that with these data the values of (3^, y, may be calculated ; and 
again, equations show conversely that if (3^, y, be given the required data for the 
problem may be immediately deduced. We infer that the necessary and sufficient 
data for determining the resultant force of a lamellar magnet, at any external point, 
by means of formulae (X.), are equivalent to a specification of the direction and mag- 
nitude of the tangential component of the intensity of magnetization at every point 
infinitely near the surface of the magnet ; and we conclude, as we did in § 78 from a 
very different process of reasoning, that besides these data, nothing but that it is 
lamellar throughout need be known of the interior distribution. 
81. The close analogy which exists between solenoidal and lamellar distributions 
of magnetism having led me to the new formulae which have just been given, it 
occurred to me that a formula (or formulae, if it were necessary here to separate the 
cases of internal and external points,) for solenoidal distributions analogous to the 
formulae (VII.) of §§ 77 for lamellar distributions might be discovered. Taking an 
analytical view of the problem (the synthetical view, although itself much more obvious, 
not showing any very obvious way of arriving at a formula of the desired kind), I 
observed that the formula -g deduced from the general expression for the 
potential, by a partial integration performed upon factors involving a, (3, y, and de- 
pending on the integrability of the function adx-\-l3d^-{-ydz, ensured by the equations 
dz dy ’ dx dz ’ dy dx ’ 
for a lamellar distribution; and I endeavoured to find a corresponding mode of 
treatment for solenoidal distributions, to consist of a partial integration, commencing 
still with factors involving a, (3, y, but depending now upon the single equation 
dx'dy'dz 
(«). 
instead of three equations required in the former process. After some fruitless 
attempts to connect this equation with the integrability of some function of two in- 
dependent variables, I fell upon the following investigation, which exactly answered 
my expectations. 
82. In virtue of the preceding equation {a), we may assume 
^ dy dz’ ^ dz dx’ 
dy 
(XIV.), 
where F, G, H are three functions to a certain extent arbitrary, which, as I have 
since found, have for their most general expressions 
F=/7' dyd.(f-%+%' 
2 o 2 
(XV.) 
