Prof. Donkin on the Theory of the Attraction of Solids. 397 
equal to the weight of half a cubic inch of water. But so small a film 
as this would form itself even if the two surfaces of the ice were only 
very imperfectly fitted to one another. If, again, by better fitting, a 
film be produced of such size and form as may be represented by a 
square film with its sides 4 inches each, the slabs will be urged toge- 
ther by a force equal to the weight of half a cube of water, of which 
the side is 4 inches; that is, the weight of 32 cubic inches of water or 
1°15 pound, which is a very considerable force. Secondly, the film of 
water existing, as it does, under less than atmospheric pressure, has 
its freezing-point raised in virtue of the reduced pressure ; and it would 
therefore freeze even at the temperature of the surrounding ice, 
namely the freezing-point for atmospheric pressure. Much more 
will it freeze in virtue of the cold given out in the melting by pressure 
of the ice at the points of contact, where, from the first two causes 
named above, the two slabs are urged against one another. 
The freezing of ice to flannel or to a worsted glove on a warm 
hand is, I consider, to be attributed partly to capillary attraction 
acting in similar ways to those just described ; but in many of the ob- 
served cases of this phenomenon there will also be direct pressures from 
the hand, or from the weight of the ice, or from other like causes, 
which will increase the rapidity of the moulding of the ice to the 
fibres of the wool. 
December 8.—Sir Benjamin C. Brodie, President, in the Chair. 
The following communication was read :— 
“On the Analytical Theory of the Attraction of Solids bounded 
by Surfaces of a Class including the Ellipsoid.” By W. F. Donkin, 
Esq., M.A., F.R.S. &c. 
The surface of which the equation is 
f(a, 9; 2, &, &)=0, Lady, hig we es CL) 
is called for convenience “‘ the surface (4, £).’” The space, or solid, 
included between the surfaces (/,, &), (A,, &), is called “ the shell 
(i t) ;” and that included between the surfaces (h, 4,), (A, k,) is 
1 
called “ the shell ( A, 2)” [This notation is borrowed, with a slight 
1 
alteration, from Mr. Cayley.] It is assumed that the equation (1) 
represents closed surfaces for all values of the parameters h, 4, within 
certain limits, and that (within these limits) the surface (A, 4) is not 
cut by either of the surfaces (h+dh, k), (h, k+dk). It is also sup- 
posed that there exists a valued, of A, for which the surface (h,,, 4) 
extends to infinity in every direction. Lastly, it is supposed that if & 
be considered a function of 2, y, z, h, by virtue of (1), the two fol- 
lowing partial differential equations are satisfied : 
Pk @k , dk 
Fabel ae #) 
dk\* dk\* , (dk\? dk 
a) +5 +(5) stil irl 
in which ¢(/) is any function of h (not involving *), and n is any 
