4 
PEOFESSOE DONKIN ON THE ATTEACTION OF SOLIDS BOKNDED 
A, A;” and let the space (or solid) included between the surfaces (Aj, A), (Ag, k) be called 
“ the shell k^ similarly, let that included between the surfaces (A, AJ, (A, Aj) be 
called “ the shell ^A, Let it also be supposed that each of the surfaces (A, A+^ZA), 
(A-b^ZA, A), is either wholly within or wholly without the surface (A, A). 
By virtue of equation (4.) either parameter may be considered a function of .r, 3/, z, and 
the other parameter. Let the function on the left of (4.) be such that when A is con- 
sidered as a function of 3/, z, A, the two following partial diiferential equations are 
satisfied : — 
d^k d'^k d^-k , 
dk\'^ I /dk\^ 1 dk „ 
s) +U) +(sj +»S/,=0j 
(0.) 
where <p(A) is a function of h not containing A, and is a constant, independent of A and A. 
The second of these equations may be put in another form thus : considering A as 
implicitly a function of y, z. A, we have 
dh I dk dh ^ Q 
whence 
or (extracting the root and employing the notation explained in art. 1) 
hence the second of equations (6.), which is 
may be changed into Q*.Q*=;w. In this form it will be actually employed, so that the 
two equations may be written as follows ; — 
m =?(/»)! 
Q..Q.=b j 
* I borrow this notation, with a slight alteration, from Mr. Catlbt. 
. . . dh ■ 
t The negative sign must be taken for the following reason : — is the ratio of corresponding variations 
dh 
of h, h, when the surface passes through a given point {x, y, z) ; now suppose that an increase of h alone, or 
of h alone, would cause a displacement of the surface, relatively to that point, of the same kind ; i. e. that 
the point would be inside the altered surface in both cases, or outside in both cases ; then Q*, Q* have the 
same sign (art. 1). But on this supposition, if h and h vary together so that the surface continues to pass 
through the point (oc, y, z), it is plain that h must increase if k decrease, and vice versa, so that ^ is 
dh 
dh 
negative. Similarly, if Q,/i, Q* have opposite signs, — is positive. The equation in the text is therefore 
dh 
always true. 
