10 
PEOFESSOE DONKIN ON THE ATTEACTION OF SOLIDS BOUNDED 
12. I shall conclude with an independent demonstration of the Theorem IV. art. 6. 
Considering h as a function of x, k, by virtue of the equation i{x, y, z, A, i^’)=0, 
we have ^ „ 
{f{h) being any function of A, and the notation being that explained in art. 1). 
Hence the condition of the possibility of satisfying the equation Dy'(A) = 0, is that 
— ^ be expressible as a function of A ; if this be the case, f(h) will be determined by the 
j'lnrs T\9z. 
We have then to show that this condition will be fulfilled if the 
f{h) Wh 
dk 
equation jijj^ — ~~ ' 
function i{x, y, z, A, k) be such that 
DU’=(p(A), Ql-\-njj=z^. 
It was proved in art. 3 that the latter of these equations gives 
Q/C Qa — ^^'5 
and also that 
from which Ave obtain 
O — O • 
Ql=—n 
dk 
d/d 
( 11 .) 
Now suppose k expressed as a function of x, y, z, A; then considering A implicitly a 
function of x, y, z, A, we have, by two differentiations with respect to a*. 
Q dk d/c dh 
dx ' dh dx 
( 12 .) 
ddk 
^ 7, , • 7 + 
ddk dh , d^k /dh\^ , dk d’^h 
dx^' “ dhdx dx'dJi^\i 
dJi\^ dk ddh 
dxj ' dh dx"^ 
dh 
Let the value of ^ derived from (12.) be introduced in the last equation, and the 
similar results be written with respect to y and z ; then we obtain by addition, 
0=D»A-2(|)-'2(^.|)+Q|§+|ot (13.) 
(using 2 to denote the sum of analogous expressions with respect to the three variables). 
Now in the first of equations (11.), namely. 
d/i\^ 
dx) 
(dk 
\dy 
+ 
/ dk\ ^ dk 
ycfe j d/i 
k is supposed to be expressed as a function of x^ y, z. A, and the equation is therefore 
identical^ for otherwise it would establish a relation betAveen x, _y, z. A, Avithout k. 
Hence we may differentiate each side with respect to A ; this gNes 
, (dk dVi \ 
d'^k 
