BY SUEFACES OF A HYPOTHETICAL CLASS INCLUDING THE ELLIPSOID. 
• ) 
fessor W. Thomson in an early volume of the Cambridge Mathematical Joui-nal (to 
which I have not at present the opportunity of referring) in some form equivalent to (if 
not identical with) that in which they are here given. They depend on the rru^st 
elementary principles, and ought to be so well known as to make a demonstration necfl- 
less ; however, I give one for the sake of completeness. Retaining the suppositions of 
the last article, we have, if P be any function of x, y, z, 
Gtg dx 
0 
(by integrating the expression on the left with respect to x, and applying a well-known 
transformation to the double integral). 
Let then u be any function of x, y, z, and put 
dfl du 
then 
d^ ^du d& ^ du 
n—u^ — e-r-. r—ii-1 
^ dy dy' dz dz 
and if the two members of the last equation be multiplied by dxdydz, and integrated 
through the space within the surface 0^, the result on the left is 
fa. 
- , d^ d& \ 
We[f*Tx'^^'d^^'^Tz J 
or, \ijp, q, T be replaced by their values, it is 
f, 
/du d^ du dd du d^ 
Q,0\dx dx'dy dy'dz dz 
da 
~\0 
but in the last of these integrals 6 has the same value 0^ throughout the integration, and 
may therefore be put outside the integral sign ; and the integral which it multiplies is 
then evidently equivalent to the triple integral \_^^^D‘^u.dxdydzy', since it would be 
obtained by one integration of each of the three terms of the latter. Hence we have, 
finally. 
JJJ D^.dxdydzf' 0'D\)dxdydzy\ (2.) 
and, subtracting fhis from the similar equation referring to another value 0^ of 0, 
\^'ji^uQsday\=i0y^'^^jyu.dxdydzy'‘—0y_^^^'D'^u. dxdydzy^-\-\_^^\i{uT>'^0—0T)hi)dxdydzyf^^ . (3.) 
which last might have been obtained at once by taking the triple integrals through the 
space included between the surfaces 0^, 0^. It is of course necessary for the validity of 
each equation, that the functions under the integral signs should not become infinite at 
any point within the limits of the integrations. 
3. Let the equation 
f(A’,y,^,/q/I)=0 (4.) 
represent closed surfaces for all values, within certain limits, of the parameters /q Ti. 
Let the surface corresponding to a particular pair of values /q be called “ the surface 
