320 
Proceedings of the Royal Society 
The double integral is therefore of constant value for all noil-closed 
surfaces having, as common boundary, a closed curve and not 
extending into space occupied by any part of the masses. To find 
its value in terms of a single integral taken round this curve, let 
V 2 r = VP . 
As P is known, the constituents of r are perfectly definite, being 
the potentials of given distributions of matter. And the substitution 
of functions of r for those of P gives us, by means of the general 
formula at the beginning of this paper, 
j^SUi/VP . ds = S/V (dpV) r , 
with the condition 
SVr - 0 . 
Again, we have obviously, as V 2 o- is necessarily a vector, 
JfS . TJvX /2 o~'ds = /S . Y<r~dp. 
Now, let cr = ^P, then 
JfS . iJJv . V 2 P ds = /S(idpV)P . 
From this 
jfUvV^ds =/V(d P V ) P . 
A particular case of this, for a curve in the plane of xy and the 
surface bounded by it, is 
jr<$* $)**-/£* -s*) 
which has obvious applications to fluid motion parallel to a plane. 
But, generally, we have also 
JfUvV 2 a-'ds = fY(dp V) . <r. 
If we take the vector of this, or if we subtract from each side the 
corresponding member of our first equation above, we have 
Jfy.TJvY 2 (T'ds = fY .(Y.dp^)a~. 
These results appear to be of considerable importance for physical 
applications, and are particularly interesting, because they involve 
the operator (indicated merely in my former paper). 
V(dpV) . 
The paper contains several applications and modifications of these 
theorems. 
