320 
Proceedings of the Roycd Society 
The double integral is therefore of constant value for all non-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, 
jfT’SUvVP . ds = S/V (dpY) r , 
with the condition 
SVr = 0 . 
Again, we have obviously, as V 2 cr- is necessarily a vector, 
JffS . UvV 2 cncZs = /S . Vcndp. 
Now, let er = fP, then 
Jf$. iTJv . V 2 P ds = /S(«ZpV)P . 
Prom this 
f/JJvV^ds =/Y(dpV) P. 
A particular case of this, for a curve in the plane of xy and the 
surface bounded by it, is 
dy — — dx 
dy , 
which has obvious applications to fluid motion parallel to a plane. 
But, generally, we have also 
ffJJvV 2 cnds = /Y(dpV) . 
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.TJvV 2 <rd8 = fY;(Y. dpV)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. 
