502 
Proceedings of the Royal Society 
these 1 now give hut one, which, however, with that in the succeed¬ 
ing Note, is quite sufficient to show their general nature. 
Let P he any scalar function of p. It is required to find the 
difference between the value of P at p, and its mean value through¬ 
out a very small sphere, of radius r and volume v , whieh has the 
extremity of p as centre. 
From what is said above, it is easy to see that we have the 
following expression for the required result:— 
where <nis the vector joining the centre of the sphere with the 
element of volume d<s, and the integration (which relates to < 7 - and 
ds alone) extends through the whole volume of the sphere. Ex¬ 
panding the exponential, we may write this expression in the 
form 
-1 fff + ■ ■ 
- - l s -™fff° * + - &c •’ 
higher terms being omitted on account of the smallness of r, the 
limit of T<t~ . 
Now, symmetry shows at once that 
If <rds = 0 . 
Also, whatever constant vector be denoted by a, 
fff(fa<r)hh= - A* Ua ) 2 ds ■ 
Since the integration extends throughout a sphere, it is obvious 
that the integral on the right is half of what we may call the 
moment of inertia of the volume about a diameter. Hence 
fff (S<rUo) a * = f ■ 
If we now write V for a, as the integration does not refer, to V, 
we have by the foregoing results (neglecting higher powers of r) 
S'T" V 
v 2 
10 
V 2 P 
) 
