504 
Proceedings of the Pioyal Society 
where <p may be any scalar function, and x depends on the cosine 
of the inclination of p to the axis. Now 
q 1 
J eh!™ = 
and by operating by we have 
so that 
But (as the interpretation of the general results is a little trouble¬ 
some) I confine myself at present to the case of a spherical shell, 
the origin being the centre and the density unity, which, while 
much simpler, sufficiently illustrates the proposed mode of treating 
the subject. I hope to return to the question, and to develope at 
length the proposed mode of solution. 
We easily see that in the above simple case, a being any con¬ 
stant vector whatever, and a being the radius of the sphere, 
ffdse Saf, = 2va f / T “* = ?!^( £ “ T “-r aT “'). 
J da V / 
Now, it appears (though I cannot say that I am yet quite satisfied 
with the logic of any of the proofs that have occurred to me) 
that we are at liberty to treat A as a has just been treated. It is 
necessary, therefore, to find the effects of such operators as TA, 
, &c., which seem to be novel, upon a scalar function of To-. 
or T, as we may for the present call it. 
Now 
2T 
(TA) 2 F = - A 2 F = F" + -Tp- , 
whence it is easy to guess at a particular form of TA. 
that it is the only one, assume 
TA = 
a_ 
dT 
+ ? 
To be sure 
