of Edinburgh, Session 1869-70. 
169 
where v is Hamilton’s operator, 
. d .d l7 d 
l dx + % + k dz 
o- is any vector-function of the position of a point, d<s an element 
of volume, ds an element of surface, v the normal at ds ; and the 
integrals are extended respectively through the content, and over 
the bounding surface, of a closed space % 
From this equation G-reen’s Theorem is deduced in the form 
fff S.vPvP^v = -ff/PyVd, + j^S.vPlWs, 
= -fffVv'Ufs + J ^PS.vP 1 Uv«fo. 
Some sections are devoted to the representation of 
///& 
(where q is any quaternion whatever) by a surface-integral, and 
the arbitrary part of the solution in the equation 
ff/rds=//ds S(U,V-1)t, 
where r is any vector, is explained. 
It is next shown that, if p be the vector of a point, a- and y as 
before, we have the equation 
f8 <T"dp = ff8.Vo-Uv.ds, 
expressing an integral taken over a limited and non-closed surface 
by another taken round its curvilinear boundary. That some such 
representation is possible is obvious from the fundamental theorem 
above, which shows that for a closed surface 
^S-v cr-Ui/.c/s = fff Sv 2 <t" ds = 0, 
and therefore the surface -integral must have the same value (with 
a mere change of sign depending on the difference between outside 
and inside ) for the two parts into which the surface is divided by 
any closed curve drawn upon it. 
Other theorems of a similar character are given, such as 
fVcndp = - ffds V.(V.UvV) 
and 
fVdp = ffds V.UvvP, 
which, in fact, contains the two preceding. 
