of Edinburgh, Session 1869 - 70 . 
169 
where v is Hamilton’s operator, 
.d_ 
dx 
• cl 7 & 
+ % + h dz 
5 
is any vector-function of the position of a point, <7$ an element 
of volume, els 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 Green's Theorem is deduced in the form 
fff S.vPyPpZs = -fffPyVd, +ffVf.vVVvds, 
= — fff Pv 2 P;<G + ff PS • vFjHc ds. 
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 
f//rck=/fdsS(V^-i)r, 
where r is any vector, is explained. 
It is next shown that, if p be the vector of a point, cn and v as 
before, we have the equation 
f S <7-dp = Jf&.X?cr{jv.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 
ff S-v trUi/.fb = fffSSj'T-d^ = 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 
fVa-clp — — ff ds V.(V.UvV) <r') 
and 
/‘P dp — ff ds V.UVvP, 
which, in fact, contains the two preceding. 
