319 
of Edinburgh, Session 1870 - 71 . 
From these equations many very singular results may be de¬ 
rived, some of which form the first part of the subject of the pre¬ 
sent communication. 
Let trbe a vector which, having continuously varying values 
over the surface in question, becomes U dp at its edge. Then 
— fTdp = ffds S .UvVcrq 
there being no vector part on the left-hand side. This gives the 
length of any closed curve in terms of an integral taken over any 
surface bounded by it. 
We have evidently 
T p dTp = — S pdp , 
whence 
fTdTp = - /PS . TJpdp = - JfdsB . UvV(PUp) . 
Hence 
fcrdTp = — ffds S . (UplHV) <r , 
for 
vu >=-■!. 
Ip 
Now if Tp be constant over the boundary, i.e., if the bounding 
curve lie on a sphere whose centre is the origin, we have for any 
surface bounded by it 
ffds S . (UpUrV)cr 0, 
whatever be the value of the vector <r. 
Again, if cr be a function of Tp only, we have 
/cr dTp = 0 
for all closed curves. Hence, whatever be the vector-function <p. 
and whatever the surface and its bounding curve, we have always 
JJ ds S . (UpUvV) o (Tp) = 0 . 
Another very simple but fundamental theorem, in addition to 
those given in the paper above referred to, may be stated as fol¬ 
lows :—Let P be the potential of masses external to a space 
Then throughout 2 we have 
V 2 P — 0 , 
so that 
//V 2 Pd? = j/SUvVP . ds = 0 . 
VOL VII. 2 T 
