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 <rbe a vector which, having continuously varying values 
over the surface in question, becomes U dp at its edge. Then 
-fTdp = Jf ds S .UvVo-'j 
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 
Tp dTp = — S pdp , 
whence 
fPdTp = - /PS . Vpdp = - ffdsS . UvV(PUp) . 
Hence 
f cndTp = - ffds S . (UpUvV) <r , 
for 
Now if Tp be constant over the boundary, *.e., if the bounding 
curve lie on a sphere whose centre is the origin, we have for any 
surface bounded by it 
ffds S . (UpUvV)o- = 0 , 
whatever be the value of the vector <r . 
Again, if cn 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 
ffds 8 . (UpTJj/V) <p (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 5. 
Then throughout 2 we have 
V 2 P = 0 , 
so that 
//V Ws = ff SUvVP . ds = 0 . 
2 T 
VOL VII. 
