786 Proceedings of the Royal Society 
With these preliminaries we see at once that (d) may be written 
f/YfY.crVvfds = -Jjpr.TV.Vcrds -JjfY.^rds +#SaV.«fe. 
Now, if r — V^^, where r is the distance between any external 
point and the element d<s , the last term on the right is the vector- 
force exerted by the magnet on a unit pole placed at the point. 
The second term on the right vanishes by Laplace’s equation, and 
the first vanishes as above if the distribution of magnetism be 
lamellar, thus giving Thomson’s result in the form of a surface 
integral. 
Another of the applications made is to Ampere’s Directrice de 
V action electrodynamique , which ( Quarterly Math. Journal , Jan. 
1860) is the vector-integral 
f Ypdp 
J iy ’ 
where dp is an element of a closed circuit, and the integration 
extends round the circuit. This leads again to the consideration 
of relations between single and double integrals. 
[Here it may be well to note that, by inadvertence, I wrote cr 
for r towards the end of the abstract of the former part of this 
paper, thus giving the result a false generalisation depending on 
the fact that r had been made subject to the condition 
S . Vr = 0 , 
while no such restriction was imposed on a~. With this restriction 
most of the results already given ( Proc . ante p. 320) are correct, 
but the general forms in the paper itself are as follows, being 
deducible at once from the first expression in the abstract : — 
jfS . U vV*<rds -jfS . Ui/VS . Ycrds = /S . Ycrdp , 
and 
JfUvWds -ffS . VvY . VP ds =/Y {d P V) P ; 
giving finally 
f/Y . U vWtfc - . UvV . YVcrds - /V . Y(dpV)<r .] 
