1888-89.] Prof. Tait on the Relation among Integrals. 257 
On the Belation among the Line, Surface, and Volume 
Integrals. By Professor Tait. 
(Read April 1, 1889.) 
The fundamental form of the Volume and Surface Integral is 
fff Vuds = ffXSvuds . 
Apply it to a space consisting of a very thin transverse slice of 
a cylinder. Let t he the thickness of the slice, A the area of one 
end, and a a unit-vector perpendicular to the plane of the end. 
The above equation gives at once 
Y(aV)u.tA = t /V.aTJvudl, 
where dl is the length of an element of the bounding curve of the 
section, and the only values of UV left are parallel to the plane of 
the section and normal to the bounding curve. If we now put p as 
the vector of a point in that curve, it is plain that 
V. aUv = JJdp , dl = Tdp , 
and the expression becomes (after division by t) 
Y(dY)uA = fudp. 
By juxtaposition of an infinite number of these infinitely small 
directed elements, a (now to be called Uv) being the normal vector 
of the area A (now to be called ds), we have at once 
ff V(UVV)wcfe = f udp , 
which is the fundamental form of the Surface and Line Integral. 
In fact, as the first of these expressions can be derived at once 
from the ordinary equation of “continuity,” so the second is merely 
the particular case corresponding to displacements confined to a 
given surface. 
VOL. xvi. 12/8/89 
n 
