618 
I- IfF^C (1) 
be the equation to one of a system of surfaces, and if the differential 
of (1) be 
S. vdg = 0 (2), 
v is a vector 'perpendicular to the surface , audits length is inversely 
proportional to the normal distance between tivo consecutive sur- 
faces. In fact (2) shows that v is perpendicular to dg, which is 
any tangent vector, thus proving the first assertion. Also, since 
in passing to a proximate surface we may write 
S . vdg = dC, we see that 
F(g+ v ~ 1 dC)=C + dC. 
This proves the latter assertion. 
It is evident from the above that if (1) be an equipotential, or an 
isothermal, surface,— v represents in direction and magnitude the 
force at any point or the flux of heat. And we see at once that if 
. d ,d 7 d 
4 ~ l Tx +} d~y + k dz ■ ■ 
. . „ d 2 d 2 d 2 
grnng <2= __ 5 ____ 2 
then v = <1 Fg. 
( 3 ), 
( 3 )\ 
( 4 >- 
This is due to Sir W. R. Hamilton ( Lectures on Quaternions , 
p. 611). 
From this it follows that the effect of the vector operation <l upon 
any scalar function of the vector of a point is to produce the vector 
which represents in magnitude and direction the most rapid change 
in the value of the function. 
Let us next consider the effect of <i upon a vector as 
cn- + j n + •££ (5). 
We have at once 
%■ 4 ?) -•(£-$-*•■ •<«>■ 
and in this semi-Cartesian form it is easy to see that — 
If represent a small vector displacement of a point situated at 
the extremity of the vector g {drawn from the origin ) 
