# Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/69

At Fig. 3 we have an illustration of curl combined with convergence.

 Fig. 3

Let us now consider the meaning of the equation

${\displaystyle V\nabla \sigma =0}$

This implies that ${\displaystyle \nabla \sigma }$ is a scalar, or that the vector ${\displaystyle \sigma }$ is the slope of some scalar function ${\displaystyle \Psi }$. These applications of the operator ${\displaystyle \nabla }$ are due to Professor Tait[1]. A more complete development of the theory is given in his paper 'On Green's and other allied Theorems'[2] to which I refer the reader for the purely Quaternion investigation of the properties of the operator ${\displaystyle \nabla }$.

26.] One of the most remarkable properties of the operator ${\displaystyle \nabla }$ is that when repeated it becomes

${\displaystyle \nabla ^{2}=-({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}})}$

an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator.

This operator is itself essentially scalar. When it acts on a scalar function the result is scalar, when it acts on a vector function the result is a vector.

If, with any point ${\displaystyle P}$ as centre, we draw a small sphere whose radius is ${\displaystyle r}$, then if ${\displaystyle q_{0}}$ is the value of ${\displaystyle q}$ at the centre, and ${\displaystyle {\bar {q}}}$ the mean value of ${\displaystyle q}$ for all points within the sphere,

${\displaystyle q_{0}-{\overline {q}}={\tfrac {1}{10}}r^{2}\nabla ^{2}q}$;

so that the value at the centre exceeds or falls short of the mean value according as ${\displaystyle \nabla ^{2}q}$ is positive or negative.

I propose therefore to call ${\displaystyle \nabla ^{2}q}$ the concentration of ${\displaystyle q}$ at the point ${\displaystyle P}$, because it indicates the excess of the value of ${\displaystyle q}$ at that point over its mean value in the neighbourhood of the point.

If ${\displaystyle q}$ is a scalar function, the method of finding its mean value is well known. If it is a vector function, we must find its mean value by the rules for integrating vector functions. The result of course is a vector.

1. Proceedings R. S. Edin., 1862
2. Trans. R. S. Edin., 1869-70.