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

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

Let us now consider the meaning of the equation

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

26.] One of the most remarkable properties of the operator $$\nabla$$ is that when repeated it becomes $$ \nabla^2 = - ( \frac{d^2}{dx^2} +\frac{d^2}{dy^2} +\frac{d^2}{dz^2} ) $$ 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 $$P$$ as centre, we draw a small sphere whose radius is $$r$$, then if $$q_0$$ is the value of $$q$$ at the centre, and $$\bar{q}$$ the mean value of $$q$$ for all points within the sphere,

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

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

If $$q$$ 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.