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

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

A Treatise on Electricity and Magnetism Volume 1 Fig3.PNG
Fig. 3

Let us now consider the meaning of the equation

This implies that is a scalar, or that the vector is the slope of some scalar function . These applications of the operator 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 .

26.] One of the most remarkable properties of the operator is that when repeated it becomes

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 as centre, we draw a small sphere whose radius is , then if is the value of at the centre, and the mean value of for all points within the sphere,

;

so that the value at the centre exceeds or falls short of the mean value according as is positive or negative.

I propose therefore to call the concentration of at the point , because it indicates the excess of the value of at that point over its mean value in the neighbourhood of the point.

If 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.