Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/38

 $$(u_\alpha, v_\alpha, w_\alpha)$$, namely the electric and magnetic ‘forces’ and the velocity of the charge of electricity. Now a vector involves direction; and direction is not concerned with what is merely at that point. It is impossible to define direction without reference to the rest of space; namely, it involves some relation to the whole of space.

Again the equations involve the spatial differential operators $$\frac{\partial}{\partial x_\alpha}$$, $$\frac{\partial}{\partial y_\alpha}$$, $$\frac{\partial}{\partial z_\alpha}$$, which enter through the symbols $$curl_\alpha$$ and $$div_\alpha$$; and they also involve the temporal differential operator $$\frac{\partial}{\partial t_\alpha}$$. The differential coefficients thus produced essentially express properties in the neighbourhood of the point $$(x_\alpha, y_\alpha, z_\alpha)$$ and of the time $$t_\alpha$$, and not merely properties at $$(x_\alpha, y_\alpha, z_\alpha, t_\alpha)$$. For a differential coefficient is a limit, and the limit of a function at a given value of its argument expresses a property of the aggregate of the values of the function corresponding to the aggregate of the values of the argument in the neighbourhood of the given value.

This is essentially the same argument as that expressed above in 1.2 for the particular case of motion. Namely, we cannot express the facts of nature as an aggregate of individual facts at points and at instants.

6.2 In the Lorentz-Maxwell equations [cf. Appendix II] there is no reference to the motion of the ether. The velocity $$(u_\alpha, v_\alpha, w_\alpha)$$ which appears in them is the velocity of the electric charge. What then are the equations of motion of the ether? Before we puzzle over this question, a preliminary doubt arises. Does the ether move?

Certainly, if science is to be based on the data included in the Lorentz-Maxwell equations, even if the