Colloquium publications.

LECTURE IV

INTEGRO-DIFFERENTIAL EQUATIONS OF THE BOCHER TYPE*
~ 1. THE GENERALIZATION OF LAPLACE'S EQUATION
46. Hypothetical Experiments as a Basis of Physics. It is regarded as experimental knowledge that in a dielectric the field
of force due to electric charges, distributed arbitrarily, is conservative; and also that the total electric induction over a closed
surface in a dielectric is 47r multiplied into the charge of electricity
inside the surface. In what sense are these laws experimental?
Obviously no experiment has ever yet been performed in which
the total work was null; and no measurements upon total
induction could be taken with such accuracy as to fix the multiplicative factor absolutely as 47r, or even to show that it was
constant.
And yet these laws are not entirely the results of more general
laws of nature, nor are they implied wholly in definitions of such
things as " charge," " dielectric," etc. It was only after Coulomb had made his direct (and inaccurate) experiments that he
was able to state the proposition: that two point charges m
and m' repel each other with a force mm'/r2. At least in some
way therefore, although there are no such things as point charges,
Coulomb's law is an experimental law.
The characteristic terms of physics-point charges, electric
density, dielectrics, conductors,-are as ideal as the points, lines,
and numbers of pure mathematics. The basic laws of any branch
of physics are stated as hypothetical experiments carried out
upon these ideal elements; e. g., " if two point charges are at
* This lecture is based on the following references:
M. Bocher, On harmonic functions in two dimensions, Proceedings of the American Academy of Science, vol. 41 (1905-06).
G. C. Evans, On the reduction of integro-differential equations, Transactions
of the American Mathematical Society, vol. 15 (1914), pp. 477-496;
Sul calcolo della funzione di Green, Rendiconti delta R. Accademia dei
Lincei, vol. 22 (1913), first semester, pp. 855-860.
C. W. Oseen; cf. p. 136, note.

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