Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/193

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527.]

FOUR ASSUMPTIONS.

161

with all our magnetic formulae. As it is difficult for the student to bear in mind whether he is to multiply or to divide by ${\sqrt {2}}$ , we shall henceforth use only the electromagnetic system, as adopted by Weber and most other writers.

Since the form and value of $Q$ have no effect on any of the experiments hitherto made, in which the active current at least is always a closed one, we may, if we please, adopt any value of $Q$ which appears to us to simplify the formulae.

Thus Ampère assumes that the force between two elements is in the line joining them. This gives $Q=0$ ,

R={\frac {1}{r^{2}}}\left({\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}-24{\frac {d^{2}r}{ds\,ds^{\prime }}}\right)\,ii\,ds\,ds^{\prime }

, ⁠

S=0

,⁠

S^{\prime }=0

.

(42)

Grassmann^[1] assumes that two elements in the same straight line have no mutual action. This gives

Q=-{\frac {1}{2r}}

,⁠

R=-{\frac {3}{2r}}{\frac {d^{2}r}{ds\,ds^{\prime }}}

,⁠

S=-{\frac {1}{2r^{2}}}{\frac {dr}{ds^{\prime }}}

, ⁠

S^{\prime }={\frac {1}{2r^{2}}}{\frac {dr}{ds}}

.

(43)

We might, if we pleased, assume that the attraction between two elements at a given distance is proportional to the cosine of the angle between them. In this case

Q=-{\frac {1}{r}}

, ⁠

R={\frac {1}{r^{2}}}\cos \epsilon

, ⁠

S=-{\frac {1}{r^{2}}}{\frac {dr}{ds^{\prime }}}

, ⁠

S^{\prime }={\frac {1}{r^{2}}}{\frac {dr}{ds}}

.

(44)

Finally, we might assume that the attraction and the oblique forces depend only on the angles which the elements make with the line joining them, and then we should have

Q=-{\frac {2}{r}}

, ⁠

R=-3{\frac {1}{r^{2}}}{\frac {dr}{ds}}{\frac {dr}{ds^{\prime }}}

, ⁠

S=-{\frac {2}{r^{2}}}{\frac {dr}{ds^{\prime }}}

, ⁠

S^{\prime }={\frac {2}{r^{2}}}{\frac {dr}{ds}}

.

(45)

527.] Of these four different assumptions that of Ampère is undoubtedly the best, since it is the only one which makes the forces on the two elements not only equal and opposite but in the straight line which joins them.

↑ Pogg., Ann. lxiv. p. 1 (1845).

VOL. II.

M

[1] Pogg., Ann. lxiv. p. 1 (1845).

[1]