Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/152

the whole system which is due to the action of the first system on the second.

If we define V as $\sum \left({\tfrac {e}{r}}\right)$, where r is the distance of the quantity e of electricity from the given point, then the equality between these two values of M may be obtained as follows, without Green Theorem—

$\sum {'}(Ve')=\sum {'}\left(\sum \left({\frac {e}{r}}\right)e'\right)=\sum \sum \left({\frac {ee'}{r}}\right)=\sum \left(\sum {'}\left({\frac {e'}{r}}\right)e\right)=\sum (V'e)$.

This mode of regarding the question belongs to what we have called the direct method, in which we begin by considering certain portions of electricity, placed at certain points of space, and acting on one another in a way depending on the distances between these points, no account being taken of any intervening medium, or of any action supposed to take place in the intervening space.

Green's Theorem, on the other hand, belongs essentially to what we have called the inverse method. The potential is not supposed to arise from the electrification by a process of summation, but the electrification is supposed to be deduced from a perfectly arbitrary function called the potential by a process of differentiation.

In the direct method, the equation is a simple extension of the law that when any force acts directly between two bodies, action and reaction are equal and opposite.

In the inverse method the two quantities are not proved directly to be equal, but each is proved equal to a third quantity, a triple integral which we must endeavour to interpret.

If we write R for the resultant electromotive force due to the potential V, and l, m, n for the direction-cosines of R, then, by Art. 71,

$-{\frac {dV}{dx}}=Rl,\quad -{\frac {dV}{dy}}=Rm,\quad -{\frac {dV}{dz}}=Rn.$

If we also write R for the force due to the second system, and l, m, n for its direction-cosines,

$-{\frac {dV'}{dx}}=R'l',\quad -{\frac {dV'}{dy}}=R'm',\quad -{\frac {dV'}{dz}}=R'n';$

and the quantity M may be written

$M={\frac {1}{4\pi }}\iiint {(KRR'\cos \epsilon )\,dx\,dy\,dz},$

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