Page:Thomson1881.djvu/15

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The kinetic energy

.

Now

,

with similar expressions for G and H.

,

with similar expressions for and . Since the particles are supposed to be very small, we shall neglect those terms in F which depend on and a'².

The part of the kinetic energy we are concerned with involves the product ee': let us first calculate that part of it arising from the product of that part of F due to e with that part of due to e'. We shall take the line joining the particle as the axis of x; and for brevity we shall denote by σ.

The coefficient of uu' in the part of the kinetic energy we are considering

:.

Now, for values of r > R,

;

.

Now, since

,

where Qn is a zonal harmonic of the nth order; and since the product of two harmonics of different degrees integrated over