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Full text of "Scientific Papers - Vi"

390                                ON  THE  ELECTRICAL CAPACITY OF                             [403
Thus (36) becomes
p,'(fc ...... (40)
When p is odd, the integral vanishes, and we fall back upon the former result ; when p is even, by (37), (38),
For example, if _p = 2,
Again, if two terms with coefficients Cp, Gq occur in Bu, we have to deal only with Jp, Jq. The integrals to be evaluated are limited to
q* dp.
If p be odd, the first and third of these vanish, and if q be odd the second and fourth. If p and q are both odd, the terms of the third order in G disappear altogether.
As appears at once from (34), (36), the last statement may be generalized. However numerous the components may be, if only odd suffixes occur, the terms of the third order disappear and (36) reduces to (26).
[1917.    Gonf. Gisotti, K 1st. Lombardo Rend. Vol. XLIX. May, 1916.
In his Kelvin lecture (Journ. Inst. El. Eng. Vol. xxxv. Dec. 1916), Dr A. Russell quotes K. Aichi as pointing out that the capacity of an ellipsoidal conductor is given very approximately by (S/4nrY, where S is the surface of the ellipsoid, and he further shows that this expression gives approximate values for the capacity in a variety of other calculable cases. As applied to an ellipsoid of revolution, his equation (6) gives
where e is the eccentricity of the generating ellipse, the plus sign relating to the prolatum and the minus to the oblatum. It may thus be of interest to obtain the formula by which u0 in (28) is expressed in terms of S rather than, as in (29), (30), by the volume of the conductor. For a reason which will presently appear it is desirable to include the cube of the particular coefficient (72. multiplied by On, so that only Jp appears, and