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Full text of "Scientific Papers - Vi"

1916]                         APPROXIMATE  SPHERES  AND  CYLINDERS                       391
In terms of u, equal to 1/r, the general formula for S is
1
o        2
By (22)       -  ^   = sin2 6 (G1Pl> + O,P2' + .. .)2 (1 - 2O>P2),
11   \CL\s J
and hence with regard to well-known properties of'Legendre's functions we find
By (41)                                 (P,3 0^ = 4/35,
and by use of the particular form of P2 we readily find
f+:d/*(l-/AB)PsPa'a = 12/35.
.'i
Accordingly
N                                 + 1 + 6r^   ar4
ir * "* r '
If we omit Cg8 and combine (45) with (28), we get
the terms in d and (72 disappearing.    When the cubes of the C"s are neg-                               1
lected, the capacity is less than \f(S/4nr)t the radius of the sphere of equal
surface.    If the surface be symmetrical with respect to the equatorial plane,
as in the case of ellipsoids, the G'a of odd order do not occur, so that the                               .t f
-earliest in (46) is <74.
For a prolatum of minor axis 2& and eccentricity e,
u* = 6-2 (l _ ^v),
whence                        u = u9 (1 - Je?2P2 + terms in e4),
so that                         Ca ==  J e2, (74 is of order e4, &c.
In like manner for an oblatum
(72 = +  02} ^ js Of order e4, &c.
In both cases the corrections according to (46) would be of order es, but we obtain a term in e6 when we retain (728.o 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