16 HYDRODYNAMICAL NOTES
where n is integral and Rn a function of r only; and in deducin, may perform the differentiations with respect to 0 (as well as wil to r) under the sign of summation, since -^ = 0 at the limits. Thus
j
dr a?r2
sm
The right-hand member of (3) may also be expressed in a serie
of the form
2&> = SCW/TT . ^n~l sin mrd/a, ..................
where n is an odd integer; and thus for all values of n we have
(? 7? «27T2 7? 4,M
UsjLUin, id i' J-^n TI" /, / -i \.,i
I ---7~7----T ' ---7 --- TT
dr~ dr a2
The general solution of (7) is
Rn = Anrn"'a + Bnr~n"/a + _
the introduction of which into (4) gives ty.
In (8) An and En are arbitrary constants to be determined by conditions of the problem. For example, we might make En> and \IT, vanish when r r^ and when r =r2, so that the fixed boundary the fluid would consist of two radii vectores and two circular arc fluid extend to the origin, we must make Bn 0; and if the bo completed by the circular arc r = 1, we have An = 0 when n is even, n is odd
A = =0
n
Thus for the fluid enclosed in a circular sector of angle « and radius
rnir/a _ r2
mr ?i - 4a a
the summation extending to all odd integral values of n.
The above formula (10) relates to the motion of uniformly rote bounded by stationary radii vectores at d = 0, 6 a. We may su containing vessel to have been rotating for a long time and that (under the influence of a very small viscosity) has acquired this r that the whole revolves like a solid body. The motion expressed that which would ensue if the rotation of the vessel were suddenl; A related problem was solved a long time since by Stokes*, who c the irrotational motion of fluid in a revolving sector. The solution problem is derivable from (10) by mere addition to the latter of i/r0 for then ^ + ^r0 satisfies V2 (^r + ^r0) = 0 ; and this is perhaps th«
* Camb. Phil. Trans. Vol. vm. p. 533 (1847) ; Math, and Phys. Papers, Vol. i.rd/a, -s ........ . ....... , ........... (4)