Bin. STOKES, ON SOME CASES OF FLUID MOTION. 117 



same radius, but whose lengths differ by several radii, might be made to oscillate in succession in a 

 fluid, at a depth sufficiently great to allow us to neglect the motion of the surface of the fluid. The 

 time of oscillation of each might then be calculated as if the cylinder oscillatetl in vacuum, acted on 

 by a moving force equal to its weight minus that of the fluid displaced, acting downwards through 

 its centre of gravity, and having its mass increased by an unknown mass collected in the axis. 

 Equating the time of oscillation so calculated to that given by observation, we should determine the 

 unknown mass. The difference of these masses would be very nearly equal to the mass which must 

 be added to that of a cylinder whose length is equal to the difference of the lengths of the first two, 

 when the motion is in two dimensions. This evidently comes to supposing that, at a distance from 

 the middle of the longer cylinder not greater than half the difference of the lengths of the two, the 

 motion may be taken as in two dimensions. The ends of the cylinders may be of any form, provided 

 that they are all of the same. They may be suspended by fine equal wires, in which case we should 

 have a compound pendulum, or attached to a rigid body oscillating above the fluid by means of 

 thin flat bars of metal, whose plane is in the plane of motion. Another way of getting rid of the 

 motion in three dimensions about the ends would be, to make those ends plane, and to fix two 

 rigid planes parallel to the plane of motion, which should be almost in contact with the ends of the 

 cylinder. 



9. Motion between two cotieentric spherical surfaces Motion of a ball pendulum enclosed 



in a spherical case. 



Let a mass of fluid be at rest, comprised between two concentric spherical surfaces. Let the 

 several points of these surfaces be moved in any manner consistent with the .condition that the 

 volume of the fluid be not changed : it is required to determine tlie initial motion at any point 

 of the mass. 



Let a, b, be the radii of the inner and outer spherical surfaces respectively ; then employino- the 

 co-ordinates r, 0, w, where r is the distance from the centre, 9 the angle which r makes with a fixed 

 line passing through the centre, w the angle which a plane passing through these two lines makes 

 with a fixed plane through the latter, the value of (p corresponding to any radius vector comprised 

 between a and b can be expanded in a converging series of Laplace's coefiicients. Let then 



«^= '^o+f, + V„+ , 



r„ being a Laplace's coefficient of the n"' order. 

 Substituting in the equation, 



dr' "^ ~^n0 Id V'" lie) '^ s.\u' 9 rf^ 



= 0, 



which cb is to satisfy, employing the equation 



J (i f ■ ^dVA 1 d' V„ 



7i{n + 1)F„ + — - — sm0--- + ^— -7~r = (9), 



sm0 d9 \ d9 J sin'9 dia- 



and then equating to zero the Laplace's coefficients of the several orders, we find 



d"rV„ 

 r — y- w (ra + ] ) r„ = 0. 



The general integral of this equation is 



c 



^'"-Cr'' + ~^^, 



where C and C' are functions of 9 and w. Substituting in the equation (!)), and equating coeffi- 

 cients of the two powers of r wliich enter into it separately to zero, we find that botli C and 

 C' satisfy it, and therefore are botli Laplace's coefficients of the w"' order. AVe have then 



0= 2''(r„r' + Z,r-"' + ") (10), 



