ON THE MOTION OF PENDULUMS. 31
which agrees with the result deduced directly from the ordinary equations of hydrodynamics*.
18. Let us now form the expression for the resultant of the pressures of the fluid on the several elements of the surface of the sphere. Let Pr be the normal, and Te the tangential, component of the pressure at any point in the direction of a plane drawn perpendicular to the radius vector. The formulae (4), (5) are general, and therefore we may replace #, y in these formulae by Co, y'y where xy y' are measured in any two rectangular directions we please. Let the plane of x' y' pass through the axis of x and the radius vector, and let the axis of oc be inclined to that of x at an angle ^, which after differentiation is made equal to 0. Then Pv Ts will become Pr, Te, respectively. We have
u = R cos (6 - *) - © sin (0 - *) , v = R sin (6 - ty + @ cos (6 - $•),
and when 0 = $•
d d d d
_ _=__ _^
dx ~~ dr ' dy rdd r } ~d% ~~~ dr ' whence
dR d® ©
-j ---dr r
In these formulse, suppose r put equal to a after differentiation. Then Pr, 2^ will be the components in the direction of r, 6 of the pressure of the sphere on the fluid. The resolved part of these in the direction of x is
Prcos<9- 70 sin 0,
which is equal and opposite to the component, in the direction of x, of the pressure of the fluid on the sphere. Let Pbe the whole force of the fluid on the sphere, which will evidently act along the axis of x. Then, observing that 2-7T&2 sin 6d9 is the area of an elementary annulus of the surface of the sphere, we get
J
(- Pr cos 6 + TQ sin 0)a sin Odd ......... (47),
o
* See Camb. Phil. Trans. Vol. vm. p. 119. [Ante, Vol. i. p, 41.]