ON THE MOTION OF PENDULUMS. * 45
gent, are at first rapidly divergent, and in calculating the numerical value of FB(r) in such a case it would be far more convenient to employ equation (88). The employment of this equation for the purpose would require the previous determination of the constant G'. It will be found however that in calculating the resultant pressure of the fluid on the cylinder, which it is the main object of the present investigation to determine, a knowledge of the value of (7 will not be required, and tha,t, even though the equation (88) be employed*.
Putting D' = 0 in (92), and eliminating G" and D" between the resulting equation and the two equations (90), we get
>..................(93J;
and we get from (83) and (84), observing that F2 (r) = Fa' (r)} and that .8 = 0,
a) — ac,--------h aFB" (a) = ac...............(94),
whence
<tc±A-*J?M ,9W
a?c-A~ F3'(a) ...........................(^}'
This equation will determine A, because if F3(a) be expressed by (87) the second member of (95) will only contain the ratio of G to D, which is given by (98), and if Fs (a) be expressed by (88) Q' will disappear, inasmuch as D' = 0.
31. Let us now form the expression for the resultant of the forces which the fluid exerts on the cylinder. Let F be the resultant of the pressures acting on a length dl of the cylinder, which will evidently be a force acting in the direction of the axis of x\ then we get in the same way as the expression (47) was obtained
F=adl (-Prcos6+Te8wO)ad0...............(96),
JO
and Pr, TQ are given in terms of R and © by the same formula (46) as before. When the right-hand members of these equations are expressed in terms of ^, there will be only one term in which
* [C' as subsequently determined will be given at the end of the paper.]