OK THE MOTION OF PENDULUMS.                       37
and, as before, JcM' is the imaginary mass which we must conceive to be collected at the centre of the sphere, in order to allow for the inertia of the fluid, and — k'M'n d%/dt the term in F on which depends the diminution in the arc of oscillation.
23.    If we suppose p! = 0, and therefore m = oo , we get from (61)
^{63)'
and, in this case, k is the same as with jSTsign changed, and &' = 0, which agrees with the result obtained directly from the ordinary equations of hydrodynamics*. If, on the other hand, we make b = oo , we arrive at the results already obtained in Art. 20. In both these cases it becomes rigorously exact to neglect in the expression for K—\ given by (59) all the terms which are not multiplied by e^v(6~^a).
If the effect of the envelope be but small, which will generally be the case, it will be convenient to calculate k and k' from the formulae (52), which apply to the case in which b = oo , and then add corrections A&, A&' due to the envelope. We get from (61)
at,    ./ ^K    ^v
which may be treated, if required, as the equation (61) was treated in the preceding article. If, however, we suppose m large, and are content to retain only the most important term in (64), we get simply
so that the correction for the envelope may be calculated as if the fluid were destitute of friction.
* See Camb. Phil. Trans. Vol. vm. p. 120.    [Ante, Vol. i. p. 41.]