ON THE MOTION OF PENDULUMS.
57
If now we put V for dg/dt, the velocity of the sphere, we get from (39), Ge^mH=V. After substituting in (125), the equation will remain unchanged when we pass from the symbolical to the real values of ^ and V, and thus (125) will be reduced to (124).
41. It appears then that by supposing the rate of alteration of the velocity of the sphere to decrease indefinitely, we obtain from the solution of the problem of Section II. the same result as was obtained in Art. 39, by treating the motion as steady. As yet, however, the method of Art. 40 is subject to a limitation from which that of Art. 39 is free. In obtaining equation (40), it was supposed that the maximum excursion of the sphere was small in comparison with its radius. Eetaining this restriction while we suppose T to become very large, we are obliged to suppose c to become very small, so that the velocity of the sphere is not merely so small that we may neglect terms depending upon its square, a restriction to which Art. 39 is also subject, but so extremely small that the space passed over by the sphere in even a long time is small in comparison with its radius.
We have seen, however, that on supposing r very large in (40) we obtain a result identical with (124), not merely a result with which (124) becomes identical when the restriction above mentioned is introduced. This leads to the supposition that the solution expressed by (40) is in fact more general than would appear from the way in which it was obtained. That such is really the case may be shewn by a slight modification of the analysis. Instead of referring the fluid to axes fixed in space, refer it to axes originating at the centre of the sphere, arid moveable with it. In the general equations of motion, the terms which contain differential coefficients taken with respect to the coordinates will remain unchanged, inasmuch as they represent the very same limiting ratios as before : it is only those in which differentiation with respect to t occurs that will be altered. If d'/dt be the symbol of differentiation with respect to t when the fluid is referred to the moveable axes, we shall have
A=^L _^| ^ • dt dt dt dx'
but the terms arising from dgjdt . d/dx are of the order of the square of the velocity, and are therefore to be neglected. Hence the