ON THE MOTION  OF  PENDULUMS.                        137
may be written /z/3 + «r, and tzr must be positive, as otherwise the mere alternate expansion and contraction, alike in all directions, of a fluid, instead of demanding the exertion of work upon it, would cause it to give out work. But if the positive constant exists, the coefficient of the squared velocity of dilatation in the transformed expression for LV in p. 69, instead of being — f//,, will be — fyw, + w, and in order that the quantity under the sign of triple integration may vanish, we must have in addition to the equations (137) on p. 70 the further equation S = 0, and the conclusion is the same as in the case of an incompressible fluid.]
additional note.   (See foot-note cut p. 77.)
[The fact that notwithstanding the great variety in the forms of Baily's pendulums, even when restricted to those to which the theory of the present paper is applicable, the results of his experiments manifest such a remarkable agreement with theory, in spite of the adoption in the calculation of a law as to the relation of p to p which we now know to be wrong, paradoxical as that fact appears at first sight, admits of being easily explained by combining two considerations, relating the one to the way in which the experiments were conducted, the other to the character of the formulae.
In the case of the 41 pendulums mentioned in Baily's paper as originally presented to the Royal Society, the high pressure under which each pendulum was swung was always the atmospheric pressure, and the low pressure did not much differ from that of 1 inch of mercury. There can be little doubt that the same practice was followed as regards the 45 additional pendulums for which the reduced results only, not the details, are given in the appendix. Hence the two pressures used would be nearly the same throughout, and nearly those measured by 30 inches and 1 inch of mercury. The ratio of the densities would be very nearly the same.
The expression (52) shows that for a sphere k is of the form
k = A + BV/   ........................(a),
where A is an abstract number, and R depends on the diameter and time of vibration of the sphere, but is constant when only the