105 



direction of the motion, a the radius of the sphere, r the time of vi- 

 bration, M' the mass of the fluid displaced, F the resultant force of 

 the fluid on the sphere, so that — F is the resistance ; then 



where 



-F=kM' d -^+k'*M' d A (2.) 



dt* * dt 



2 + 2\2iray 2 \2ta?J 2ffa 2 ' 



The effect of a fluid on the time of vibration depends on the term 

 which involves k ; the effect on the arc of vibration depends on the 

 term which involves k'. 



The expression for F has precisely the same form (2.) in the case 

 of a cylinder, but k and k' are certain transcendental functions of 

 (ju/ t)i a- 1 (a here denoting the radius of the cylinder), which the 

 author has tabulated. 



The value of [xJ having been determined for air, or any given fluid, 

 by one experiment giving the effect of the fluid either on the time of 

 vibration, or on the arc of vibration, of any one pendulum consisting 

 either of a sphere suspended by a fine wire, or of a long cylindrical 

 rod, or of a combination of a sphere and a rod, the formulae which 

 follow from (2.) ought to make known the effect of the fluid both on 

 the time and on the arc of vibration of all pendulums of the above 

 forms. The agreement of theory with the experiments of Baily re- 

 lating to the effect of the air on the time of vibration of pendulums is 

 remarkably close. Even the rate of diminution of the arc of vibration, 

 the observation of which held quite a subordinate place in Baily's 

 experiments, agreed with the rate calculated from theory as closely 

 as could reasonably have been expected. 



The value of the index of friction of water was deduced by the au- 

 thor from some experiments of Coulomb's on the decrement of the 

 arc of oscillation of discs which performed extremely slow oscilla- 

 tions in their own plane by the force of torsion. When this value 

 was substituted in the expression for the time of vibration of a 

 sphere, the result was found to agree almost exactly with Bessel's 

 experiments on the time of vibration of a sphere swung in water. 



As a limiting case of the problem of a ball pendulum, the author 

 has deduced the resistance of a fluid to a sphere moving uniformly 

 under such circumstances that the square of the velocity may be neg- 

 lected. The resistance thus determined proves to be proportional, 

 not to the surface, but to the radius of the sphere ; and therefore 

 the quotient of the resistance divided by the mass increases very ra- 

 pidly as the radius decreases. Accordingly, the terminal velocity of 

 a minute globule of water descending through the air depends almost 

 wholly on the internal friction of air. Since the index of friction is 

 known from Baily's pendulum experiments, the terminal velocity can 

 be calculated numerically for a globule of given diameter. The ve- 

 locity thus calculated proves to be so small, in the case of globules 



