338 Cambridge Philosophical Society. 



The second and third of the general equations are not written down, 

 because they mny be supplied by symmetry. In these equations p is 

 the density, p the mean of the normal pressures in the direction of any 

 three rectangular planes passing through the point of which x, y, z 

 are the coordinates ; u is the velocity in the direction of x, t the time, 

 and ju.' a certain constant, depending upon the nature of the fluid, 

 which the author proposes to call the index of friction. 



The author has succeeded in obtaining the solution of equations 

 (1.) in the two cases of a sphere and of an infinite cylinder. The 

 latter may be a])plied to the case of a pendulum consisting of a long 

 cylindrical rod, by treating each element of the rod as belonging to 

 an infinite cylinder oscillating with the same linear velocity. The 

 following is the solution in the case of a sphere, so far as relates to 

 the resultant action of the fluid on the sphere. 



Let be the abscissa of the centre of the sphere, measured in the 

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

 bration, i\l' the mass of the fluid displaced, F the resultant force of 

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



_F=/tM'^-!l+/l'*M'^, (-2.) 



dr^ r dt 



where 



A=L.^fi^y. /t-^fi^y+gi^. . . (3.) 



The eff'ect of a fluid on the time of vibration depends on the term 

 which involves k ; the eflfect 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 A' are certain transcendental functions of 

 (yj t)ia-^ (a here denoting the radius of the cylinder), which the 

 author has tabulated. 



The value of jw-' 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 late 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 



