OF FLUIDS ON THE MOTION OF PENDULUMS. [57] 



Now the general character of the motion must be nearly the same whether the velocity 

 of the cylinder be constant, or vary slowly with the time, so that it does not vary materially 

 when the cylinder passes through a space equal to a small multiple of its radius. To 

 return to the problem considered in Section III., it would seem that when the radius of the 

 cylinder is very small, the motion which would be expressed by the formulas of that Section 

 would be unstable. This might very well be the case with the fine wires used in supporting 

 the spheres employed in pendulum experiments. If so, the quantity of fluid carried by the 

 wire would be diminished, portions being continually left behind and forming eddies. The 

 resistance to the wire would on the whole be increased, and would moreover approximate to a 

 resistance which would be a function of the velocity. Hence, so far as depends on the wire, the 

 arc of oscillation would be more affected by the resistance of the air than would follow from 

 the formulae of Section III. Whether the effect on the time of oscillation would be greater 

 or less than that expressed by the formula? is difficult to say, because the increase of 

 resistance would tend to increase the effect on the time of vibration, while on the other hand 

 the approximation of the law of resistance to that of a function of the velocity would tend to 

 diminish it. 



Section V. 



On the effect of internal friction in causing the motion of a Jiuid to subside. Applica- 

 tion to the case of oscillatory waves. 



49. We have already had instances of the effect of friction in causing a gradual subsi- 

 dence in the motion of a solid oscillating in a fluid ; but a result may easily be obtained 

 from the equations of motion in their most general shape, which shews very clearly the 

 effect of friction in continually consuming a portion of the work of the forces acting on the 

 fluid. 



Let Pj, P 2 , P 3 be the three normal, and T lt T 2 , T 3 the three tangential pressures 

 in the direction of three rectangular planes parallel to the co-ordinate planes, and let D be 

 the symbol of differentiation with respect to t when the particle and not the point of space 

 remains the same. Then the general equations applicable to a heterogeneous fluid, (the equa- 

 tions (10) of my former paper,) are 



(Du \ dP, dT„ dT 2 



with the two other equations which may be written down from symmetry. The pressures 

 Pi, &c. are given by the equations 



_ [du .\ tdv dw\ 



**-'-*$:*)• ri =-^U + ^)' • • • < is3 > 



and four other similar equations. In these equations 



. du dv dw 



3d = — + — + -p- (134) 



dx dy d% 



Vol. IX. Part II. 32 



