OF FLUIDS ON THE MOTION OF PENDULUMS. [35] 



which may be treated, if required, as the equation (6l) was treated in the preceding article. 

 If, however, we suppose m large, and are content to retain only the most important term in 

 (64), we get simply 



Afr = — ~± -, Ak' = 0, (65) 



2 (b 3 - a 3 ) 



so that the correction for the envelope may be calculated as if the fluid were destitute of 

 friction. 



Section III. 



Solution of the equations in the case of an infinite cylinder oscillating in an unlimited 

 mass of fluid, in a direction perpendicular to its axis. 



24. Suppose a long cylindrical rod suspended at a point in its axis, and made to oscillate 

 as a pendulum in an unlimited mass of fluid. The resistance experienced by any element 

 of the cylinder comprised between two parallel planes drawn perpendicular to the axis will 

 manifestly be very nearly the same as if the element belonged to an infinite cylinder oscillating 

 with the same linear velocity. For an element situated very near either extremity of the rod, 

 the resistance thus determined would, no doubt, be sensibly erroneous ; but as the diameter of 

 the rod is supposed to be but small in comparison with its length, it will be easily seen that the 

 error thus introduced must be extremely small. 



Imagine then an infinite cylinder to oscillate in a fluid, in a direction perpendicular to its 

 axis, so that the motion takes place in two dimensions, and let it be required to determine the 

 motion of the fluid. The mode of solution of this problem will require no explanation, being 

 identical in principle with that which has been already adopted in the case of a sphere. In 

 the present instance the problem will be found somewhat easier, up to the formation of the 

 equations analogous to (33) and (34), after which it will become much more difficult. 



25. Let a plane drawn perpendicular to the axis of the cylinder be taken for the plane of 

 xy, the origin being situated in the mean position of the axis of the cylinder, and the axis of 

 x being measured in the direction of the cylinder's motion. The general equations (2), (3) 

 become in this case 



dp r<Pu d°u\ du 



d~x = lL [dx 1 + lhf) " P ~dt' 



) (66) 



dp (d?v d?v\ dv 



d^ = M [dx 1 + df) " P dt' 



du dv 



dx dy v ' 



By virtue of (67), udy-vdx is an exact differential. Let then 



, udy -vdx = d% (68) 



29—2 



