[16] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 



them to the case of a pendulum they may be a good deal simplified without the neglect of any 

 quantities which it would be important to retain. In the first place the motion is supposed 

 very small, on which account it will be allowable to neglect the terms which involve the 

 square of the velocity. In the second place, the nature of the motion that we have got to 

 deal with is such that the compressibility of the fluid has very little influence on the result, so 

 that we may treat the fluid as incompressible, and consequently omit the last terms in the 

 equations. Lastly, the forces X, Y, Z are in the present case the components of the force of 

 gravity, and if we write 



p + II + pf(Xd.v + Ydy + Zd%) 



for p, we may omit the terms X, Y, Z. 



If x be measured vertically downwards from a horizontal plane drawn in the neighbourhood 

 of the pendulum, and if g be the force of gravity, f(Xdoe + Ydy + Zdss) = g%, the arbitrary 

 constant, or arbitrary function of the time if it should be found necessary to suppose it to be 

 such, being included in II. The part of the whole force acting on the pendulum which 

 depends on the terms II + gp% is simply a force equal to the weight of the fluid displaced, 

 and acting vertically upwards through the centre of gravity of the volume. 



When simplified in the manner just explained, the equations such as (l) become 



dp fcFu dru d*u\ du 



d~x = >l {dw* + df + d^i ~ P ~di' 



dp td*v d?v d?v\ dv 



dy = ,i Id** + df + ~d?) " P It ' 

 dp icPw d 2 w d'wA dw 



d~i = fl (dx* + df + dx*) ~ p dt\ 



which, with the equation of continuity, 



du dv dw 



j- + -7- + — - 0, (3) 



dx ay dz 



are the only equations which have to be satisfied at all points of the fluid, and at all instants 

 of time. 



In applying equations (2) to a particular pendulum experiment, we may suppose ft con- 

 stant ; but in order to compare experiments made in summer with experiments made in winter, 

 or experiments made under a high barometer with experiments made under a low, it will be 

 requisite to regard p as a quantity which may vary with the temperature and pressure of the 

 fluid. As far as the result of a single experiment*, which has been already mentioned, 

 performed with a single elastic fluid, namely air, justifies us in drawing such a general 

 conclusion, we may assert that for a given fluid at a given temperature p. varies as p. 



2. For the formation of the equations such as (1), I must refer to my former paper; 



(2) 



* The first of the experiments described in Col. Sabine's 

 paper, in which the gauge stood as high as 7 inches, leads to 

 the same conclusion ; but as the vacuum apparatus had not yet 



been made stanch it is perhaps hardly safe to trust this 

 experiment in a question of such delicacy. 



