OF FLUIDS ON THE MOTION OF PENDULUMS. [61] 



Equating the two expressions for the decrement of vis viva, putting for m its value 27rX _I , 

 where \ is the length of a wave, replacing fj. by fj!p, integrating, and supposing c to be the 

 initial value of c, we get 



16irV< 

 C = C €~ X s . 



It will presently appear that the value of -^/V for water is about - 0564, an inch and a 

 second being the units of space and time. Suppose first that X is two inches, and t ten seconds. 

 Then l6irV*k" 2 = 1-256 > and c : c :: 1 : 0-2848, so that the height of the waves, which varies 

 as c, is only about a quarter of what it was. Accordingly, the ripples excited on a small pool 

 by a puff of wind rapidly subside when the exciting cause ceases to act. 



Now suppose that \ is 40 fathoms or 2880 inches, and that t is 86400 seconds or a whole 

 day. In this case l67r 2 M '£X~ 2 is equal to only - 005232, so that by the end of an entire day, in 

 which time waves of this length would travel 574 English miles, the height would be diminished 

 by little more than the one two hundredth part in consequence of friction. Accordingly, the 

 lono- swells of the ocean are but little allayed by friction, and at last break on some shore 

 situated at the distance of perhaps hundreds of miles from the region where they were first 

 'excited. 



52. It is worthy of remark, that in the case of a homogeneous incompressible fluid, 

 whenever udx + vdy + wdss is an exact differential, not only are the ordinary equations of 

 fluid motion satisfied*, but the equations obtained when friction is taken into account are 

 satisfied likewise. It is only the equations of condition which belong to the boundaries of the 

 fluid that are violated. Hence any kind of motion which is possible according to the ordinary 

 equations, and which is such that udx + vdy + wdx is an exact differential, is possible 

 likewise when friction is taken into account, provided we suppose a certain system of normal 

 and tangential pressures to act at the boundaries of the fluid, so as to satisfy the equations of 

 condition. The requisite system of pressures is given by the system of equations (133). 

 Since fx. disappears from the general equations (l), it follows that p is the same function as 

 before. But in the first case the system of pressures at the surface was P, = P 2 = P 3 — p, 

 T y = T 2 = T z = 0. Hence if A Pi &c. be the additional pressures arising from friction, we get 

 from (133), observing that £=0, and that udx + vdy + wdss is an exact differential dtp, 



d ! d> d?d> (Pd> 



AP >=" 2 ' jl -^. A ^=- 2 ^3±' AP 3 =-2m-^, • 0*2) 



A7' 1 =-2 M —', Ar 2 =-2 M -^|-, AT 3 = -*M-T-7- • 043) 

 dy dss dzdx dxdy 



Let dS be an element of the bounding surface, /', m, n' the direction-cosines of the normal 

 drawn outwards, AP, AQ, A R the components in the direction of x, y, z of the additional 



* It is here supposed that the forces X, Y, Z are ouch that Xdx t Ydy ->- Zdx is an exact differential. 



