72 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS
where c, w, n are three constants, of which the last two are connected by a relation which it is not necessary to write down. We may continue to employ this equation as a near approximation when friction is taken into account, provided we suppose c, instead of being constant, to be a parameter which varies slowly with the time. Let V be the vis viva of a given portion of the fluid at the end of the time t, then
V=pc*m*Jff6-*mydx)dydz...............(141).
But by means of the expression given in Art. 49, we get for the ]oss of vis viva* during the time dt, observing that in the present case /Lt is constant, w = 0, 8 = 0, and udx + vdy = d<j>, where <j> is independent of z,
which becomes, on substituting for <j> its value, S/4C2m4 dtfjje~2my dx dy dz.
But we get from (141) for the decrement of vis viva of the same mass arising from the variation of the parameter c
Equating the two expressions for the decrement of vis viva, putting for m its value 27rX~1, where X is the length of a wave, replacing /JL by ///>, integrating, and supposing c0 to be the initial value of c, we get
It will presently appear that the value of ^/pf for water is about 0*0564, an inch and a second being the units of space and
* [There is an oversight here, which M. Boussinescq has pointed out (Memoir es des Savans Strangers, Tome xxiv. No. 2, p. 34). I should have said "the loss of energy." Now in a series of waves of small disturbance the total energy is half kinetic and half potential. Hence the reduction of energy consequent upon a reduction in the amplitude falls half on the kinetic and half on the potential energy. Hence the reduction of the kinetic energy or vis viva is only half of that given by the formula in the text, and therefore the expression for dc/dt is twice what it ought to be. Hence the numerical coefficient in the index of the exponential should be 8 instead of 16; and retaining the same numerical data as in the examples, we should have for the ripples c : c0 :: 1 : 0-5337, and the height of the long waves would be reduced in a day by little more than the one four-hundredth part.]