132     ON THE  EFFECT OF THE  INTERNAL  FRICTION  OF FLUIDS
and t~t^f for t in (18*7), and integrating with respect to t'.   We thus get
To apply this result to the case of an oscillating disk, let
d9 _r     —_r]?(ify foe the velocity of any annulus, and G the moment of
Ctu
the whole force of the fluid on the disk.    Then
o   / ix/c/o \    i
r I -r—  dr; o     \«#/o
_an(j f™£\ _wiH be got from (188) by substituting rF(t) for /(£).
We find thus
Q- = — ^Tr// . pa^41   F^f (t — £,) -77............(189).
Jo                 v^i
If we suppose the angular velocity of the disk to be expressed by A sin nt, where A is constant, we must put F(t) = A sin nt in (189), and we should then get after integration the same expression for Q as was obtained in Art. 8 by a much simpler process. Suppose, however, that previously to the epoch from which t is measured the disk was at rest, and that the subsequent angular velocity is expressed by A_t sin nt, where A_t is a slowly varying function of t. Then
F(t) = 0 when t < 0,        F(t) = A_t sin nt when t > 0. On substituting in (189) we get
A_t-t, cosn(t — tfj) ~,* ......(190).
Now treating A_t as a slowly varying parameter, we get from a formula given by Mr Airy, and obtained by the method of the variation of parameters,
J a       n
.....................(191),
where / denotes the moment of inertia. In the expression for G we may replace A_t__ti under the integral sign by A_t outside it, because A_t is supposed to vary so slowly that A_t-_tl does not much differ from A_t while ^ is small enough to render the integral of