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2J 1 
U — Uq +'Ui^ — k ./j \/(l “ l^~)> 
V = v„ + \V = - 
We see then that the effects of evaporation and precipitation will be to cause a 
steady flow of water, not by means of undercurrents only, but by currents sensibly 
uniform throughout the depth, towards the equator ; in addition to these currents 
the cause in question will give rise to longitudinal currents, not of a steady character, 
but increasing uniformly with the time, and these will be accompanied by an appro¬ 
priate continuous deformation of the free surface. Were it not for viscosity these 
currents would increase without limit and ultimately endang’er the stability of the 
system, but under the action of dissipative forces a steady state must ultimately be 
attained, in which the rate at which the-currents are generated exactly balances that 
at which they are destroyed. Thus, suppose the type of motion set up is such that 
if left to itself it would be reduced in the ratio 1 : e in a period r. If U denote 
the velocity of any jaarticle, the law of variation of U under the influence of vdscous 
forces is then 
0U/0^-f U/7 = 0, 
whereas, if there be no viscosity and the system is subjected to such a disturbance as 
we have been dealing with, the velocity varies according to the law 
0U/0i =/, 
wdiere _/is constant. Equating the rate of increase of the velocity without viscosity 
to the rate of decrease under the influence of dissipative force, we And that the 
ultimate state is deflned by ... 
U/r=/, or U=/r. 
Thus, if the disturbing influence tends to set up one of the possible types of motion 
of which the system is capable under viscosity, the ultimate velocity of any particle 
will be that which it would acquire in a period equal to the modulus of decay of the 
type of motion in question. 
By wey of numerical illustration, take a year as the unit of time and an inch as 
the unit of length, and suppose a = 40. This will imply an annuaTrainfall at the 
poles vv^hich exceeds evaporation by 40 inches, and an annual rainfall at the equator 
which is less than evaporation by 20 inches. Further, suppose hja — o-gVo- Then 
JJ - -1 X 2890 X 40 ya y (1 - [x~). 
U wflll be numerically greatest when jx" = or in latitude 45°, and the 
greatest value of U is 28,900. The maximum latitudinal-velocity will therefore 
