Jati, 2, 1879] 



NATURE 



20^ 



state of motion that prevails after an infinitely long time 

 the distribution of the velocity is the same in a thin fluid 

 like water and in a thick fluid like syrup. In the fixed 

 state of motion the influence of friction is shown by the 

 participation of all the strata in the motion which is im- 

 parted from without to the surface alone. Dependence 

 on the coefficient of friction takes place only on the con- 

 sideration of motions that vary with the time, and affords 

 a measure for the depth of penetration of a surface- 

 impulse within a given time. 



The formula which gives the velocity at the depth x of 

 a mass of water originally at rest when for the time / 

 the surface has been kept at a constant velocity Wo, has 

 naturally a less simple form than the formula which was 

 found for steady motions. (The formula is the same as 

 that which determines the propagation of heat in a solid 

 wall whose one side is kept at a te^nperature Wo", whilst 

 the other remains at o".) From this formula results the 

 simple law that any velocity whatever between o and zt/o 

 prevails at different times at depths which are related to 

 one another as the square roots of the times. I have used 

 the formula to compute the time that a point at the depth 

 of 100 metres requires to attain half the surface velocity, 

 i.e., J Wo. The coefficient of friction of the water was 

 assumed according to O. E. Meyer's determination, at 

 0*0144, in which centimetres and seconds are the units of 

 calculation. The result was that 239 years are required 

 for the layer of water 100 metres deep to assume the 

 half of the surface velocity. If it be asked what length 

 of time is required for one-tenth of the surface velocity to 

 penetrate to that depth, the answer is 41 years. Accord- 

 ingly, the same velocities will be attained at a depth of 

 10 metres after 2'39 and o'4i years respectively. In a 

 more viscous fluid the resulting numbers would be 

 smaller. 



These numbers are well calculated to give an idea 

 of the slow rate at which changes of motion are pro- 

 pagated dowTiwards. For the numbers computed for 

 the propagation of a given surface motion, hold likewise 

 for the penetration of a change of the motion from the 

 surface downwards, whose influence is simply added to 

 the already existing motion. A steady current, therefore, 

 whose velocity diminishes linearly according to the depth, 

 will sustain only an extremely slight alteration (except in 

 the strata nearest the surface) from passing changes of 

 motion that affect the surface, e.g. from contrary winds 

 or storms. There will prevail, rather, at every deep- 

 lying point of this current, a mean velocity that changes 

 only very slightly according to the time, and which is de- 

 termined by the mean velocity at the surface. This latter 

 velocity has the direction of the prevailing wind, accord- 

 ing to whose strength it varies by a law that cannot be 

 more accurately settled. 



If the surface velocity varies periodically according to 

 the time, as is the case with all winds that depend on 

 seasons and the hours of the day, then, after this periodic 

 state has lasted an infinitely long time, the velocity at all 

 depths is a periodic function of the time of similar period, 

 but such that the amount of variation decreases rapidly 

 according to the depth and that the occurrence of the 

 maxima and minima is delayed proportionally to the 

 depth. At a depth of 10 metres the amount of the yearlv 

 oscillation is already diminished to less than J^th ; at a 

 depth of 100 metres it is beyond observation ; at this 

 depth the velocity is that corresponding to the steady 

 state when the mean annual velocity is given to the sur- 

 face. When the depths decrease in arithmetical propor- 

 tion, the amounts of the oscillation decrease in geome- 

 trical proportion such that at four depths :»%, x^ x^ x^, 

 •which stand in the relation 



the amounts D^, D._, D^, D^ stand in relation 



A maximum and the following minimum of the annual 

 oscillation always exist at the same time at a vertical dis- 

 tance of u '9 metres. 



To give a conception of the time that a constant 

 surface-velocity which begins at the time t = o requires, 

 in order to bring the interior of an ocean 4,000 metres 

 deep, which was previously at rest, to the state of steady 

 motion, the following numbers will serve : — After 10,000 

 years there prevails at the half-depth, />., at jr = 2,000 

 metres, just the velocity o'oy]w^. Since, according to 

 the already-stated formula, in the steady state the velocity 

 o'5zfo must prevail at this point, it is easily seen how far 

 the ocean is still removed after 10,000 years from the 

 steady state. After 100,000 years the velocity at the 

 depth stated is already o'46ia/o, therefore very near the 

 definitive value. After 200,000 years it differs only by 

 two units in the third decimal place. 



Among the results we have found, particular emphasis, 

 is to be laid on two, which seem more or less to contra- 

 dict the views which have prevailed up to this time. In 

 the first place, the steady motion arising in the interior of 

 an unlimited stratum of water from an unvarying surface 

 velocity makes itself felt w^ith linearly decreasing velocity 

 down to the bottom. Hitherto the view frequently ex- 

 pressed was, that the influence of surface currents, e.e^.y 

 the drift caused in the tropical ocean by the trade winds,, 

 reached only to very moderate depths. Secondly, it was 

 found that all variations according to time, whether 

 periodic or aperiodic, of the forces acting on the surface, 

 propagate themselves downwards with extraordinary 

 slo\\-ness, the periodic in very quickly decreasing amount. 

 Taking both statements together, it follows that the 

 movement of the chief part of a stratum of water exposed, 

 to periodically varying surface forces is determined by 

 the mean velocity of the surface, and that the periodic 

 variations are observable only in a comparatively thin 

 surface stratum. From this it is obvious that hitherto 

 the influence of the friction was undervalued in one 

 direction, in so far, namely, as it was believed that its 

 influence need not be considered as penetrating so deep, 

 but in another direction it was overvalued, as too great 

 an influence was wont to be ascribed to friction in respect 

 of the propagation of varying current motions Its effect 

 was also very much overvalued in another point, viz., in 

 respect of the action of a bank on a stream flowing along 

 it. If, I repeat, the whole surface is kept at a constant 

 velocity, then also in the current bounded at the side the 

 distribution of velocity in the steady state is independent 

 of the co-efficient of friction. Beyond that, the influence 

 of the banks on the distribution of velocity is exceedingly 

 slight. 



A further result is that two steady currents flowing 

 parallel to one another, but in opposite directions, in a 

 fluid-stratum of constant depth, may very well graze one 

 another without mutual disturbance. Their surface of 

 division is then a vertical plane parallel to their directioa 

 in which the velocity o prevails, and which, therefore, 

 stands to each current in the relation of a solid bank. 



We have already shown nimierically how extraordinarily 

 slow the velocity existing at the surface is propagated 

 downwards when the interior was previously at rest. 

 Hence it may be concluded, vice versa, that when every 

 point of the whole mass of fluid has at a given moment a. 

 given velocity varying according to the depth, and when 

 from the same moment onwards the surface remains at 

 rest, the effect of this initial state vanishes equally slowly,, 

 i.e., the ocean passes into the state of rest with the same, 

 slowness with which in the first case the surface-velocity 

 was propagated into the interior. In fact the formulae 

 show that the times for the increase and decrease of the 

 same fraction of the given velocity are expressed by the 

 same number. 



If from some cause or other strong currents had been 

 generated in the ocean, say 10,000 years ago, these 



