ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
257 
araouni to 28,900 indies, or about lialf-a-mile, per annum. This velocity will be cpiite 
inappreciable to observation. 
The amount of longitudinal velocity generated in the course of a year in -any 
latitude is given by the formula 
aClU! 
— \J?). 
This will be greatest when p,® = f, that is, in latitude 55° nearly, and its greatest 
value corresponds to a velocity of about of a mile per hour. The maximum 
current velocity due to this cause would therefore amount to about four miles per 
hour if the modulus of decay is as long as 20 years. 
The most crucial test to which we can subject the theory of ocean currents here 
put forward will consist in the evaluation of the moduli of decay for the types ol 
motion concerned. This I have endeavoured to do, but as the work involves 
analytical considerations of a somewhat different character from those which occur 
in the present work, I have deemed it advisable to present the results in a separate 
paper.* These results, so far as they are applicable, seem to point to a modulus of 
decay far in excess of the 20 years here required, but the mathematical difficulties 
have compelled me, in dealing with friction, to leave the rotation entirely out of 
account. It appears that in the simpler system so treated, the types of free current 
motion are far more arbitrary in character than those at which we have arrived by 
including the rotation. This arises from the fact that our rotating system will be 
cajoable of a large number of free oscillatory motions besides those which we have 
examined in which the periods of oscillation always bear a finite ratio to the period 
of rotation. As the period of rotation is lengthened, the period of each of these 
types of oscillation is prolonged, and the possible forms of steady motion where 
there is no rotation must include the limiting forms of each of these types of 
oscillatory motion. 
The moduli of decay of the free current motions when there is rotation may 
therefore be very different in value from those obtained in the paper referred to, but 
it does not seem to me that they could be much less in order of magnitude than the 
moduli of decay of the principal types of free oscillation. If this prove to be the case, 
the estimate of 20 years, which we have taken for the modulus of decay, will not be 
so excessive as might at first sight appear. 
Of course, if the v/ater does not cover the whole earth, or if the depth be not 
uniform, the currents due to the rotation will follow the free stream-lines defined by 
equation (38) instead of following the parallels of latitude. The rotation will, no 
doubt, produce its maximum effect when the stream-lines in question coincide with 
the parallels of latitude, but this circumstance does not alter our main conclusion as 
to the adequacy of evaporation and other such causes to generate currents quite 
comparable with those known to exist in the ocean. 
* Read before the London Mathematical Society, December 10th, 189G. 
HDCCGXCYII.—A. 2 L 
