r 201 1 
IX. On the Apj^Iication of Harmonic Analysis to the Dynamical Theory of the 
lides. — VKvi I. On Laplace’s “Oscillations of the First Species,'' and on the 
Dynamics of Ocean Currents. 
By S. S. Hough, M.A., Fellow of St. John's College, and Isaac Newton Student in 
the University of Cambridge. 
Cominunicated by Professor G. H. Darwin, F.R.S. 
Received Mai’cli 12,—Read April 8 , 1897. 
The earliest attempt to subject the Theory of the Tides to a rigorous dynamical 
treatment was given by Laplace in the first and fourth books of tlie ‘ Mecanique 
Celeste.’ The subject has since been treated by Airy,"^ Kelvin,! Darwin,^ Lamb,§ 
and other writers, but with the exception of the extension of Laplace’s results to 
include the theory of the long-period tides, but little practical advance has been made 
with the subject, in spite of the enormous increase in the power of the mathematical 
resources at our disposal, and the problem has remained in very much the same 
condition as it was left by Laplace. This arises no doubt partly from the difficulties 
inherent to the subject, but partly from the form in which the theory was originally 
presented by Laplace in the ‘ Mecanique Celeste,’ which has been described by Airy 
as “ perhaps on the whole more obscure than any other pait of the same extent in 
that work.” The obscurity complained of does not however seem to have beer, 
entirely removed by Laplace’s successors, and it was the fact that every presentment 
of the theory with which I was acquainted offered some points of difficulty, that in 
the first instance led me to take up the problem ab initio, partly with the purpose of 
allaying the doubts which had arisen in my own mind as to the validity of certain 
approximations employed by Laplace and adopted by his successors, and partly in 
the hope that I might be able to extend the results of Laplace to meet more fully 
the case presented by the circumstances actually existent in Nature. 
Up to the present I have been unable to free the problem to any extent from the 
limitations which have been imposed by previous writers, and consequently it wmuld 
be futile to claim that the results I am now able to put forward materially advance 
* ‘ Encyc. Metropolitana ’; Art., “ Tides and Waves,” Section III. 
t ‘ Phil. Mag.,’ 1875, vol. 50. 
t ‘ Encyc. Britannica ’ (9th edition) ; Art., “ Tides.” 
§ ‘ Hydrodynamics,’ chapter viii. 
MDCCCXCYII.- A. 2 D 
20.7.97 
