204 
MR. S. S. HOUGH OH THE APPLICATIOH OF HARMOHIC 
G. W. Hill, in his ‘ Researches on the Lunar Theory but in the present instance 
we are able to avoid the difficulties involved in the use of these infinite determinants, 
in that the forms of determinant which occur are those which are associated with 
continued fractions. 
§ G deals wdth the analytical discussion of the deduction of the period-equation. 
The method is based on a paper by Lord Kelvin, t in wdiich the author defends the 
procedure of Laplace against certain allegations to which it had been subjected 
by Airy, but I have endeavoured to present the arguments in a somervliat different 
light, so as to bring out more clearly the analogy between our problem and the 
general problem of vibrating systems with finite freedom. 
In 7-10 I have given illustrations of the method of solving the period-equation 
numerically, and of the subsequent determination of the type of motion for the 
different fundamental modes. As the ground covered in these sections is almost 
entirely new, I have devoted considerable time and labour to the arithmetical 
determination of the periods and types of the principal oscillations for a system 
comparable with the earth in magnitude. The results are tabulated in these 
sections. 
§ 11 deals briefly with the forced tides of long period due to the moon in an ocean 
of uniform depth. The results agree with those previously obtained by other 
methods, but differ from them in analytical form. In § 12 I have given illustrations 
of a means of extending the method of numerical computation to cases where the 
law of depth is less restricted in character. 
The consideration of forced tides of very long period, dealt with in § 13, points to 
the existence of free oscillations of infinitely long period. This, I believe, was 
first noted by Professor Lami3,| but the application of the dynamical equations for 
the tides to the treatment of these free oscillations has not been previously carried 
out. The types of motio]i in C[uestion appear to be of considerable Importance, as 
they throw light on a phenomenon which in the past has been the subject of 
considerable controversy. The difficulties wdiich have been met wdth in attempts to 
account for the existence of ocean currents all seem to me to arise from an over¬ 
estimate of the eflects of viscosity on the motion of the sea. The large-scale ocean 
currents have been attributed by Sir John Herschel§ and others entirely to the 
influence of the ‘'trade” and other prevailing winds, wdiich give rise to slow steady 
motions vdiich, in the absence of friction, wmuld remain permanent even w^ere the 
originating' cause entirely to cease. The difficulty in accepting this view^ arises from 
the assumption that such currents wmuld succumb to the influence of frictional 
* ‘ Acta IMatliematica,’ vol. 8. 
t ‘ Pliil. Mag.,’ 1875, p. 227. 
t ‘ Hydrodynamics,’ § 198. 
§ ‘Physical Geograph}'^,’ §§ 57-60. 
