140 ME. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



a very lengthy analysis similar to that used by the latter writer. The comparative 

 simplicity of these results seemed, however, to point to the fact that they might be 

 more easily obtained by less pretentious means. The deduction of the formulae in 

 the former paper from the differential equation of LAPLACE presented no serious 

 difficulties, but in attempting to apply a like method to obtain the more general 

 formulae of the present paper, I found that formidable obstacles had to be overcome. 

 The method of integration now adopted seems to leave little to be desired for sim- 

 plicity, considering its generality, but the fact that it has been built up partly by 

 working forwards from the differential equation, and partly by working backwards 

 from the results, must account for the apparent artificiality of the procedure. 



In all previous attempts at the solution of the dynamical equations for the tides, 

 the integration has been effected by assuming that the expression for the tide-height 

 could be expressed by an infinite series of terms of known form associated with 

 undetermined numerical coefficients. The differential equations then lead to a 

 difference-relation between a certain number of these coefficients from which their 

 numerical values are to be evaluated. The numerical determination of the coefficients 

 will be facilitated when this difference-relation contains as few terms as possible. 

 Now it is found in the present paper that, without imposing any restriction on the 

 period of the disturbing force, if the form we assign to the terms of the series for the 

 tide-height is that of the tesseral harmonics or LAPLACE'S functions, a linear relation 

 involving three successive coefficients only may be deduced, provided that the law of 

 depth is such that both the internal and external surfaces of the ocean are spheroids 

 of revolution about the polar axis. This however appears to be the most general law 

 of depth which can be employed without obtaining more than three successive 

 coefficients in the linear relation in question, and consequently our discussion deals 

 only with cases where the law of depth is subject to this limitation. 



In 1 I have collected the principal properties of the functions used in the 

 analysis. These properties are for the most part well known, but in consideration of 

 the want of agreement in the notation employed by different writers, I have thought 

 it best to briefly prove such of them as are required in preference to giving references 

 to places where they may be found. Moreover I have thus been enabled to write 

 the results in the exact form required for subsequent application. 



2-4 deal with the integration of the differential equations and the deduction of 

 the linear equations (31), (40) connecting the coefficients in the expansion of the 

 tide-height. These equations, the analogy of which with the equations (23), (23A) 

 of Part I. will be at once apparent, constitute the chief analytical results of the 

 paper, and the remainder is occupied with the application of these formulas to 

 the discussion of the free and forced vibrations on lines similar to those adopted 

 in Part I. 



5-11 treat of the free oscillations, the discussion being confined to the case 

 where the depth is uniform. A period-equation is obtained, and an approximate 



