Mr. W. Ferrel on Laplace's Theory of the Tides. 183 



equation (3), in the expression of u, equation (10) in Sir 

 William Thomson's paper; and then by putting u = we 

 should have an equation which must be satisfied where 

 cos 8 = 0, in order to satisfy the condition above. Now it is 

 readily seen from mere inspection that equation (10) with 

 u = is satisfied at the equator by any expression of a in 

 which the terms contain a factor which is any power whatever 

 of x~ sin 0, 6 being the polar distance ; and hence the expres- 

 sion of a in equation (3) satisfies it, not only with Laplace's 

 value of K 4 , but with any assumed value of K 4 whatever. The 

 condition, therefore, that the meridional component of motion 

 must vanish at the equator does not determine K 4 , and the 

 argument in favour of Laplace's process entirely fails. 



At the close of § 9 Sir William Thomson makes an im- 

 portant remark in italics, the truth of which must be conceded ; 

 and it therefore renders erroneous certain views entertained 

 by Airy and adopted by myself in the case of no friction with 

 regard to the indeterminate character of the problem ; for it 

 furnishes a condition for determining K 4 in all cases except 

 that of the critical depths of the sea which make the period of 

 the free tidal wave coincide exactly with that of the forced 

 wave. The expression of a, equation (3), is composed of two 

 parts, one depending upon and vanishing with the tidal force, 

 and the other depending upon K 4 , which is arbitrary and 

 cannot be determined from any condition contained within 

 the differential equation solved, but must, as is the case with 

 all such arbitrary constants, if determined at all, be determined 

 by some external condition not contained in it. In order to 

 determine K 4 , therefore, we must know what the value of a is 

 when the force vanishes ; and then, putting this value for a in 

 equation (3), we have an equation for determining K 4 , since 

 all the terms then in the second member of the equation de- 

 pending upon the force vanish, and the others are all known 

 functions of K 4 by reason of the relations of equation (6). 

 Now, if we can have no semidiurnal free tidal wave except in 

 the case of the critical depth of the sea just referred to, or if 

 in this latter case such tides are destroyed by friction, we evi- 

 dently have a=0 when the force vanishes ; and with a = the 

 equation can be satisfied only with K 4 =0. In the case of 

 the critical depth, however, where there is no friction, a 

 becomes arbitrary, as the amplitudes of all free tidal waves 

 are ; and then K 4 becomes likewise arbitrary. Hence we have 

 K 4 =0 for all cases except that for the critical value of the 

 depth. This is the value which I have maintained K 4 should 

 have in the case of nature, in which free oscillations are de- 

 stroyed by friction (Tidal Ecsearches, § 159) ; but if in the 



