184 Mr. W. Ferrel on a controverted Point in 



case of no friction there can be no semidiurnal tide except for 

 the critical depth (for we are not considering any other), then 

 we must still have K 4 = in my expression of a, and instead 

 of confining the expression with this value of K 4 to cases only 

 when there is friction, it must be extended to all cases except 

 that in which the sea has the critical depth referred to and is 

 without friction, in which case K 4 is arbitrary. Sir William 

 Thomson, therefore, while defending Laplace's process of de- 

 termining this constant, has furnished us a condition for deter- 

 mining it, which shows that we must either have K 4 = or 

 arbitrary, and not equal to the value given by Laplace's 

 process. 



For the same reasons as in the preceding case, the similar 

 arbitrary constant A x in the expression of a' which I have ob- 

 tained in the solution of the equations for the diurnal tides 

 (Tidal Researches, eq. 297), must be made zero in all cases, 

 except that of the critical depth where there is no friction, in 

 which case alone it becomes arbitrary. Hence my tidal ex- 

 pression in this case, as in the case of semidiurnal tides, with 

 Ai = 0, only needs to be extended so as to embrace all cases 

 without friction, except that for the critical depth. With 

 A x = 0, the expression a! does not make the diurnal tide vanish 

 in all parts of the earth as Laplace's result does, and the ex- 

 pressions for the motions of the sea do not make them infinite 

 in the case in which the rotatory motion of the earth vanishes. 



It seems at first view plausible that the expression of a in 

 the semidiurnal tide should be general, including both the 

 free and the forced tidal waves, so that when we put the terms 

 in the expression depending upon the tidal forces equal to 

 zero the remaining part of the expression should represent 

 arbitrary free oscillations ; and this would be true if in satis- 

 fying the original differential equations there were not an as- 

 sumption made which does not hold for the free waves. In 

 satisfying the equations, it is assumed that they are satisfied 

 with an oscillation which has a period which is the same as 

 that of the forces ; and this is certainly true of the forced tidal 

 wave, but not generally true for the free tidal wave ; for if we 

 were to commence solving the original tidal equations for the 

 case of no disturbing force of moon or sun, it would certainly 

 be begging the question mostly to assume that they must give 

 semidiurnal oscillations. The expression of a, therefore, with 

 the arbitrary K 4 cannot be extended so as to embrace the free 

 oscillations, except for the case of the critical depth where 

 there is no friction ; for where an assumption is made in the 

 solution, the results obtained from that solution cannot be ex- 

 tended so as to include cases in which the assumption cannot 

 be shown to hold. 



