ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 179 



difference of 5 minutes between these periods will be sufficient to reduce the tide to 

 less than ten times the corresponding equilibrium tide. It seems then that the tides 

 will not tend to become excessively large unless there is very close agreement with 

 the period of one of the free oscillations. 



The critical depths for which the forced tides here treated of become infinite are 

 those for which a period of free oscillation coincides exactly with 12 hours. They 

 may be ascertained by putting A. = 2co in the period-equation for the free oscillations 

 and treating this equation as an equation for the determination of h. The roots may 

 be found by trial and error as in 9, the approximate values with which to commence 

 the trials being suggested by the discussion already given. The two largest roots are 

 found to be given by 



a 3 = 0-10049, %/4w 2 a 2 = 0'02545, 



and the corresponding critical depths are about 29,182 feet and 7375 feet. 



We have hitherto supposed that p/ar = 048093, but for purposes of comparison I 

 have also examined the case where /3/cr = 0, that is where the mutual attraction of 

 the waters is neglected. The series for in this case become 



= Cr; [1-0927 Pj + l'91817P'i - 0-66909 PS + 0'1070lP; - 0'0103GP; 



+ 0-OOOG83Pi, - 0-000033Pf, + O'OOOOOI PI, - . . .] 



= <; [- 1-0733P:] + 0'24502P'i - 0'02790P; + 0'00185P; - O'OOOOSOPjo 



+ 0-000002 Pj, - . . ] 



= 1&-, [9'34370P^ 0-70311 Pj + 0'03449P^ - O'OOlOePjj + 0-000022P'f - . . .] 

 = ? [17739P; - 0-05750P; + 0'00132Ps - 0'00020Pi + . . .] 



or the depths 7260, 14,520, 29,040, 58,080 feet respectively. From these series we 

 deduce as the ratio of the tide to the equilibrium tide at the equator the four values 



' - 7-4343, - 1-8208, + H'2595, + 1-9236, 



results which agree, except in the third case, with the numbers given by Professor 

 LAMB* deduced from the numerical formulae of LAPLACE. 



It will be seen that in three cases out of the four here considered the effect of the 

 mutual gravitation of the waters is to increase the ratio of the tide to the equilibrium 

 tide. In two of the cases the sign is also reversed. This of course results from 

 the fact that, whereas when p/tr = 0-18093 one of the periods of free oscillation is 



* By a carefnl re-computation of the semi-diurnal tide for the case /3=10 (notation of Professor LAMB) 

 I find the following series more accurate than that given for f/H" 1 : 



S~ + 6-1915 -4 + 3-2447 + 0-7234 ,< 8 + 0-0919 1- 10 + 0-0076 J* + 0-0004 *" + . . . 

 This series reduces to H'2595 when v = 1, thus agreeing with the result obtained above, 



2 A 2 



