ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 181 



and when A<7/4cu 3 a 5 = j, 



= | [1-7646 P; - 0-06057 Pi + 0'001447 PI - 0'000022 Pi + . . .]. 



From these series we find for the ratio of the tide-heights to the equilibrium tide- 

 heights at the equator the four values 



2-4187, 1-8000, +11-0725, + 1'9225. 



On comparison of these numbers with those obtained for the solar tides in the 

 preceding section, we see that for a depth of 7260 feet the solar tides will be direct 

 while the lunar tides will be inverted, the opposite being the case when the depth is 

 29,040 feet. This is, of course, due to the fact that in each of these cases there is a 

 period of free oscillation intermediate between twelve solar (or, more strictly, sidereal) 

 hours and twelve lunar hours. The critical depths for which the lunar tides become 

 infinite are found to be 26,044 feet and 6448 feet. 



Consequently this phenomenon will occur if the depth of the ocean be between 

 29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. An important 

 consequence would be that for depths lying between these limits the usual pheno- 

 mena of spring and neap tides would be reversed, the higher tides occurring when 

 the moon is in quadrature, and the lower at new and full moon. * 



There appears then to be a considerable range of depth comparable with the mean 

 depth of the ocean over which the reversal of the spring and neap tide phenomena 

 would take place, but in that the actual tides are highest in the neighbourhood of 

 new and full moon we conclude that the effective depth of the ocean does not lie 

 within this range, and that none of the periods of free oscillation of the actual ocean 

 lie between twelve solar hours and twelve lunar hours. The true effective depth is 

 almost certainly less than 26,044 feet, and therefore both solar and lunar tides will 

 be in the main inverted, though the configuration of the land and of the ocean bed 

 will probably give rise to considerable variations of phase in different places. 



The shortest period of free oscillation of the second class for the case s = 2 approxi- 

 mates to, but is in excess of, three days. But if n denotes the moon's mean orbital 

 motion, the speed of the lunar semi-diurnal tide is 



2 (w n). 

 If we equate this to ^ <w, we obtain 



n = | to. 



Hence, if the moon's orbital motion were accelerated, or the earth's rotation 

 retarded, until the month and day were in a ratio less than 6 : 5, it would be possible 

 for the period of the lunar semi-diurnal tide to confound itself with one of the periods 

 of the oscillations of the second class, and the tides would then tend to become very 

 large. 



* C,f. KELVIN, ' Popular Lectures.' vol. 2, p. 22 (footnote). 



