1886.] Equilibrium Theory of Tides for the Continents. 307 



doubling coalesce on the equator, and a fourth pair coalesce at the 

 antipodes of the third pair ; lastly, in the case of the tides of long 

 period the circles of evanescent tide tend to coalesce with the circles 

 of doubled tide, in latitudes 35° 16' N. and S. 



We are now in a position to state the results of Thomson's corrected 

 theory by comparison with Bernouilli's theory. 



Consider the semi-diurnal tide on an ocean-covered globe, then at 

 the four points on a single meridian great circle which correspond to 

 the points of evanescence on the partially covered globe, the tide has 

 the same height ; and at any point on the partially covered globe the 

 semi-diurnal tide is the excess (interpreted algebraically) of the tide 

 at the corresponding point on the ocean-covered globe above that at 

 the four points. 



A similar statement holds good for the diurnal and tides of long 

 period. 



By laborious quadratures Mr. Turner has evaluated in Part II the 

 five definite integrals on which the corrections to the equilibrium 

 theory, as applied to the earth, depend. 



The values found show that the points of evanescent semi-diurnal 

 tide are only distant about 9° from the N. and S. poles ; and that of 

 the four points of evanescent diurnal tide two are close to the equator, 

 one close to the IS", pole, and the other close to the S. pole ; lastly, 

 that the latitudes of evanescent tide of long period are 34° N". and S., 

 and are thus but little affected by the land. 



Thus in all cases the points of evanescence are situated near the 

 places where the tides vanish when there is no land. It follows, 

 therefore, that the correction to the equilibrium theory for land is of 

 no importance. 



G. H. D. 



II. 



For the evaluation of the five definite integrals, called by Sir 

 William Thomson gi, J, jl, and represented in the present paper 

 by functions of the latitudes and longitudes X Q , \, X 2 , and Z l5 Z 2 , respec- 

 tively similar in form to the functions of the " running " latitude and 

 longitude to be integrated, it is necessary to assume some redistribu- 

 tion of the land on the earth's surface, differing as little as possible 

 from the real distribution, and yet with a coast line amenable to 

 mathematical treatment. The integrals are to be taken over the 

 whole ocean, but since the value of any of them taken over the whole 

 sphere is zero, the part of any due to the sea is equal to the part due 

 to the land with its sign changed ; and since there is less land than 

 sea, it will be more convenient to integrate over the land, and then 

 change the sign. 



Unless specially mentioned, we shall hereafter assume that the 

 integration is taken over the land. 



