136 Mr 0. Fisher, On the condition [Jan. 29, 



Now, if any tide exists, supposing the water to be of mean 

 depth h, and y to be depth of the wave below its mean level, we 

 shall have for the distance from the earth's centre to the surface 

 of the water, 



r = a[l+{l-fi')e] + h-y. 



Substituting this value, we have for the potential of the 

 deformed earth at the surface of the water, 



g[a[l-{^-,.')e]-h-\-y]+gae{^-,.') + r{l-f.'), 



or g{a-h+y) + r{^-fj.^). 



If we differentiate this with respect to aO, it will give the 

 horizontal attraction of the deformed earth in the direction away 

 from the moon, which, since /a^ = cos^^, will be 



- sin 2^. 

 a 



Again, the moon's potential at the same point is 



This, when differentiated, gives the moon's attraction on the 

 same particle of water in the same direction as before 



- - sin 2^. 

 a 



These two forces being equal and of opposite signs will balance 

 one another, and the water will not be disturbed, and there will 

 be no tide. 



It follows from the above — in fact it is almost self-evident — 

 that in order that the argument for rigidity derived from the 

 ocean tides should be complete, the surface of the earth should, if 

 the interior is liquid, be deformed to the exact shape that the 

 moon's attraction would produce in a liquid globe ; or, in other 

 words, the mean form of the surface of the solid crust should be, 

 under the attractive force, one of the equipotential surfaces, just as 

 the ocean surface would be one of them, if not interfered with by 

 land. But it seems certain that in nature the solid surface would 

 not so conform, because on the hypothesis of liquidity there must 

 needs be deep depressions, which I have called roots of mountains, 

 answering to the elevations, and extending to many times their 

 height down into the heavier liquid in order to support them. The 

 presence of these roots explains the observed deficiency of gravity 

 beneath elevated tracts. Now seeing that a tide is caused by the 

 local accumulation of liquid by differential horizontal flow, these 

 roots would so break up and deflect the tide wave in the substratum, 



