of a Liquid Substratum beneath the Earth's Crust. 219 

 whence 



y = W _ C os 28 cos (2co + 28). 



(5) Suppose c to be the maximum value of y when friction 

 is not taken account of. Then 



ajjb 



And the maximum value of y when friction acts will be 



c' — c cos 28 (4) 



Equations (3) and (4) show that, as friction is increased, 28 

 tends towards 90° or 8 towards 45°; while at the same time c r } 

 the maximum tide above mean level, diminishes to zero. 

 Hence as friction (or viscosity) increases, the vertex of the 

 tidal spheroid moves eastward, the ellipticity of the tidal sphe- 

 roid simultaneously decreasing, until, when friction is infinite, 

 its vertex reaches 45° east of the moon, and the tide disappears 

 altogether. The same general result appears from Mr. Dar- 

 win's table (p. 16)* to hold good in the case of bodily tides, 

 if there should be such in the earth. 



(6) "We notice that our result is independent of the density 

 of the liquid, and that the weight of a floating crust, if con- 

 sidered flexible, would not affect it — the reason being that 

 such a crust would aid in depressing the hollows just as 

 much as it would hinder the elevation of the ridges. It would 

 have an effect analogous to an additional load to the bob of 

 a pendulum. 



The coefficient of k in the denominator of the expression for 

 c shows that the term may be neglected, although k itself be 

 not small. For a is the space over which the wave travels in 

 one second, while a is the radius of the earth. 



Neglecting this term, c and likewise d are positive or nega- 

 tive according as 



« 2 

 k > or < — 

 9 



a 2 

 Consequently, when- k < — there will be low tide under the 



* " On the Bodily Tides of Viscous and Semi-elastic Spheroids, and on 

 the Ocean-Tides upon a yielding Nucleus," Phil, Trans. Eoy. Soc. part i. 

 1879. 



