NEW ADDENDUM. 43 



If, in applying the expression («), the symbolic fraction, for a first approximation, 

 be omitted, we have, according to above assumption of discrepancy, e = le,. This 

 value of £ will be in excess ; hence, the thickness of crust deduced from it will err 

 the other way, and a determination on this basis will give a thickness which must, 

 in fact, be exceeded. 



The limit of solidity, proceeding inwards, may and probably docs depend upon 

 both temperature and pressure. Isothermal surfaces Prof. Hopkins finds to have 

 incrcasbuj ellipticities. Surfaces of equal pressure, deduced from the hypothetical 



law of density, A^-^^, have diminishing ellipticities, and if qh^ — 150° the above 

 b 



law agrees sufficiently well with the actual ellipticity and ratio of surface to mean 

 density of the earth. This law for e = |fi demands a thickness of crust of j the 

 radius, or 1000 miles. This is a minimum, since the actual surface of solidifica- 

 tion (lying between this and the corresponding isothermal surface) would have 

 greater (and hence too great) ellipticity. 



Before commenting upon this application, and upon the real meaning of tlie 

 formula, I return to the case of homogeneousness. Some of tlie results arrived at 

 by the analysis of Prof. Hopkins may be illustrated by the following considera- 

 tions : The fluid spheroid, treated of p. 36, is subjected, by the attraction of the 

 sun, to the distortion expressed by (47). This distortion, as shown by the form 

 of the expression, is equivalent to an exceedingly sliglit rotational displacement^ 

 of figure about an equatorial axis, such as would be caused by displacing through a 

 still more minute angle the planes of diurnal rotation. It is one of the beautiful 

 results of the analysis to show tliat the change in the direction of the centrifugal force 

 due to this slight obliquity of the planes of rotation is equivalent to turning forces at 

 all points of the fluid exactly proportional to their distances from the equatorial axis. 



Let now a rigid shell, exactly conforming internally to the external surface of 

 the fluid, be applied, and the Avhole turned back until the planes of rotation are 

 restored to perpendicularity to their axis ; the precessional efl'ect of the attracting 

 body now operates upon the whole mass ; for there are no longer counteracting 



tidal protuberances. If we take that part of (47) which is due to the direct action 



o a 

 of the sun, viz., — sin Q cos sin X cos ?. cos cj (for, the protuberances being repressed 



' <7 

 by the shell, the pressures on its interior which replace tliera will arise only from 

 the direct action), and estimate it as a pressure and calculate the elementary couples 

 for an internal ellipticity, e, we shall find the integral couple (and tliis corresponds 

 with Prof. Hopkins' result) to be identical with (48) viz., exactly that due to the 



' The required angle of the displacement is the height of the tidal wave (47), for o = 0, divided 

 (If 



by — for an ellipse of ellipticity, e, (2e sin \ cos ji). Prof Hopkins shows that the corresponding 



divergence of the planes from perpendicularity develops a couple = — « n'e multiplied by the sine 



15 



"4 S 



of twice this arc (or twice the arc itself). Performing the operations we get, ~-^ ne — - sin $ cos 0, in 



l.T 



which we have the solar conple (ID) and (4S), which causes the displnconicnt, since e = -: (C — A.) 



