Fry — The Centre of Gravit//^ Sj'c 148 



the three values of k corresponcUng to the three values of h are zero, we 

 obtain that /t,, Ih, li^ must each satisfy 



fa hh + h r ^ 



pdah - — ^^ ^ pcthlcc = 0, and pcla^h - h{bhj - 2h^) 



pct'da = 0. 



Thus, if h = Afi + Bfi is the general solution of the differential equation 

 for the ease n = \, the arbitrary constants A and B are determined by 

 equations of the form 



K,A + K.,B = 0, K(A + K:B = 0, 



where K^, K/, K^, K/ are constants depending on the mass of the earth 



pci'da, h, a, and the law of the density of the liquid. Hence (except 



for specially arranged values of these latter quantities) it must follow that 

 A=B=Q. 



In this case oi n = 1 we might also have made use of the fact that A = -. 

 so that Ki = and the first equation reduces to K2B = 0, so that ^ = 0, and 



lb 



hence A = unless for special values of pa~da, h, a, and the law of density 



J 

 of the liquid. Thus, 



hi =^ hi = hi = ; .•. yi = 0, and 



(p - p()da^y] = for ^ =^ rj,, or rja, or 173; 



therefore the origin is the centre of gravity of the earth on the supposition 

 that the density is diminished at every point by p^ the density of the liquid 

 next the earth. Also Fi = is easily seen to be the condition that the 

 origin should be the centre of gravity of any surface of equal pressure or 

 density in the liquid. Accordingly, all the statements made above have 

 been proved. 



Again, if we neglect the action of the distant spherical bodies, and 

 assume, as must then be the case, that the axis of rotation is a principal 

 axis, it follows that it and two perpendicular axes are the principal axes 

 for the total mass at the centre of gravity of the whole mass. We can 

 now prove that these axes are also principal axes for the volume enclosed 

 by any surface of equal pressure or density in the liquid, and that they 

 are principal axes for the earth on the supposition that its density is 

 diminished at every point by the density of the liquid next the earth. 

 Thus given the density of the liquid next the earth, the axis of rotation 

 must be one of three known axes. 



