142 Proceedings of the Royal Irish Acadein//. 



When there is no heterogeneous earth, so that the matter is entirely 

 liquid, we integrate from to a in the first of the steps by which we derived 

 (3) from the differential equation, and from a to a in the second step, and so 

 combining the result with (1) we obtain the single condition 



{2n+l)\, 



M-da — - 



■ , ^ -, (2n + l)k 

 pda"^'h - — — = 0, 



on the assumption, however, that ah = when a = 0, as it must. 



The last condition is of course the same as that obtained by putting a = a 

 in the condition (1), and is the condition used by Laplace in the case of 

 matter entirely liquid. 



From the conditions (3), we readily derive some general theorems, which 

 I proceed now to state and prove. 



Granting that the axis of rotation must pass through the centre of gravity 

 of the whole mass, we can take the origin to be that point, and can show that 

 it must also be the centre of gravity of the volume of each equal pressure or 

 density surface in the liquid, and also that it is the centre of gravity of the 

 solid earth on the supposition that the density of the earth is diminished at 

 every point by the density p^ of the liquid next to the earth. Thus the 

 position of this point through which the axis of rotation must pass is known 

 when the law of density of the covering liquid is known ; it depends only on 

 the density of the liquid next the earth ; and no matter what this density is, 

 the point lies on the line joining the centre of gravity of the solid earth to the 

 centre of gravity of its volume. 



The proof of this theorem is as follows : — The condition that the centre of 

 gravity of the whole mass may coincide with the origin is easily seen to be 



fa 



pda^h = 0, by which we mean 



pda^n + p,h\K - ^o) + /orfftVi = 0, 



when in it we substitute for h, x] any one of the values hi, h^, or h^ of li, with 

 a corresponding value »ji, r/o, or ijg of r/, where hi, h^, Ih are the unknown 

 coefficients in Yi for the liquid, and rji, rj2, »;3 the known coefficients in Yi for 

 the earth. 



Thus combining the above, which may be written 



-h /'a 



{p - po)da^ii + pjj^h'o + pda% - 0, 



Jo J'-* 



with the result of putting n = 1 in the equations (3), and remembering that 



