144 



Proceedings of the Royal Irish Academy. 



Taking 



Y^ = lix[ir - y) + \/\ - fJL^ . fx (h cos (j) + Ih sin ^) + (1 - /.r) {hi cos 2(p + A5 sin 2^) 



in the liquid witli a corresponding meaning for i/i, jjo, jja, j/^, 7j5 in the solid 

 earth, the condition that the axes should be principal ones is easily seen to be 



by which we mean 



pda/h = 0, 



pda'i] + pj)' (A-o - j?o) + 



pda/h = 0, 



for the three cases when h and jj = Jh, rj^ or A3, 773 or h^, i]„. Combining 

 this result with the conditions (3) for n = 2, and remembering that 

 \, h, ^4 and Jc^ = 0, we get that A2, h, and Ih must satisfy the equations 



pda% - 6^ {hJi/ - oJiq) pcrda = 0, 



,dh 



hh; + 2Ji, 



pa^da = 0. 



Hence, if h = Afi + Bf is the general solution of the differential equation 

 for K when w = 2, A = B = ^ when A = Ao or 7^.3 or A 5, unless for special 



pa-da, h, a, and the law of the density of the liquid. It now 



values of 



follows that 



(p ~ Po)f^'^^^'J = 0> when r] = i]2 or 7/3 or j/s, or that the axes 



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. 

 Also, as Ao = h^ = A 5 = 0, it foil ws that the axes are also principal axes 

 for the volume enclosed by any surface of equal density or pressure in the 

 liquid. 



In a similar way it is now easily seen that the condition that the term F„ 

 should not appear at all in the equation of an equal pressure surface is 



ip-p,)da^^''Y^ = 0. 



All these results are easily proved directly for the special case when the 

 liquid is homogeneous. 



These theorems, as to the centre of gravity and principal axes of a surface 

 of equal pressure, are proved by Laplace only for the special case of the 

 external free surface. 



