Fluidity of the Mass of the Earth. 275 
It is this last statement which I controvert. In my paper 
in your Number for May last (page 329, line 23), I assert that 
equation (13) “assumes that the law of density and ellipticity is 
continuous throughout the whole mass, solid and fluid, the solid 
parts lying in strata of the form and density they would have if 
they were wholly fluid.” 
38. Professor Haughton, in your Number now received, replies 
to my reasoning by showing that he has differentiated equa- 
tion (12) right. This I never called in question. ‘The ques- 
tion at issue ” between us is not, as his “ mathematical friend ” 
states, “to determine a rule for differentiating this equation 
[viz. (12)] with regard to a.” What I assert is, that certain 
terms of the differentiated equation will not cancel each other 
so as to produce equation (13), wnless we make such an assump- 
tion as involves this principle,—That the same law of density 
and ellipticity belongs to the solid and fluid parts. This I will 
now show more fully. 
4. I would first, however, observe that equations (12), (13) 
apply equally to the solid and fluid portions of the mass. Equation 
(12) expresses the law, that the resultant of all the forces acting 
on any particle is at right angles to the layer or surface in which 
the particle lies. This law is essential to the equilibrium of the 
fluid part. It is also tacitly taken to be true for the solid parts 
by Professor Haughton. For he differentiates equation (12) 
with respect to a, and therefore assumes that equation (12) holds 
at the immediate neighbourhood, on both sides, of the surface to 
which a belougs. In the case in which this surface is the 
bounding surface between the solid and fluid parts, the mass is 
solid on one side and fluid on the other. Hence equation (12) 
applies to both the solid and fluid portions. 
5. To banish the integrals from equation (12) and obtain 
equation (13), we must multiply by a’, differentiate with respect 
to a, divide by a‘, and differentiate again. The result is equa- 
tion (13). The first differentiation produces, from the second 
= ; and, from the third term, a term + }pa® = 
It is assumed that these terms cancel each other. So also the 
second differentiation produces, from the second term, a term 
term, a term—jpa° 
+S; and, from the third term, a term ep These are 
assumed to be equal to each other ; that is, & is assumed to 
have the same value on both sides of the surface of which a is the 
mean radius. This will be the case when this surface is one of 
the layers wholly within the fluid, or wholly within the solid 
part, even though the laws of density and ellipticity of the fluid 
