Arrangement of the Mass of the Earth. 345 



graph, and neglect small quantities of the second order ; 



.-. const. = -(l- e(^-^) ) + ^ + -jf +.. . . + 2 -(l-/^ 2 )- 



Equate Laplace's functions of the same order on both sides, 

 observing that \-yfi is one of the second order, and 



a. const. =E + ^mE, W 1= =0, W a =Ea«(e-jro)(£ -/*«), 

 W 3 =0, W 4 =0, .... 



v E , / m\E« 2 n 9 



From this formula is derived an expression for the value of 

 gravity, which 



= -, cent, force, == , m -§ (1 —v?) ; 



dr dr a 2 v ^ ' 



and therefore by pendulum experiments the value of e may be 

 found. 



This formula for V is not so general as it appears to be. It 

 is true only for a certain arrangement of the mass E; for it has 

 been obtained upon the supposition that a part of the mass at 

 the surface (it may be to a very small depth) was once fluid, or 

 is arranged as if so now. Through this portion of the mass, 

 then, the arrangement or law of density must be according to 

 the fluid law. From the manner in which E enters into the 

 above expression for V, viz. as the total mass, without any appa- 

 rent reference to the manner in which that mass is arranged, it 

 may be thought by some that although the exterior part follows 

 the fluid law, the other parts may follow any law, so long as the 

 total mass is the same*. That this is an erroneous view I pro- 

 ceed to show. The equilibrium of a fluid in a vessel is unaffected 

 by the form or density of the material of the vessel, unless we 

 take into account the attraction of the parts of the vessel. This 

 is the case with the earth, the solid part of which may be regarded 

 as a vessel holding the portion which is fluid by hypothesis, and 

 altogether regulating its form by the attraction of its several 

 parts. If the whole were fluid, the equation of equilibrium with 

 which the calculation begins would appertain directly to the 



* It is the fact of my using the value of V above given in two papers in 

 the Philosophical Magazine for December 1862 and Supplement, giving 

 Tests of the Fluid Theory, which has led to my writing the present paper. 

 As E enters into V without any apparent condition as to the distribution 

 of the mass, it might be thought that the changes of that distribution 

 which I suggested as tests ought not to have produced any effect on the 

 pendulum ; which would be directly contrary to the results of my papers. 



Phil. Mag. S. 4. Vol. 26. No. 176. Nov. 1863. 2 A 



