344 Archdeacon Pratt on the nearly Spherical 



small ; and therefore p* is a function of r 1 + a , 

 some small function of p! and co' ; and /-faV = constant will 

 be the general equation to the layers of equal density ; and these, 

 then, are all nearly spherical, because u . v 1 is small. 



II. If the exterior surface be a spheroid of equilibrium, the 

 layers are also spheroids of equilibrium, and follow the fluid law, 

 whether they are or have been fluid or not. 



3. If the surface is one of fluid equilibrium, then 



constant = V + iw 2 (l-^V=:V + f - 3 (l-/* 2 )r 2 , 



& a 



where m=Q^-§, E= the earth's mass. 



By differentiating the value of V first given in paragraph 1, it 

 is not difficult to show that 



dfiV 1 * Idfi J-T l-> a dm* " 



dKrV 

 T dr* * 



Let V be expanded in a series of Laplace's functions, and terms 

 of the ith order on the two sides equated. Then remember- 

 ing that by Laplace's equation the first side will be equal also to 

 — f(t + 1)V« we have 



/7 2 rV 

 r-^—iii+DV^O. 



The solution of this is Vt=W i r _i + Z i r i , where W f and Z; are 

 independent of r. The complete value of V becomes 



Now V must evidently equal zero when r is infinitely great : 

 hence Z =0, Z^O, Z 2 =0, . . . Also the greater r is, the more 

 nearly equal are the distances of the attracted point from the 

 various points of the earth's mass ; and ultimately as r is increased, 

 the potential must become E~r. Hence 



V= B + W 1+ W 2+ 



The first term of this is the value V would have if the earth's 

 mass were arranged in exactly spherical shells. This is not 

 exactly the state of the earth, but is very nearly so, as I have 

 already shown. Hence W v W 2 .... are all small quantities. 



Let r=a(l-\- e(i— yu, 2 )) be the equation to the ellipse gene- 

 rating the spheroid of equilibrium of the surface. Put this and 

 V in the equation of equilibrium at the beginning of this para- 



