On the nearly Spherical Arrangement of the Mass of the Earth. 343 



the point on the surface is 



f 1 C 2 " C p'r 12 dfJ day' dr f . 

 J-iJo Jo \r* + r'*-2rr'p\* 

 where 



P=jjl/jJ+ v/l— /jL q \/I—fju !2 cos(<a — co'). 

 By expansion this becomes 



r i p2iT /V CM Ji+2 -} 



Y= J_ J. Jo p ' l^ Po+ - • • + 7* P,+ • • ' J***'"*'' 



where P . . . P f . . . are Laplace's coefficients. 



Put p'=R/ + /3. U 7 , where R/ is a function of r' only, inde- 

 pendent of fJ and co ! , and U' is a function of all three, r', fJ, co 1 ; 

 /3 is a constant. What I am going to prove is, that /3 is a small 

 fraction of the order of a. Suppose that 



f V«+ V^= 0(rO -f /3f{r',fjJ,co f ) = <f)(a + uu ! ) + £^(« + aw VX) 



=A + "B(m' + . . . +<+ . .) +£(^0+ • • • ++H • • )> 

 these being series of Laplace's functions. Then by a well-known 

 property of these functions, M being a constant, 



v= f , JT {? m+ • • • + &i ( aBM '<+w) + • -}^'» u > 



- ? M+ • • • + (3iTT)^ &**+*« + • • • 



by another property ; U{ and ^ being the same as the same 

 functions with an accent, fi and co being put for juJ and co'. 



Now gravity is the attraction of the earth's mass diminished 

 by the centrifugal force, which is always extremely small. Since, 

 then, gravity very nearly equals the attraction of the mass, and 

 its direction is very nearly towards the earth's centre (being a 

 normal to the spheroid of equilibrium of small ellipticity), it 

 follows that 



r d/n r*y\—ij? dco 



which are the attractions in the meridian and prime vertical at 

 right angles to r, must both be very small quantities. Hence 



aB du L+ dp daB *j pdp 

 dfju djj, dco dco 



are both very small, and this whatever values p and co have — 

 that is, for all points on the earth's surface. This cannot be the 

 case unless /3 be a very small fraction, of the same order as a. 

 Hence the term in p ! which depends upon ^ and co ! is very 



