404 Professors Perry and Ayrton on a neio 



the surface of the sphere ; and the force in the direction z is 

 the same as would be the force in the same direction due to a 



P + m 2 /PS 2 \ 

 distribution having a density aw or aw ( n). 



Regarding the force as being due to such a distribution, 

 CCawdS PS 2 n_ _CC(Twnd& n^ 

 J J PS 2 ' n 'PS J J PS 2 'PS 



Now the first integral is 1 1 p ^ , which we know has a 



value equal to the potential inside the sphere due to a uniform 

 distribution of density aw over the surface, and is therefore a 

 constant, Airaio. So that the entire force in the direction z is 

 4:7raw minus the force in direction z due to a distribution of at- 

 tracting matter of density naw over the surface of the sphere. 

 Now it is easy to show that a distribution of attracting 

 matter of a density proportional to n or to A + C cos over 

 the surface of a sphere will give 



X=0, 



Y=0, 



Z = a constant ; 



therefore all that is necessary is to determine the value of this 

 constant. We neglect the term A, because a uniform distri- 

 bution produces a constant potential, or a zero force in all 

 directions ; the distribution C cos being a zonal harmonic, 

 produces a potential inside the earth, 



V = —5— r cos u 1} 

 6 



47r n 

 = T Cz; 



dV Air 

 so that the force Z which equals -7- is -^- C. Thus for the 



47T 



distribution C cos we have the force -5- C ; so that the con- 

 stant force above mentioned, Airaw, requires the distribution 

 Saw cos 0. From this we must subtract the distribution naw, 

 or aw (cos 0—r cos 0i), giving us for the total distribution of 

 attracting matter over the surface of the sphere a density 



2aw cos + awr cos 0\ ; 

 but the latter term means a uniform distribution, producing 

 therefore no internal force, and may therefore be neglected. 

 And the first term is a zonal spherical surface harmonic; there- 



