218 



c/»e*+='^ 



+ 



PHILOSOPHICAL TRANSACTIONS 



cg'e* + ^? 



+ 



+ 



[anno 1738. 



\ +p X3+P l + 9X3 + g 2 + J» + ?x3+f 2 + p + gx3+</ 



+ =- 



+ 



+ 



1 +p X 3 +;> X 5 +p \-\-qx3 + qxS + q 2 + P + ? X 3 + p x 3 + 5 



+ 



8c/g«c 



«+^ + f 



+ 



2 + ;) + ?x3 + gx5 + 9 

 8 4.4;>cg/'Ae^+^ + ? 



+ 



4 + ^pcfye"- + ^Z" 



+ 



l+px3+px5+p 

 8 + 4?c^Ae*+''+? 



+ 



4 + 2qeg'^>.e* + ? 



3 + ?x5 + ?Xl+/> 

 for the total gravity of 



S + p + q X 3 + px 5 +p i + p -\-q X 3 + q X 5 + q 



any column cn, having regard only to the attraction. 



29. If in this expression we make a = O, we shall have the gravity of the 

 column at the pole. 



30. And if we make x := «, we shall have the aggregate of the attractions of 

 the column at the equator. 



31. Now because the column on is in equilibrio with the column cb; it 

 follows from thence, that if we subtract the weight of the column cb, from 

 the aggregate of the attractions of the column on, the residue must be equal 

 to the sum of the centrifugal forces of the column cn. Now to endue our 

 spheroids with this property, we will resume the expression of the centrifugal 

 force in e, which we found art. 14, which will give 



( ^^^ — ^ + z==^-z= — ) - , for that part of the centrifugal force which 



3 + px5+/) 3 + ?x5 + ?e 

 acts according to cm, in any place m, by expunging the terms in which aa 



would be found. This value being multiplied by r, and by the density, will 

 give, when we have taken the fluent, ^^ ' =^ 



+ 



8c/ge'+^ + ?A 



+ 



2+px3+/)X5 + p Z+p X 3-\-q X 5^ q 



cge^ £qj. jj^g gm^ Qf jjjg centrifugal 



2+?x3+px5+p 2+qx3+qx5+q 



forces of the column cn, still expunging those terms in which either met or M 

 are found. 



mu , • .u- • W I+ipcA^+'^A , 8 + lpcfge^+f + f>. 



Then makmg this expression equal to =rrt — ~ — ' —- 



l+pX3 + pX 5+p 2+p + qx3+px5+p 



+ __EI} ^<fg'"'^L^!^_ + rr7gcg^e^+^_^ ^j^j^j^ jg ^^g difference of the 

 2 +p + 9 X 3 + ? X 5 + 9 1+9x3+9X5+9 



weight of the column at the pole cb, from the sum of the attractions of the 

 column CN, we shall have the equation 



PPff_ , 2pg/g 



1 +px 2+p X 3"+7 X T'+p T+P+ qx 3 +p X 5+p X 2 + q 



