234 Wisconsin Academy of Sciences^ Arts and Letters. 



ured from the polar axis. The following is au equivolent expression with 

 the elliptic angle (O) ni.asured. from an equatorial radius: 



Att.=?ir 1+^ h— ^ h cos^'04-- h^— — '1- cos'^ O— ilv' cos" O-fh^ (") +"1 

 A-'L 5 5 35 350 50 J 



Let a represent an equatorial radius varying from O to A, then for mass 

 interior to any liyer M varies as (a) cube, hence int-'rior attraction varies 

 as a. For iuterior attraction a can be substituted for M divided by A\ 

 h in the above expression for attraction equals A — B divided by A. To 

 put the expression in shape so that h may represent unity divided by A, 

 the constant terms l+|h-fi|h-l-etc., so far as used in the computation 

 must be made unity. The same result is obtained by not making this 

 change, providing due allowances are made for A — B not being equal to 

 unity. 



Att = a [1-1 h cos^ O — iV_y h" cos^ O — i^ h^ cos* O — h-^ („) - J. 



The difference in attraction for any point on the polar radius and same 

 layer point on the equatorial radius is: 



{i 



h + 3-5^'^+35h'^+'J 



When the rotating fluid body is in equilibrium the mass from a layer in 

 any equatorial cone must just balance the mass from same layer in either 

 of the polar cones. As these masses vary as their distances from the cen- 

 ter of the body, gravity in the plane of the equator and in the polar axis 

 must vary inversely as same distances, and the centrifugal force or repul- 

 sion fr im rotation in the plane of the equator must, in case of equilibrium, 

 be expressed by, 



a. h— a 



• 16 \ /4 6 X 



(5^^+3-5l^'^+«) = ^(-^-35l^-^-") 



The repulsion from rotation at any point in or oi the rotating body in 

 direction parallel to the equatorial plane is per demonstration (Art. 25), 



/4 6 4 \ 



Eep. = a cos O (^g- h- 35 h^^g-g h^- „ ) 



Gravity is the third sid^of a triangle in which the two sides (attraction 

 and repulsion of rotation) and the included angle [(O— z)-|-(<^'— w)] are 

 given. The third side or 



Gravity == I Atty-^+Rep.--2 Ait.XRep.XCos [(0-z)+(a-w)J ^ ^ 

 Gravity = a Fl-h cos'^ O+^lV^ cos- 0-+ h^' cos" 0-l-hs(„)-,, 1 



