212 Wisconsin Academy of Sciences., Arts and Letters. 



sin* ^ =--- cos 2 5+- cos= 2 S-. 



2'^ 2 2' 



cos* 3= - + -COS 25+ - cos= 25. 



2- 2 2^ 



2 2^ 2' 



The first term ( - I of the above result is true for any value given to 5 

 from 0° to 45°; therefore - is the average for that term. It is now required 

 to find the average of the succession of terms, - cos^ 2 5 + - cos'' 25, + - 

 cos 2 5„ + etc., to 5 for 45°. When 5 becomes 45°, 2 5 becomes 90° ; therefore 

 these terms can be put in shape as follows: - [cos^ 2 5+ cos- (90°— 2 5) + 

 cos" 25, + cos- (90°-25i) + cos^ 2 5 g + cos'^ (90°-253)+ etc., to 25 equal 



45°], divided by the number of terms. The average value, then, for cos^ 25 

 is -J. The average value of 



1 +^„ cos'2 25 = i + i X i = ■& = A 

 2- 2- 2.4 



By like computation the average value of 



Sin«5 + cos«5 1.3 „ . . 15_3.5 



"48~2.4.6" 



24 6 8 10 ' ®**^"' ^'^^ higher powers. 



For each of the infinitisimal wedges of an oblate ellipsoid, mass and dis- 

 tance squared are constant factors. The expression then for the attraction 

 of an oblate ellipsoid on a particle outside in the plane of the equator is: 



(l+^X^n=E^ + 2^x|a^E^ + etc.). 



M •. . 1 .3 

 D 



For a prolate ellipsoid, etc. : 



M' • . 1 3 „_ „ 3 3 



V ( ^—ixr^'-^2A^~,^'^' -'''■)' 



