228 Wisconsin Academy of Sciences, Arts and Letters. 

 The expression for the whole ellipsoid becomes: 



M 



Att, = -Y^(l— I n- E'^ cos- 0+x% n-^ E- sin- O— ). 



M 



M= :, — =M (1+* E' sin- O cos- 0-|-J- E« sin- O cos* O+f W sin' 



cos- tr^ ^ ' - 



O cos- 0+„). 



M 



b. Att.=-j^(l— I n- E- CO.- 0+f^ n'^ E- sin- O— f n'^ E* cos* O+f n' E* cos* 



O— |y E* sin'2 O cos- O+f n- E* sin- O cos" O+f g- n* E* sin^ O- 

 O cos- O+J. 



Att. 



M 



=-jy^\ 1—1 n- E'^ (1-1 sin- O)— f E* (n^— f n*+f sin^ 0-f sin* O 



—4 n- sin^ 0+3 n- sin* 0+if J n* sin- O— |f n* sin* 0)+„] 



23. To find the increase ia attraction in passing on the surface of an 

 oblate ellipsoid from the equator to the poles. 



When the attracted particle is at the surface of the Ellipsoid n equals 

 unity and D equals B,, and expression (b) Art. 22 becomes 



Att =~ (1-f E^' cos^ O + -iV E'^ sin'-^ 0-„). 

 B" 



b;^=a'^ (1-e-^ cos2 o). 



Att=— (1+t E-* cos- 0+A E-^ siu-^ O+g^ E^ cos^ 0^-/^ E^ sin^ O + 



A'2 



— - E* sin'^ O cos- O +,) 

 175 



(a) Att = -,(1+ f E^^- iV E-2 sin- 0+ /^ E^-jW E* sin= O-^^o- E^ sin" 0+ J 

 At the poles angle O becomes zero, and 



(b) Att =^, (1 + I W +|ii E* + etc) 



At the equator angle O becomes ninety degrees, and 



(c) Att =^0 C 1 + ^ E-^+ lii W-\- liM Eo+etc . ) 

 ^ -^'V 10 ■ 7.2.4 9.2.4.6 / 



Subtract expression (a) from (b) and the difference is, 



=A. [s " -»' ° + '^' ="' ° {m + 5-0 ='°' °)+ ■ ]■ 



The increase in attraction thea from the equator to the poles varies as 

 the square of the sine of the elliptic angle, exact for the second power of 

 excentricity or first power of ellipticity and very nearly true to the sixth 

 power of eceentricity or the third power of ellipticity. 



