206 Wisconsin Academy of Sciences^ Arts and Letters. 



Angle C a P equals (90°-a), and angle a C P equals 90°-(3-z). Let 



C P equal D. Then in triangle C a P, Pa=D ^"^ (^-^)- 



cos a 



Attraction of mass (m cos 3) in direction Pa on P = y <^os jj ,cos^ a 



D^ cos^ P-z) 



.,, ,. „ • J- i- T^ ^ m COS 5 cos (5 + y) cos 2ar 



Attraction of mass in direction P C = j^^s v -t j / ^yj!= 



D- cos^ (-S'-z) 



In triangle C c g, Diagram 3, 



A cos 5 

 cos (S-z) = -^ 



In triangle C B, g, Diagram 3. 

 B cos 5 



cos (S- + y) 



B, 



M A,** B cos hx 



Attraction, etc., = "n2~X a2 r 



A ^ gs 



A = B; cos- a = A^ B^ or -^.-^-=^„ ^ 



' A* B,- cos ^ a 



111 Ti^ 



Attraction of mass (m cos 3) in direction P C on P =-yp^X"^rT 



In case then of an oblate ellipsoid, 

 Attraction of mass (m cos 3) in 



Likewise for a prolate ellipsoid, 



na' A^ 

 Attraction ot mass (m cos ^) etc., =^pry^x^xT 



■R3 133 3 



:^=^ |=(14-E,-sin25) ^ 



B/ B\l+E>in^3) 



(l+E;sin25)~W-- E/sin-3-f— E/sin^ - ?:He «sin«5-f-etc. 

 2 2.4 2.4.6 ' 



-:^=(1-E=sin25)~ W 4- ?E2sin25 + 5:^ E ^ sin ^ ^ + ?i--lE«sin«:& + etc. 

 A,3 2 2.4 2.4.6 



For oblate spheroid, 



Attraction of mass (m cos5)&c=— (l-^_E/sin'^ + ?l^E/sin*3- etc. 



D--'' 2 2.4 ' 



For prolate ellipsoid, 



Attraction of mass (m' cos3)&c.=^(l+ ^ E^sin^^^- ^i^E-'sin^j^+etc, 



D;^ 2 2.4 



12. In the above expressions for attraction any power of E or E, is a 

 constant for any part or the whole of the same ellipsoid, while any power 

 of sin 3^ Taries from to 1. In case of the oblate ellipsoid having a constant 

 semi minor axis (B) mass (m) varies as A^, or as the mass of the whole ellip- 

 soid till the ellipsoid becomes a sphere. The same relation is true of the 



