The Variation in Attraction Due to the Attracting Bodies. 207 



prolate spheroid. The coefficients, then, for eccentricity square, fourth 

 power, etc., for tlie ellipsoid can be correctly computed from a sphere. 



If we should represent the mass of the sphere by unity, and compute 

 the fractional masses cut from the sphere by rotating a system of chords 

 comprising an infinite number, extending from A to C and perpendicular 

 to radius A C, and multiply each of the fractions by its requisite sin- 3^, 

 the sum of the products would be the value for sin- 5 for the whole of an 

 ellipsoid. The same method is likewise true for computing the average 

 value for any power of sin 5. Let sin- ip-i, sin'* ipi, sin^ V'c, etc., be the 

 average values for sin'-^ ^, sin* 5, sin^ 3, etc., for the whole of a homegen- 

 eous ellipsoid with the attracted particle at the surface. 



By using certain expedients wonderfully abbreviating the just described 

 system of computation, I get the exact numerical values for sin^^'o, sin*^^, 

 sin^^g, &c., and also for cos ipi> cos-ip.^, cos^^g, &c. Which are: 



*Sin2z^'2= A. 

 5 



a. 7. 9 



2. 4. 6. 8 

 Sin^^'s = ^ „ g j^ , Sec. for higher powers. 



Cos^i=A. 

 4 



Cos2^'o=A. 

 o 



COS3^'3=A. ■ 



Cos* ^4= — &c. for higher powers. 



The expression for the attraction of the whole mass (M) of an oblate 

 ellipsoid on a particle at the pole reduces to : 



^Yi- .^..e;- +1e,*- Ae,^+ etc ") . 



W\ 5 ' 7 ' 9 ' / 



The expression, &c., for the prolate ellipsoid becomes: 



M:(l+|>E+AE.+ |E. + etc). 



T B A 



13. Let rj or g be represented by n. Since in a right angled triangle 



having a constant perpendicular while its hypothneuse and base increase 

 in length, the sine of the angle at the base diminishes as the hypotheuuse in- 

 creases, and as each and all the angles 3 that make up the angle ^'j are 



*How these results are obtained is fully explained in the investigations 

 on the subject referred to at tlie beginning of this paper. 



