The Variation in Attraction Due to the Attracting Bodies. 353 



raatical expressions are attainable, easy to integrate, while 

 by the old method complex expression are unavoidable, re- 

 quiring the ability of a Legendre, a Laplace, or an Ivory to 

 integrate. 



In Article 12 I have gis^en the fractional numbers for in- 

 tegrating the chord elements of attraction for an ellipsoid. 

 If every man and women, from Adam and Eve to the last 

 born of their posterity, had been engaged in the work of 

 making the computations for these fractional numbers by 

 the method outlined in Article 12, the end in view would 

 now be unattained. Such being the impossibility in the 

 way of that method, I Avill now give my other method in 

 brief. It is as follows: 



In diagram 1, Art. 3, let particle P be at the surface of the 

 sphere having center C, in line P C of the diagram, then 

 secant line P d a g becomes chord Pag and equals to chord 

 k 1 h or 2 r cos ^. Likewise chord p n o equals 2 r cos ^. 

 Let radius A C perpendicular to the axis of rotation be di- 

 vided into n parts by a system of chords drawn parallel to 

 axis P C, and also let these chords vary in length from chord 

 to chord by a common difference 2 r divided by n. Let lines 

 p n o and k 1 h be any two adjoining chords of the sj'stem. 

 The mass cut from the sphere by a rotation of these chords 

 equals I .t r' (cos'S^ — cos'-^,). As the increments to the 

 cosines used in this computation vary from cos. to cos. as 2 r 

 divided by n, cos. ^ may be represented by any simple va- 

 riable quantity (y). When n becomes infinite or 2 r divided 

 bv n, infinitisimal, then the differential mass or, 

 dm = ;f 7f r" 3 y- dy. 



Multiply each chord element of mass making up the 

 sphere by its requisite cos ^, cos'-^, cos'-^, and so on to cos. 

 having an exponent infinite. It is now required to intigrate 

 the expressions y d m, y-' d m, y' d m, etc., between the limits 

 zero and unity. 



jlydim = \7t V' fl 3 r d y = A TT r« X I 

 fl y-dm = | 7tr^ Jj 3 y' dy=| tt r^ x f. 

 Jl y'' d m = J TT r» J ^ 3 y5 d y = | tt r^ X f . 



