440 DR W. PEDDIE ON 



pull exerted by all groups which broke first in the range £' — £ is 



Fkvtf - i)- r * . . 2nrdr - I TrTevaHaU 2 ) r ■ 



Thus the total outward force due to those groups whose breaking-range £ is less than 

 r<p is 



IvfotfiaWi , l 1V> = IwWYa V)!s . — , . 

 4 a m(m-l) 2 4 2 m 2 



The single force, equivalent to this, acting at a distance a from the axis, is 



\irlivd 2 (a 2 ^. i- (4) 



The outward force due to groups which broke first between and <p is 



J \h-(d - «£) . 2irrdr . vr(6 -</.) = hrkva\a\6 - </>) 2 ] . 



n 



Referred to a this becomes 



\wliva\a\6 - <£) 2 ] (5) 



The whole inward force due to unbroken groups is 



f (N - vr6) . 2irrdr . frr</» = 2TrM 2 {a<t>) \®~\ "(«#)] . 



J 



When referred to distance a this becomes 



27rJia 2 (a<f>) 

 The total inward force is therefore 



}*-£«]. • 



. . (6) 



1 7r/,:Na 2 (a<i) - -w&va^a 8 * 2 ) • 2^ - 5 nkva^aW 2 ) (7) 



2 5 1 m 2 5 



By comparison of the expressions (3) and (7) we see that token, in the inward motion, 

 tlie range is less than half its maximum value, the inward force is less than the inward 

 force at the same stage on the outward motion by an amount which depends only on the 

 square of the maximum range. 



I 



