Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 101 



(R-rf 



dcp^ iR::7y +3«i',- ^^.^^ 



d(p ^'d(p) 



{RKr,-r^RO. (i?, 



(R - ry 

 R'r,-r'R, (R'r,-r'R,Y 



(R - r)' "^ ■ (R - ry " 



If then, a be the angle, the common normal makes with the vertical in the position of equili- 

 brium ; since «=«,, there results, 



dy, Rr 



sin a . 



d<p R -r 



d'y, Rr' . R^r^ - r^'R. 



dx, Rr 



= cos a ■ 



d(p R-r 



d'w, . Rr^ R^r, - r<R. 



<P~v, Rr' . „ R'r.-r'R, . R'r,-t^R, 



TJi = - cos a . -—: - sin a . it r . — — ; 2 sin a . r . 



d0' (R- ry (R - ry (R - ry 



R'r^-r'R, (R^r^-rR^y 



+ cos a. —rn TT — + 3 cos a . 



(R - ry (R - r) 



which values being substituted above, we have finally 



Rr 



A'l = o , JCi = — sin a . 



R^r' 



3 ' 



F, = , Fa = cos a -= a , 



R-r 



„ R -2r . R'r^ - r'R, 



JL^-cosa. Rr . —- - a - sin a . —7^ ^ 



{R - ry (R - r) 



R-2r R'r,-r'R, 



K. = sin„.i?r.^-^— ^ + cos„.-^^— ^, 



Rr „ „ R-2r . R'r^ 



r, = a- cos a . y, + Rr- cos a . 77; r-, + sin a . ■ 



R-r (_R-ry {R-ry' 



r'R,-R'r, R'r.,-r'R^ (R'r, - r-R,y 



+ 3 r . sin a . -— —- + cos a . ——- — — + 3 cos a . ^^ — -— — i^ . 



{R - ry (R - ry {R - ry 



Length of pendulum = ^... (24), 



cos a . -= a 



R-r 



