96 arR. holditch, on small finite oscillations. 



This last result compared with the former, shews that an increase of the angle of vibration is 

 attended with a diminution of the arc, and vice versa. 



Example. In an ellipse, R = — j- ^—. — — -j > 



'^ ' n (1 — e sm m)8 



/?_ = ^^.sin0.cos0.(l -e'=sin'6»)-i, 

 a 



R^ = £i^ . (1 - e' sin= 9) -* . (cos= 9 + e" sin' 9 - sin^ 9 + ie" sm' 9 . cos= 6), 

 a 



and by substituting these values in (18), 



-'Vf{-f[— ©'])■ 



If the ellipse become a circle, R' = ah, 



and r=. V^.fl + ^]. 



If the axis of a cycloid be inclined at 9 to the vertical, 



R - 2a cos 0, 

 i?, = - 2o sin 9, 

 R^ = —2a cos 9. 

 L —2a cos 9, 



/ 2a COS. 9 f _A0Vtan^\ 



-TT V ^ • (^1 j2 j ■ 



o 

 Increase of angle of ascent = - - . tan 9 . i\9'. 



The time of oscillation in a cycloid therefore decreases, as the arc increases, when the axis is not 

 vertical. 



If a central force kf(S), varying according to any law act on a particle in a given curve, 

 the co-ordinates of the centre of force being a, /3 ; then, taking the arc for the independent variable, 



Let ^^ = «, and •M=0(^); 



.-. U = k.(f>(x).zi = 0, at the position of equilibrium, 

 Ui =k.<l}(x).z2, 

 Us = k.(j)(z) .Xs, 

 U3 = Sk(f>i (xr) .zl + k(j){z).Zi; 



but z = (w-aY+ (y-/3'); 



•■■ -1 = 2 . (a? - a) .— + 2 . (y - /3) . -p , and if 9 be the angle made by the normal with the axis 



c '^^ ■ n , dy 

 of .V, -— = sm9 and — ^ = cos : 

 as ds 



.: «, = 2 . (,i' - a . sin + 2 . (y - /3) . cos 9, 



