Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 93 



v/(/3 - «) . (7 + sr) a I 2 \p o/j 



a (. V;) re/ 1 + a?'^ J 



Ti™ of decent. V^.{(|-t).f-2,.„-V'| + »}, 



ascent ■■ 



but tan-' V^ = tan-^ \/l + - . /3 = - + ^ ; 

 p O 4 4ffl 



.-. time of descent. Vf-jj* (i^-;)-/5) <"). 



....:.... ..ee„.= Vf.{|-(i-3.^) „.). 



Excess of time of ascent over the time of descent, or 



-v/!•(T-^■''=^/^•(ll-^■'^ <■=). 



which is remarkable as not involving tt. 



The excess of the arc or angle of ascent = 'y-/3 = -.i3^ = — - ■ B^ O*)- 



^ " ^ a SU^ 



These results are on the supposition that the displacement of the system was by an increase 

 of the independent variable ; in the opposite case, the odd differential coefficients of V and the 

 even ones of U must have their signs changed. 



Example. Two bodies m and m, , moving in a circle and connected by a rod subtending 

 an angle 4 a at the centre are acted upon by a repulsive force in the circumference, varying as 

 the n"' power of the distance. 



Let 2 be the angle at the centre between the radii passing through .S" the centre of force 

 and m, .-. 2 + 2 0, = 4 a, 



P = k .{9, a sin Q)", and p = 2 a sin ; 



U = 



Pdp + P,dpi U 



dd 



, or — — — —7; = m sin" cos — m^ sin" 0, cos 0,, 



mds' + m.dsl 

 V -j^^ =4re=(m + m,)- 



If the bodies are equal, V = Sma", 



(7, = 8 ma^k . (2 a sin a)"'' . (n cos'^ a - sin- a), 



and L = 



A; . (2 a sin a)""' . (sin' a-n cos* a) ' 



^ /L { Aa" r^ ^„ (w-l).(ra-2) 2n-2 -| 1 



g \ 256 L sm'' a M cos^ a - sin~ « J J 



where Aa is the whole angle of oscillation. 



