Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 95 



and r = . V-.(t +^-. L^(^.qT:;^„.) + r6^ 4(/rQ-^ay)J |- 

 If the planes include a right angle, and the centre of gravity be in the middle of the rod, 



L = a + ~ , 

 a 



both of which are independent of the inclination of the planes to the horizon. 



If a particle move in a curve, by a constant force {g), making a given angle rf> with the 

 axis of X ; then, 



dp cos (bdx + sin (bdy . , „, ds 



""-^Tu-^- du -^•-'"('^-^^ri7.' 



where 9 is the angle made by the normal with the axis of a; ; and making this the variable, 

 V = R', where R is the radius of curvature ; 



F, = ZR] + 1.RR., 

 U = g . sm {<p - 6) . R, 



U, = ~ g cos ((j) - 6) . R + g . sin (cp - 6) . R„ 



U.^ = - g .iin ((p ~ 6) . R -2gcos (cp -9) . B, + g . sin ((p - 6) ■ R^, 

 U3=gcos{(p -9) .R - 3g sin {<p - 9) . R^ - 3g cos (cp - 9) . R^ + g . sin (<p - 9) . R, 



In the position of equilibrium U = 0, or (p — 9=0; 



■■■ U,= -gR, 

 U,= -2gR„ 

 U,= -gR- SgR,. 



gV 

 Hence L = — rr-= R, and 



— t/i 



T= TT 



q o Art 



Excess of time of ascent = ' . 



sVgR 



Excess oi angle or ascent = 



3R 



If the arc be made the independent variable, 



--'^^^{■-'■(;^-1^-iS)l <")■ 



i/Xcess of time = - -7=^ ■ 



3VgR 



-- - - /?, . As-^ 



lixcess of arc = = . 



aR 



