Mb. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 103 



r„^/:J.{,..,..(,-«-^.±.l.-)}. 



If a body be suspended by an axis whose radius is r on a circular support, whose radius is R ; 

 and ffl, be the distance of the centre of gravity below the point of support, 



k' + a^ 



^= Tr ' 



cos a ■ T=, -(- a 



R - r 



and, if the pendulum be suspended on another axis, the radius of which is r, , and be isochro- 

 nous with respect to these axes ; 



L = 



Rr, Rr 



cos a ■ -rr cos a . 1- o, - a 



R - Ti R - r 



a' - a? 



and if the axes are equal, L = — = o, + a , 



«! — a 



1 13 r . Rr cos a 



and k' = (a + a,) . — + aa^, 



R — T 



and therefore if Kater's pendulum be supported on a concave or convex surface, the length is 

 independent of the curvature of the surface. 



Rr 



1{ A = — , and it rests on the first axis, 



R ~ T 



^" " ^ ^- r ^ ^ ■ lie ^ 4rZ. (./ + a) -^ —l6?-(A+V)-^ir 

 which is not independent of a, unless R is infinite, and therefore A = r, in which case. 



On Sliding Bodies. 



When a body oscillates by sliding contact on a horizontal plane, X and its differential 

 coefficients vanish, and by (15), 



The equation (2) becomes Y = w sin Q + {a - y) . cos 9, since 0, = 0, and y, = 0, and .-. (p = B ; 

 = x . cos 9 — {a — y) . smd + 

 = <v . cos 9 — (a - y) . sin 9, 



(if di/ 



.-. Y^ = x . cos9 — (a - 2/) . sin + sin . —^ - cos 9 .— ^ 



cl9 du 



dsc (111 



Y.^= - ajsin9 - (a - y) . cos 9 + cos . — -r + sin . — - , 



d9 d9 



