Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



99 



\/- 2A:r . {0(( 



and as A;r is the whole quantity of attracting matter, the time is the same while kr is, and 

 therefore if the matter be in the form of a ring, the above is still the time of oscillation. 



If the force = k . S\ then (piS") = , and 



•\/2 



\/-rk.{n+ ]).o"-''l 640^ J' 



On Rockins Bodies. 



In the position of equilibrium, the centre G of the rocking 

 body, will be in the same vertical line as the point of support ; 

 that is, when A is at A^, AG will be vertical. 

 Let AG =a, AN= y, NP = x, 

 A,N, = y,, N,P = x„ 

 and PO being a common normal, 



let AOP = e, and A.O^P = 0,; 

 then (p the angle rocked through =0 — 0,; 

 and if X, Y be the co-ordinates of G, measured from A^, 

 X = a', — a: . cos <p + (a — y) . sin (p] 

 V = yi + x . sin <p + (a — y) . cos (p) ' 



(22), 



. dy d.v 



also, sm d = —r- , cos 9 = -— , and ds = ds, ; 

 ds ds 



dX dxt dx . dy 



-r— = -r; r~ • cos (b + x .sincb . sin d) + (o - «) .cos d) 



d(p d<p d<p ^ ^ d<p T ^ ■>' T 



— cos Q, 



ds 



+ {a - y') . cos <p 



- cos Q . cos (b \- X . sin (p - sin sin (h 



d(p "^ d0 ^ ^ 



= a? . sin ^ + (a — y) . cos 0, 



d Y dy, d.v . dy 



-rr = -r— + -; — . sm m + x . cos d) . cos d> - (a ~ y) . sm (b 



d(p dcp dcp " ^ d0 ^ ^ 



• r, ^^ n ■ . ds . ds 



= sm y, . -—- + cos . sin . -— - + ,t; . cos - sm . cos rf) . (a ~ v') ■ sin d) 



d^ ^ d^ ^ ^ d0 ^ "'^ ^ 



= a? cos (p — (a — y) . sin ; 



dX 

 °^ d^=^-2'' 

 dF 



(23). 



To find X, Y and their differential coefficients with respect to (p ; we have, from (22) 



X=0, 

 Y=a, 



N2 



