90 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



.-. mds^ + m,ds] + ... = (Vo+ V,z + Fj . ^ + ...jdz", 



and (F„.r,...n.J)^^t7,(.--/3=) + fA.^\r3.^'. (.). 



For a first approximation, K- j^= ^i • C^' - /3'"')> f>"om which it appears that -^vanishes when 



ds 

 sr = ± /3, and since s = /x// iti)du = R («) = R{a ± ^) = /?„ (a) ± B, (a) • ^, ^ vanishes, when 



s = R (a) ± R (a) . /3 ; which shews that each body of the system vibrates to an equal distance on 

 each side of its position of equilibrium. 



/^^o ^ / »n >// (af + m, f (a)' + ... 

 The time of oscillation = 7r V ~yr~ = ■"■ • 'V 7~r^ , . . • 



m \lr (a)° + m,f (aY + ... 

 And L the length of the pendulum = - g- 



mfi (a) + »Ji^, (a) + ... 



gdy 

 In the case of gravity, Pdp = -gdy, and /(?<) = - — , 



mds^ + m,rf«; + ... 



and .-. L= — -^ ^ — ; (■■>;, 



ma y + m^a'y^ + ... 



the position of equilibrium being determined from mdy + m^dyi+ ... = 0. 



If the body be rio-id, and X, Y the co-ordinates of its centre of gravity, and Mk" the moment 

 of inertia about the centre, and Q the angle of rotation be made the independent variable ; then 



ds' ds\ ,,,, ., dJf + dY' 



»n ^^ + »»i ;7^ + ... = Mir + M . 



= Mk- + M 



dJC 



![¥ ' 



dX' 



*' + dS^ 



and .-. i = _4L (6). 



a" 1 



~d¥ 



mds- + m^ds\ + ... 

 When a rigid body oscillates about a point of suspension, the expression ,„^3y ^ ,„_^-y^ ^ 



becomes L = "'^' "^ "''^' "*' "• , the point of suspension being made the origin (7). 



{m + m, + ...) Y 

 The equation (4) for the purpose of integration may be more conveniently put under the form 



— .(p + qz + rz^) =a.{fi' -!>') + b. (^' - z") + c . (/3' - «') 

 df 



= (li -z).{(a + bx + n^ + c/30 . (« + /3) + 6^'}- 



Assume (a + bz + cz' + r(3') . (z + /3) + 6/3= = (a + 6^ + c^ + c/3=) . (s; + /3 + ^^= + e^';::), 



.-. (a + 6.r + CZ-' + cfi') . (5 + ez) = b, 



or. aS + bSz + aez = 6, omitting the squares of /3 and z ; 



.-. nS=b, and 6^ + oe = 0, 



