VIII. On Small Fiiiite Oscillations. By the Rev. H. Holditch, Fellow of Cuius 

 College, and of the Cambridge Philosophical Society. 



[Read May \5, 1843.] 



The system of bodies here considered is supposed to be such, that their position, and the forces 

 acting upon them in that position, depend upon a single variable ; and the object is to find general 

 expressions, which may be applied to any particular case, without performing any integration, for 

 the length of the isochronous pendulum and the time of oscillation, rocking or sliding, when the 

 body or system of bodies is slightly disturbed from its position of equilibrium, the approximation 

 including the square of the variation of the independent variable. 



By the principle of vis viva, 



mu° + THiDj + ... =2 m jPdp + 2mifP^dpi + ... 

 Let u be the independent variable, and a its value when the system is in equilibrium, and a + ;» its 

 value at the end of the time t ; also let /3 be the value of z at the beginning of the motion, when 

 the system is disturbed and left to the action of the forces upon it. 



Pdp 



Let -^ =/(w)=/(a + ^), 



P^dp, 



~d^ " ^ ^''^ = (a + Sf), 



J -,. mPdp + ntiPidp, + . 



nd C7 = — 1— i-! 



dtt 



.-. Uo = mf (a) + wi, (p (a) + 

 U, = m/i (a) + m, <pi (a) + 

 U2 = m/2 (a) + mi 02(a) + 



(1)> 



and mPdp + in^P^dp^ + ... = iUo + UiX + U-^ + ... Irfw (2); 



but, when the system is in equilibrium, u = a and C7, = 0, or mf{a) f m,(p(a) + ... = 0, which 

 determines its position when at rest, and as du = dz, the integration of (2) will give 



m f Pdp + mJPidp, + ... = U,.^^^ + U, .^^f^ + U,.^^+ ... 



2.3 2.3.4 



Agam, let ^ = ^^ (w) = ^/' (a + x), 



. „ mds' + m.ds': + ... 



and V = ! — ! 



du' 



Vol. VIII. Part I. M 



W; 



