94 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS 



When the body is rigid, the general expressions may be put under more convenient forms : for 

 if tl Id ffe ntial coefficifnts be taken with respect to the angle of rotation round the centre of 

 Iravitv Jr and }' being its co-ordinates, and Mk^- the moment of mert.a round the centre, 

 ^ ■' V =Mk' + M{X\+Y\), 



F, = 23f(A',X,+ F,K), 

 V, = 'iM {Xl + X^X,+ Y\ + r, F3). 

 And U = - MY,, 



r/, = - MY,, 



U,= - MY,, 

 U,= - MY, ; 

 ... V, = M (k' +Xl), F, = 2 MX^X, , and K = 2 3/ (Xl + X,X, + YD ; 

 /r + X', 



.-. L = 



C = 



xl + x,x, + r| 

 Hk-' + xi) 



X,X,Y, 



Y, 



^1 ^ rt 



16 r, i'ik' + xfy 



5 Fl 



+ — . 



.(15), 



(16). 



4,Y^ik^ + Xl) is' Y- 

 In the case of a particle, L = yr , 



X, Y.! 



16 F, 



X,Y, 



+ -.^, 



(17). 



iX,Y, iS'Yl' 

 Example. A rod oscillates upon two planes, inclined at the angles a and a, to the horiz 

 the centre of gravity being at the distances a and a, from the extremities of the rod. 



Here ^ = ^ sin + B cos 9, 

 r = Msin9+ Ncos9, 

 where 6 is the inclination of the rod to the horizon, and 





S = 



(a + a,) . cos a cos a, 



sin (a + a,) 

 a cos a sin Oj — a, sin a cos oi 



a sin a cos oi — Bi cos a sui a, 



sin (a + a,) 



and A'^: 



(a + a)i . sin a sin a, 



sin = - 



sin (a + a,) ' " sin (a + a,) 



.-. F, = J»f cos - JV sin = 0, 



M _ ^ N 



and 



i9 = - 



y/M' + N" y/M' + N' 



(a + a.) . (a sin-a, + fl!i sin-a) 



Let M' + N'= . ,, —^ - ««> = 



sin (a + Oi) 



and ^i./+BiV= ^°-^''->-^°.?;°'"r""'°^ = ^' 



a sin'' (a + a,) 



Q, 



aoi 



then ;5r, = - X3 = -^ 



F^ = - F, = v/Q 

 P 



X = 



a/Q 



, , o'a? + Qk' 

 , and Z, = ' . — ; 



