102 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



and T = TT sj — . (1 + C . Ad)^), where C is to be determined by the substitution of X., F, &c. 



g 

 in (16). 



If the pendulum be suspended by a point, r and its differential coefficients vanish, and 



^1 = - ^3 = - Kj = Fi = a, and .-. C = 1 , 



It) 



L = a -{ , measuring a downwards, and 



a 



-'V^f(-¥)- 



g 



Tf -o r ' RrKjR-^r) R'r.,-r'R, 



' 10 l6(R-rY.{Rr-{R-r).a} l6 (R - rf . {Rr - {R - r) . a\ '"^ " 



16 (R - ry .{Rr-(R-r).a\4,iR- rf . {k' + a«) "^ 48 " (ff - r)' . {/?r - (fl - r) . ap ' 

 If iU and r be constant, or the curves be circles, 



R^r . (r - a cos a) 1 cos a. (Sr'R- I'R') a' sin^ a R- r* 



^ " 4,{R- rf . (fc" + a^) "'' Te "*" l6(R-ry. \Rr cos a - (R - r) .a\ ~ i(R- rf . (k- + a')' 

 a sin= a . R'r" . (R - 2r) 5 sin^ a . {R^r - 2 r-Rf 



4 (fl- r)^ |i?r COS a- (i? - r).ffl I . (&■> + a') 48 * (ft - r)* . {cosa..Rr- (i? -r).a} 

 If R and r be constant, and also a = ; 



16 4(i^-r)^ (ft^ + ffl^) l6(fl-rf . JSr- (iZ-r).o} *• ''' 



Ex. One sphere within another, 



L = -.iR-r) 



5 



'-^^('-^^•l6(i^) 



Ex. If an ellipse whose semiaxis a is horizontal rock within another ellipse whose semiaxis 

 «, is also horizontal, 



{k' + b').(alb-a'b{) 



L = 



a''a\ — alb' + a'bb 



an' a' 3a'e^ 



(b' + a'e' sm' 0)1 b o' 



a^a\ e' J_ a'a'^bb,. (a', - a') 



4 (a^6 - a'fti)" . (fc' + 6') ' 16 " l6(a?6 - 0=6,)^ . (b'al - a°bb^ - a'al) 

 3a*a* a*elb^ - a\e'b 



+ 



16 ■ (0^6 - 0^6,)' • K/*' - a'bb, - a'a\) 

 Ifjthe bowl becomes a plane, 



_ (fc ' + 6') . 6 



