98 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



= li.f^^dr . sin . d9 .dc^.p"'^. (w - r .cosO) + ... since j)^ = r^ + w^ - 2rxcosd. 

 Let k = nr"*^ dr. sin 0d9.d^; 

 .: U = kp—^ix -cosG) + ... 



U, = k .(n- I) .p"-^. {,v- r . cos 6)- + Ap""' + ... 



= n. r'"+°"^'dr sin0. dO. d<p\(7i - l) .cos-0 + l}, wlien *• = ; 



which, being integrated from = to 6 = tt ; from (p = to = Stt, and from r = r, to r = r^, 

 we have finally 



^" 3.(m + w + 2)-^'^^ '^' ■'' 



and r = ^== x/ 37r.(m + n+2) 



or if Jf be the mass of the hollow sphere; 



^ (w + 2) . (m + 3) . 3/. (rr"'"' - r7+'+^)' 

 If the force varies as the distance, T ■■ 



If the force varies inversely as the square, T is infinite. 



If m + « + 2 = 0, ! = log - 1 . 



Wi + w + 2 ° V-i/ 



r 



. 3.(n-l).log- 



And if 77! + 3 = 0, T = TT V ■ *" 



2 



(« + 2). J/.(,-r' -rp')' 

 (2). To find the correction for the time of oscillation, we have 

 U, = k .{n - l).{n - 3) . (ri - 5) . p-'-\{x - r . cos 6)' + 6k .(n - \) . {n - S) . p"' . {.v - r cosBf 



+ 3k. {n - l).p"-\ 

 or, making a? = 0, there results for the attracting particle 



{r^ = |u . (n - 1) . r'"+"-' dr .sinO-dO. d<p[{n - S) . (w - 5) . cos^ 9 + 6 . {n - 3) . cos^ 9 + s]. 



drdcp. 



which, integrated between 6= (0, tt), is 2,u . — ^-^ ^-—^ iZ ,."!+»- 



5 



and again between <^ = (0, 2 tt) and r = (r, , rj) is for the hollow sphere 



_ -tM-^M.(w- !).(« + 2) 



5.(m + n) ^"^^ ' ^' 



• T = ^ A./ 3. (m + n + 2). (rr' - r^^^) /, _ 3^^ n.( 7i- l). (m + w + 2) ^r" - <'"" I 

 ^ (« + 2).(TO + 3).ilif.(r^"+^-r^+»+=)-\' 80 ■ m + n •^»+»+2_^»+n+2|- 



If the sphere be solid and density uniform, and the force inversely as the square of the dis- 



ance, T = \/—. 



If r equal forces '2.kl.^{^'), be placed in the angles of a regular polygon, the time of oscilla- 

 on of a particle at the centre will be found to be 



