92 Mn. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



the other terras vanishing as 71 is not greater than 2. 



The limits of » being ^ and - 7, those of w are 03 and 0, and 



r dai , . ,. .. (2W -3).(2W-5)...3.1 TT 



/ between these limits = — -r — r- -7 : — - . - ; 



^(1 + .7'=)" (2n-2) . (2m -4) ...4.2 2 



J^/(l3-^).(y+^) I 2 2 4.2J 



= - TT . {/3" - n^'-> . O + ^ . i8=) + n . '1^. 3/3-} 





Hence /- 



V^(/3-«).(7 + «) 





-^"v/(/3-«).(7+^) 



= — TT . ■ 



ifol" j 2 ■ Up Sp" 2a 4pa 8a^ 2j3a 2aV j' 



and as 



or 



a/, _^' ^a'l 2aj r + 2a^j - ^ o-r^ Ua'' 2j'^J 

 at 4 \p a 4p^ 2pa 4o"/ J ' 



or, restoring the quantities U and V, and their differential coefficients, 



. = .x/'" 



-^•(i + C/30 (9), 



and i. = "^ (10), 



where L and t are expressed by quantities and their differentials. 



The times of descent to the position of equilibrium and of ascent from it, will be found 

 by the integration of equation (8) between the limits, x = (/3, 0), and « = (0, - 7) ; but as the 

 first powers of the arc will appear, it will be sufficient to integrate 



