Mr. holditch, on small finite oscillations. 91 



b . b- 



or 



, ^ = - , and e = ■ ; 



id therefore (a + fi* + car" + c/3^) . (« + /3 + - /3°- ; /3-jr), 



o a 



or (a + 6«r + c«= + c/30 . l+-/3.</3 + sr.l--.j3 



^ a\ /^,„ f, _* «^\ 



ft 



'V^)\ 



differs only from the factor (o -\- bz + as^ + cfi') . (z + fi) + 6/3" by quantities of the fourth de- 

 gree of /3 and z : or 



^. (p + 9^ + rz^) = (fi-z). [l +^(iy^(i + z. (l - ~. ^y^.(a + bz + cz^ + cfi^) 

 is true to the fourth order of those quantities ; and the limits of the oscillation of the system are, 



z = (i, and z = 7 — , or - 7. 



l--.fi 

 a 



Again, as /3 = 7 . (l - -./3j 



/3 + ;^ . (1 - ^ . /3] = (7 + ^) . (1 - ^ . /?) , and 

 dz^ / j2 \ 



and if a + c/3' = a 



d« 1 /p +qz + rz' 



dt = 



\/(/3 -z).(y + z) ■ / 1^' a + bz +cz' 



The position of equilibrium must be a stable one, and therefore JmPdp + Jm^P^dp^ + ... 



a maximum, or f/, is negative, and .•. a = a + c/B' = — 6^ + c/3° is positive, and .-. also '\/ - 



is positive. 



Expanding therefore the last term of the above expression, 



v/(/3-^).(7+«) / _6'/3^ I V2p 2ay V2p Sp' 2o 4/,a 8a^/ J 



a" 

 to be integrated between a: = /3 and z = — y, excluding the powers of /3 above the second. 



z" dz 



For this purpose, taking - —— ' ^ , let v/(/3 - ») • (7 + !») = (/3 - *■) . .r, 



V (p - sr) . (7 + xr) 



■ ■■ X = -' , and - . = -„ , 



1 + '"' v/(/3 - «) . (7 + ») 1 + •" 



, «°dg (;3af' - 7)" . 2dai _ Zdai f^ _ fi + y \ 



sfdz (/3j?' - 7)° . 2da? Zd/v 1^ P + y Y 



- «) . (7 + .jr) ° (1 + t^)"-^' rr^" V 1 + •''V 



M 2 



