1883.] "Apparently Neutral" Equilibrium. 415 



10. In the case of one sphere resting on another it is very 

 easy to form the equation of energy, and thence determine the 

 value of in. In the case of one symmetrical body resting on 

 another, whether on the summit or not, we have, if be the angle 

 by which G overhangs the vertical through the point of support, 



g df ~U«7. 6 



in the critical case ; therefore 



d*s_ g (i.lY*(dV\ . 

 df 6* 2 Vp p'J UW ' 



in which the value of (-7-3 ) is given by Mr Routh in the Quarterly 



Journal of Mathematics, XI. p. 100 ; and thus the value of m is 

 determined. The nature of the equilibrium in the critical case 

 appears to have been first discussed by Dr Curtis, Q. J. IX. p. 42. 



11. The exception to this critical case again, is that in which 

 five positions of equilibrium come together. Then the equation 

 of motion is of the form 



dtr r 



Hence t = \ / — . / - . . 



And on writing d> 2 = ■=— '-'- a , 



y = 3*tan^, 



this reduces to (vide Bertand's Integral Calculus) 



t=. /I A 1 f d * 



V ft ' /S 2 ■ 3 i J J l _ Bi n t 15 o gin 2^ ' 



and a quarter period is given by 



1 



7 T =2. T x 159814, 



^/3 2 



which varies inversely as the square of the amplitude. 



