414 



Mr Larmor, On Critical or 



[May 28, 



Now write cosiJr = — , and use the results for differentiating 

 P 

 F that are given in Cay ley's Elliptic Functions, § 73. We find, 

 on putting <£ = 0, that a quarter period of the oscillation is given by 



^ ho 



ffW/3 



in which F X E X stand for the complete elliptic integrals of the first 

 and second orders to modulus sin45°, and e = c/m/3 2 . 

 Also 



t = T- 



ff-m' 2 /3 



(l + i<0 



F+ e (E— IF) — \e sin-v/r cosyjr 



e cos\|/< 



+ 



2 sini/r Jl-% sin' 2 -v/r_ 



=T- 



K 



ghn*fi 



e 



F+eE-^0j/3 2 -0 2 + 



e/3d 



j2{?-&)\' 



in which F, E are the functions of arc cos -5 to modulus sin 45°, 

 whose values can be taken at once from Legendre's tables. 



9. The disturbance produced by the small term proportional 

 to the distance is represented by the terms multiplied by e. And, 

 in particular, if e — 0, so that the equilibrium is critical, we have 



T- 



,. 1"[^, sin 45° 



g' 2 m^/3 



t= 1 K 1 \ F (%, sin 45°) -in arc cos -5, sin45 

 g-m-fi { \* / V P 



where m = a/4, /3 = amplitude of excursion. 



In this case, therefore, the period of an oscillation varies in- 

 versely as its amplitude, and is equal to 



-^- x 1-85407. 



If the plane on which the motion takes place be frictionless, 

 k will be the radius of gyration about an axis through the centre 

 of gravity : and the same will be the case in the problem of hydro- 

 static oscillations. 



