6 Stephenson, New Type of Dynamical Stability. 



ment ; whence it follows that the impulsive moment in 

 any cycle from rest to rest at A is always directed towards 

 the central line of motion AB, and varies directly as the 

 displacement — the conditions determining a mean motion 

 of simple oscillation. 



Such considerations tend to familiarise the motion, 

 but the quantitative results may be obtained by a more 

 analytical method, which will now be given for the sake 

 of comparison. 



Let be the inclination of PC to AB on starting from 

 A after the impulse, and w the angular velocity at the 

 same time* ; also let 0i, Wi be the values of these quantities 

 after the impulse at B, and ^-^^ w the values after the 

 succeeding impulse at A. 



On starting from A the velocity of the mass centre 



has components 



( V- /zwsiny, /^wcosO) 



along and perpendicular to AB. After the impulse at B 

 the velocity is 



(- F-/^wiSinf^i, //wicosyj). 

 Hence if (A', Y) the impulse at B 



XI M= -2V- /liwiSinOi - wsinG) 

 V/A/= /i{w j^cosdi - w cos d). 

 Again, considering the motion about the mass centre 

 we have 



Mk\u)i - w) = (.Y sin 6, - Fees d,)/i 

 .'. k-{toi - w) = - 2 V/isindi - /i-{u>i - uj)cos{Oi - d), 

 and to the required degree of approximation 



(/^- + ^■-j(wi - w) = - 2 V/id^ . . . (i.) 



Similarly, considering the succeeding impact at A we 

 have 



{/r + A'){w.,-uj,)=2FM, . . . (li.) 



.\{/r + /i%o.,-io) =2F/i{e,-6,) . . (iii.) 



* It must be remembered that w here has not the same meaning as in the 

 preceding. 



