ENKROY ACCELERATIONS, A STUDY IN ENERGY, &C. 289 



The conditions may otherwise be expressed by saying that for ail 

 values of h, tlie expressions 



, iiiy , vi.^ }? , w-j m^ A 



[(X,+i^)2] ,-[t,+p], , m,[p] 



2[{l\+R){X,—Ii)l M , M ,m,[^;+p] + m,[^,-^p] 

 and 



. [p] , b^ , — '^H [s 2 + P] — «^2 [?1 + P] 



shall be of tlie sarae sign. 



The conditions of stability may be obtained by assuming that the 

 values of \j\'\ \^>\"\ [^'i^'a] differ from their equilibrium value by har- 

 monie functions of the time of period 2 7r\p. We thus obtaiu 



i/;//,— [£,+/:], , w.jM 



[p] , [p] ,/^«i^/'2— '^'-oKi+p]— '^''i[?2+p] 



as the équation for determining p-, and the condition for stability is 

 that this équation must hâve real roots. The condition of stability the- 

 refore now ditfers from the condition that energy-equilibrium may be 

 possible. 



Example 3. — A single ^article moving in any field of force m a 

 plane. 



Let V be the poteutial of the fiekl, and to avoid introducing the con- 

 stant w. iuto the équations suppose the particle to be of unit mass. Let 

 u, 11 be the velocity components and let 



_ d'^r _J-V 

 dx dy ' dy'^ 



19 



