The conditions may otherwise be expressed by saying tliat for all values of the expressions
0 , mx , m.l >r , >», m2 /.
[(^■14"-ft)2] 7 — [si-WL ^ > m\ [p]
[(■^2 ft)2] , 0 [?2+PL m1 W
2[(A", -f-ft)(A"2—ft)], [p] , [p] , [?»+p] + «»i [si + p] and
— fêi+p], 0 » w-i W
0 » — [?2 +p] 7 ml [p] [p] , [p] > ml [?2 + p] w'-2[?l+p]
sliall be of the same sign.
The conditions of stability may be obtained by assuming tliat the values of [f,2], [?'22], ] differ from their equilibrium value by harmonie funetions of the time of period 2 vjp. We thus obtain
iP2mt —[?i +P], 0 m, [p]
n , —[?2+p], m2[p] =0
[p] , [p] , [£, +p]—»», [?2 +p]
as the equation for determining p2, and the condition for stability is tliat tliis equation must have real roots. The condition of stability tlierefore now diti'ers from the condition tliat energy-equilibrium may be possible.
Exajiple 3. — A single partiele moving in any field of force in a plane.
Let V be the potential of the field, and to avoid introducing the constant m into the equations suppose the partiele to be, of unit mass. Let w, v be the velocity componeiits and let
dV dV_ ,_d2F _ d- V ,d2V
da-' dy' ' dx2' * dx dy' dy2