146 REPORT—1899. 
the period being, for example, 27. The question is now propounded, 
whether the system admits of periodic solutions differing but little from 
this, when p has values which, though very small, are different from zero, 
M. Poincaré finds the answer. By choosing, as initial values of the co- 
ordinates, certain functions of p, it is in general possible to obtain such 
periodic solutions. 
It is further shown that as p varies, periodic solutions disappear im 
pairs in the same way as the real roots of algebraic equations, This 
happens when a certain functional determinant is zero. 
Poincaré next proceeds to define the characteristic exponents of a 
periodic solution. 
Suppose that a periodic solution, as above, has been found. Consider 
a motion differing but little from this, and defined by 
x=o(t) +6: hoa teay tc W??), 
where the quantities — are supposed to be so small that their squares and 
products can be neglected. 
Then to determine the £’s we have 
ey. sip OG Wis tan ie 
dt =XE, te,’ V—" ayeoae n). 
As these are linear differential equations of a definite type, with 
periodic coefficients, £; will be a sum of 7 quantities, each of the form > 
e%'S,,, where the quantities S,, are periodic functions of ¢ with the same 
period as ¢,(¢), and the ~ constants a, are the roots of a certain algebraical 
equation. 
The constants a are called the characteristic exponents of the periodic 
solution. If they are purely imaginary the é’s will remain small, and the 
periodic solution is stable ; if not, the solution is unstable. 
If two of the characteristic exponents are equal, the form of the 
solution is altered, as the terms of the form te“ now appear. 
When the original equations have the Hamiltonian canonical form, 
the characteristic exponents can be arranged in pairs, the exponents in 
each pair being numerically equal, but of contrary signs. The 7 values of 
a2 are called the coefficients of stability of the periodic solution considered ; 
if they are all real, negative, and distinct there is stability. Whén the 
Hamiltonian function does not involve ¢, one of the m coefficients of 
stability is zero, so two of the characteristic exponents are zero. 
The author now shows that the functional determinant already men- 
tioned vanishes only when one of the characteristic exponents of the 
original periodic solution is zero ; the theorem already given can thus be 
put in the more precise form. 
If a set of equations, which depend on a parameter p, admit of a peri- 
odic solution when p=0, for which no one of the characteristic exponents is 
zero, then they also admit of a periodic solution for small values of pu. 
Poincaré then turns to the periodic solutions of the differential equa- 
tions of dynamics. For greater definiteness, the system is supposed to 
have three degrees of freedom ; the equations are taken in the form— 
dx, oF dy;_ 6F 
day OR ayy _ OF sg 
aa ae Sa 
and F is supposed to be a uniform function of the «’s and w’s independent 
