150 REPORT—1899. 
Poincaré now goes on to discuss the question of the convergency of 
Lindstedt’s series. He takes the differential equations in the form 
da,__6F 
dx, _«F dy, oF dy, _¢F 
Gi tye. Or. ey, at ca, dt ex, 
where 
F=K)+ B F, 
and F, is a function of #, and a, only. The a’s and y’s are regarded as 
functions of w, and w,, where 
Wy=AjyE+H,, Wy=ob +a, 
and where 
«=0 
qd 
Y= t SY: 7 (== 10 2) 
, <1 
A= pe AL | 
«=0 
The author sketches Lindstedt’s result, that the constants \* can be so 
determined that these expressions for x, and y; (in which x and y; are 
periodic functions of w, and w,) may formally satisfy the above differential 
equations, with an error of the order p**’. 
The series are first assumed to be uniformly convergent ; and it is 
shown that if this assumption were true, all the characteristic exponents 
of the periodic solutions (which arise when \, is commensurable with i.) 
would be zero. Since this is not the case, the assumption must be false ; 
and thus the result is obtained, that Lindstedt’s series do not converge 
uniformly for all values of the arbitrary constants of integration which 
they contain. 
The author next discusses the nature of the integrals of the differential 
equations, other than those integrals which are already known. 
The system 
Oa, aN. diy. OW on peta 
di dy, dt éx, Geraied 
has an integral 
F(x, %o; Yi, Yo) = constant. 
Suppose, if possible, that another integral exists of the form 
(x), La, Yi, Yo) = constant, 
where ¢ is a uniform analytic function of p, 7, %, Y\, 2, and is periodic 
in y, and ys, of period 27. It is proved that in this case the equations 
6F _OF oF oF 
8a, da _ oy) dy. 
must be satisfied at all points of all periodic solutions. It is then further 
