148 REPORT—1899. 
define a known periodic solution. 
Put w=, +E, 
The system now becomes a set of m equations to determine the é’s ; if 
these are solved on the supposition that squares and products of the és 
can be neglected, the solution is of the form 
E,= Aye"); + Ace™ poi + eee $A,67ntbyis 
where the A’s are constants of integration, the a’s are the characteristic 
exponents, and the ¢’s are periodic functions of ¢. In order to solve the 
equations when products of the 2’s are not neglected, assume 
E= bit Mbait «++ EM Pri 
The equations determining the 7’s can now be written down ; it is 
proved that they can be solved by assuming each of the quantities 7 to be 
a series of ascending powers of A,e™’, A,e™,...A,e; the <A’s being 
constants of integration, and the coefticients being periodic functions of ¢. 
In general, some of the quantities a will have their real parts negative. 
The other a’s can be got rid of in the expression for £ by taking the cor- 
responding A’s to be zero. Then, when ¢ increases indefinitely, €; will 
diminish indefinitely ; in other words, the solutions thus obtained approxi- 
mate more and more closely to the original periodic solution as the time_ 
increases ; they are called asymptotic solutions. 
Similarly, another class of solutions can be obtained which differ widely 
from the periodic solution when t=-+0co, but approximate to it for 
t=—co, These form a second kind of asymptotic solutions. 
In the case n=2, the solution can be represented by the locus of a 
point whose coordinates are 
ecost, e = sint, a. 
Tf the solution is periodic, this locus is a closed curve in space, and 
the solutions asymptotic to it are represented by curves asymptotic to this 
curve. The aggregate of these asymptotic curves is called an asymptotic 
surface ; there will of course be two asymptotic surfaces corresponding 
to t=co and t=—co respectively, and each of them passes through the 
periodic curve, 
M. Poincaré then discusses the case in which the original equations 
ey ee ee et) 
contain a parameter p. It is shown that, when the X’s and the 
characteristic exponents a of the periodic solution are expansible in 
powers of », the coordinates of a point describing an asymptotic solution 
can also be expressed as series of ascending powers of p. 
The theory of asymptotic solutions is then specially developed for 
the differential equations of dynamics. The system is taken in the form 
oF dy; oF 
been =— I=1, 2,... 
di ty? dé &x; caso 
where F is expansible in powers of p. 
It has already been shown that the characteristic exponents a are © 
developable in powers of /p, and are all zero when p=0. It is now 
