LINEAR DIFFERENTIAL EQUATIONS. 
27 
more into detail concerning these results, I will say that no 
advance of a fundamental or far-reaching nature has been 
made by this extension. Indeed, it seems to me that none 
can be expected, since considerations drawn from the theory 
of invariants and the group theory place definite limitations 
on the applicability of this method of research. 
Poincare * constructed a fundamental system with inte¬ 
grals of the form: 
2/» O) = «“ +“■ + Jr.) • (H) 
The «, I, and a’s are constants, and a and X have different 
values, in general, for each fundamental integral. The co¬ 
efficients, the a’s, are determined directly by substituting the 
series in the given equation. In general, only divergent 
series result, which converge absolutely for the one value, 
x = oo . They are therefore said to represent the integrals of 
the equation asymptotically, and are aptly termed asymptotic 
solutions. It may be of interest to note that solutions of this 
sort were proven by Poincare to exist for the problem of three 
bodies. He also demonstrated the non-existence of solutions 
of Fuchs’s form for this problem, except in cases already long 
known. 
Thome,f who has investigated equations some of whose 
integrals are regular and the others irregular, arrived at 
irregular integrals having much the same form as Poincare’s 
asymptotic solutions, and having identical properties. 
Systems of fundamental integrals analogous to those just 
mentioned were also deduced by Poincare J by means of an 
application of Laplace’s transformation. He substituted the 
integral 
f v (z) e“ dz (I) 
c 
for y and determined v (z) subject to the condition that the inte- 
* Acta Matliematica, vol. vii. 
f Crelle’s Journal, vols. 74, 75, and 76. 
t American Journal of Mathematics, vol. vii. 
