* 
LINEAR DIFFERENTIAL EQUATIONS. 35 
have been constructed for still fewer cases. It is true systems 
have been deduced composed of the so-called asymptotic solu¬ 
tions, but, as we have seen, these series are, in general, di¬ 
vergent. While much important work has been done in 
recent years toward establishing the legitimacy of the use of 
divergent series, much yet remains to be done before they can 
take their place in analysis on the same footing with con¬ 
vergent series. Borel * has given an excellent summary of the 
most important results in this extensive field in his memoir 
recently awarded a prize by the French Academy. 
The importance of the application of the ideas of the group 
theory to linear differential equations can hardly be over¬ 
estimated. The study of the linear group provides us with 
a means of arriving at the relations existing between the 
functions defined by the linear equations and the algebraic 
and Abelian functions, as well as the relations of these func¬ 
tions with one another. The results, being at once applicable 
to all linear equations, are the most general obtainable. The 
theory also enables many demonstrations to be arithmetized 
which have hitherto depended on intuitive concepts. This 
arithmetization of mathematics is now admitted to be a step 
of essential importance, since the methods of research inau¬ 
gurated by Weierstrass and his followers have come to be 
generally understood, and the inadequacy of our intuitive 
concepts to enable us to form correct ideas of the true nature 
of functions has been generally recognized. 
* Ann ales de l’Ecole Normale, 1899. 
