28 
RADELFINGER. 
gral (I) satisfy the differential equation when taken around the 
singular points of v (z). This transformation has been much 
used and with great success in the study of Fuchsian equa¬ 
tions, many properties of these being more easily deduced 
by its use than by any other method of analysis. One of its 
applications in this connection is to calculate the constants 
in the linear relations subsisting between members of differ¬ 
ent fundamental systems. A good example of this use is 
given in Goursat’s thesis, reprinted in Craig’s treatise.* 
Group theory .—I now take up the consideration of the most 
important advance that has been made in our theory in recent 
years. This advance consists in the extension and application 
of the ideas of the Galois group theory to the theory of linear 
differential equations. This advance is due in the main to two 
members of the French school, Picard and Vessiot. Picard’s 
results were first made known to the world in a short note 
published in 1883,t but he subsequently amplified his work, 
and his first discoveries, together with much additional 
matter, was made the subject of a memoir which appeared in 
18874 He has several times recurred to the subject in short 
notes published from time to time in the Comptes Rendus. § 
Vessiot, who may be said to have completed Picard’s work, 
embodied his researches in his thesis published in 1892.|| 
More recently still, in 1896, Picard If has given a good but 
brief presentation of the theory. He has also recast many 
of Vessiot’s propositions and placed the foundations of the 
theory on a firm arithmetical basis. 
Both these mathematicians, more particularly Vessiot, base 
their work on results obtained by the late Sophus Lie, the 
great originator of the theory of continuous groups, an idea 
which promises to become the dominant idea in modern 
* Craig (Thomas): Differential equations with uniform coefficients. 
New York, 1889. 
t Comptes Rendus, April, 1883. 
X Annales de la Faculte des Sciences de Tolouse, 1887. 
\ Comptes Rendus, October, 1891, and December, 1895. 
|| Annales de PEcole Normale, 1892, p. 197. 
^[Trait6 d’Analyse. Paris, 1896, vol. iii. 
