34 
RADELFINGER. 
He proves that: 
(h) The necessary and sufficient conditions that must be ful¬ 
filled in order that a linear differential equation may be solvable 
by a succession of quadratures is that its group be of the form 
■hi — u n 
Y 2 a 21 y x -f- a 22 y 2) 
= a 31 y x -j- a. i2 y 2 -f- a 33 2 / 3? 
h n = a nl y 1 + a n2 y 2 + a nS y a . .. a na y n . 
Lie had previously discovered this group and named it the 
integrable group. In terms of the group theory the theorem 
may be stated: 
(i) The necessary and sufficient condition which must be ful¬ 
filled in order that a linear differential equation may be solvable 
by quadratures is that its group be an integrable one. 
Also: 
(j) If an equation is integrable by quadratures its group is an 
integrable one. 
It follows from these theorems that the case in which a 
general linear differential equation is integrable by quadra¬ 
tures is a very special one. 
Conclusion .—The advance made in the theory of linear 
differential equations is considerable, yet much remains to be 
done. As it stands today the theory may be regarded as 
fairly complete, only for Fuchs’s special class of equations. 
Comparatively little has been accomplished in the theory of 
those equations which have irregular integrals. These con¬ 
stitute much the larger class. The existence of an integral 
represented by a series, with a circle of convergence about 
an ordinary point, has been established, but this series has 
been calculated in but few cases. Fundamental systems com¬ 
posed of solutions represented by series, with circles of con¬ 
vergence described about a singular point, systems so essential 
for a complete study of the function defined by the equation 
