LINEAR DIFFERENTIAL EQUATIONS. 
33 
both of which are solvable by quadratures. The second is 
the well-known equation of Riccati. It has an integral in 
common with the first. 
The subgroup of three parameters can take on eight differ¬ 
ent forms. It will be sufficient to state that the equations 
corresponding to these reduced groups are of a less general 
form than the irreducible equation of four parameters. The 
most general form of the equation of the second degree, in¬ 
tegrate by quadratures, corresponds to a subgroup of three 
parameters. 
Fuchs * has obtained a very beautiful result pertaining to 
equations of the third order which may be appropriately men¬ 
tioned here, since it has been restated in terms of the group 
theory by Picard.f Fuchs showed that if a rational relation 
of the form, 
R (2/12/2 2/3) = 0, 
subsists between the integrals of an equation of the third 
order, two cases present themselves: 
First. If the degree of the relation R (; y 1 y 2 y 3 ) is higher than 
the second, the equation of the third order is reduced to one 
solvable by quadratures. 
Second. If the relation R (y x y 2 y 3 ) is of the second degree, 
the equation of the third order is reduced to one of the second 
order. 
Lastly, I consider the celebrated problem of the solution 
of linear differential equations by means of quadratures. 
This problem is analogous to the more famous one in algebra, 
viz., the solution of equations by means of radicals. The 
latter problem was first shown in its true light by means of 
the theory of groups. To Yessiott belongs the honor of 
first determining the conditions to be satisfied in order that 
a linear differential equation may be solvable by quadratures. 
* Acta Mathematica, vol. i. 
fTraite d’ Analyse, vol. iii, p. 550. 
X Annales de l’Ecole Normale, 1892, p. 197. 
