32 
RADELFINGER. 
The corresponding equation is : 
g+ Pl ( z)g + 2U*)2/ = °. 
In the case of the subgroup of one independent parameter 
the group may have two different forms. The first is: 
Y 1 = y 1 o m , 
Y 2 = y 2 o», 
in which m and n are integers. The corresponding equation 
of the second order breaks up into two reduced equations of 
the form 
in which R (a) denotes a rational function of x , equations 
obviously solvable by a simple quadrature. The second form 
of the group of one parameter is: 
Yi = y» 
Y=yi 6 i + Vv 
In this case one of the integrals is rational and the other is 
given by an equation of the form 
^l + R, (x)y = R,(x), 
an equation solvable by a quadrature. 
The subgroup of two parameters has the form: 
Yi = 2/n 
Y\ — 2 /i ^1 2/2* 
The corresponding equations are: 
g + *,(«) S = 0, 
