36 ALF GULDBERG. M.-N. Kl. 
and this will represent either one or two systems of equations, according 
as the quadratic for determining 4 has equal or unequal roots. That 
these systems are relevant to the solution of the given equation may 
be shown as above. | 
We have now finally only to consider the case in which Æ=o, 
G=o, F +o. As the treatment of this case is quite analogous with 
the case just examined, we shall here only give the results. 
Case I. Let E=o, G=o, F + 0, and suppose D +o: We then 
obtain the following system: 
AN (AN AY 
«(9-27 
Me af _ 
=) +DL_1,Dz =o, 
where A, and 4, are the roots of the quadratic equation 
D? — CDI + BD + FH=o0, 
and where 
0 
Case II. Letting £=0, G=o, F + 0, and supposing D— 0, 
we obtain the systems 
7 =o 
Y-uß)- 
F(Z Rone ce ae 
EAE Fe 
where u is determined by the quadratic equation 
Hu? — Cu+F=o, 
and where 
A= 0. F 
That these systems are relevant to the solution of the given equation 
may be shown as before, 
