( 692 ) 
The possibility of the solutions (2), (7), (12) and (13) depends on 
the sign of 2,. In all these cases cos (a + 8) and cosa’ are of the 
same sign. Thus if we put 
Q — U; Es Zin u, B, 
a, 
we find that for positive values of Q the solutions (2) and (12) 
can exist, (7) and (13) being impossible; for negative values of Q 
(2) and (12) become impossible, but (7) and (13) are possible. We 
find for these solutions: 
Solution (2) Solution (12) 
ge | Hi Ee Ev 
l= 0 i= 0 {== 180 f= 180 
A= i= Ae == 160 4; = 160 
x positive. % negative. 
Solution (7) Solution (13) 
= 180% + == 0" £180" oe de 
L, == 180 Er 0 is 180 
ds An A, == Lee dts) 
x positive. x negative. 
All four solutions have K = O°. 
For the solutions (2) and (12) we find 
Ss Jog se 
E, B &, 2a,A 
and for (7) and (13) 
Beg Ge 
E, B é, 2a, A 
For Q=0 (or, if higher orders of ¢; are taken into account, for 
a value of © in the neighbourhood of the value for which Q = 0) 
[Hs 
we have e,=0. The solutions (2) and (7) then become identical, 
and similarly (12) and (13). We thus find that the two cases (2) 
and (7) form together one continuous family, which exists for all 
my, 
values of . The same thing is true of (12) and (13). 
Ms, 
Thus all that has been said above regarding the existence of the 
periodic solutions has now been proved. It remains to investigate 
their stability. For that purpose we must form the equation (2). We 
introduce the notations : 
l—s be Orde, eR 
Tien, ) Oo eo 
