dr de 1 d'r 4 dx a et 0 9 
ge iter Soha oh ham ar eit ak off Be 
being a linear differential equation of the Sd order with constant 
coefficients. When we substitute «= ert, we get for the solution of — 
p the third degree equation : 
1 a 
pt opt + ap + (2-4) =o 
q q 
Interesting from our point of view are only the cases of physical 
signitication, i.e. damped vibrations. In order to yield them it is 
necessary that this 3'¢ degree equation has one real negative root 
and two conjugate complex roots, the real part of which is negative '). 
Condition for this equation having one real and two conjugate 
complex roots is that its discriminant D is positive, hence D> 0, 
in which 
at A AG tte a' 
Dee EEE ple ce 
rt “) 
1) It is necessary to make a remark on the energy. 
The equation of § 2 
Ce. oe daz 
aa toa tte Aged. o> Oe AER alef) 
yields multiplied by = 
da dx ee dx 
TR 
+c (EE) + ane =0 
or 
da \? dx gend De dx d*x dx 
a) trl) =F (240 tia +4(F)).@ 
ae i 
What stands behind 5 must be the energy (only «> Aq can have signifi- 
cation). This term gives the potential energy on the supposition that there is an 
elastic potential 4 (~—Aq)x?, and then an elastic force (c<—Agq)x, which is not 
the case here. It seems to me that the interpretation will be as follows: the 
lefthand member gives the work done on the system, the righthand member 
the change of energy. How must this work done of the lefthand side be imagined? 
dx \?2 
In the first place the work — zal d a) done by a frictional force proportional 
dx : dix? 
to eA and then in the second place also the work — q Ee ‚ done by a force 
d°x Hen 
proportional to =a Then of course not only the potential energy that exists with 
regard to the force (~—Ag)x appears in the righthand member for the potential 
