1042 
where 
D=b" (R'— R) cos b" (R'—R) — (b'* RR + 1)sinb'"(R—R); (56) 
u is thus real *). 
The form of expression (56) suggests the case of stationary waves; 
for a given value of 6’, w becomes zero for definite values of 7, 
which means that nodes would be formed at definite distances from 
the centre. This cannot bappen, however, between & and £': that 
nodes cannot exist in this region becomes clear at once, if we con- 
sider that at constant value of 6" the number of nodes would have 
to increase with diminishing values of A or increasing values of R’, 
which is impossible, as at 7 = #’ there is always a node and at 
y= R there can never be one. 
The absence of nodes is due to the fact, that there is a definite 
limit, which is a function of R’—R, which 6’’ cannot exceed, so 
that « cannot become zero for any possible value of 7. For « must 
never become intinite, therefore never zero, and 6" (#'—R) must 
always remain smaller than the smallest value which makes /) zero. 
Now it is clear, that, if 6” (R’—&) always remains so small, that D 
cannot become zero, this cannot happen either with the numerator 
of u. 
Hence it appears already that, whereas, for #’—Rf infinitely 
Ò 
small, 6” and thus also — could assume all possible values in the 
aperiodic region, this is no longer the case for finite values of R’'—R, 
and that for R’'—R—=o 6" becomes zero, which completely agrees 
with what was found before. 
5. As regards LZ, this quantity assumes the following form in the 
aperiodic region: 
7 aR U 2 U " LA d 
Li UPRR + 3 (R—R)] 8" cos" (ER) — 
— [8b'°R'R — DPR + 3] sin b" (R—R)}, - … . (57) 
which is found from (24’) by putting 6 = 6"? or otherwise directly 
by means of (56) according to § 8 (Comm. 1484). This expression 
is again real, as it should be according to (26). 
1) This conclusion can also be drawn from equation (11), As a matter of fact 
the expression (56) can be derived directly from (11) or (12). Jn this connection 
it may be noted, that the case discussed by Zemprin in Ann. d. Phys. (4) 19 p. 
783. 1906 is actually that of an aperiodic motion. His equation (14) is thus iden- 
tical with our equation (56) except the small error pointed out in the previous 
communication (148), § 18 note). 
