903 
fall of the velocity during the action of the repulsive force — 
expressed as function of the time — is now less great than in fig. 
1“, so that the descending branch will be much more horizontal. 
The natural consequence of this is, that the time average gets much 
nearer to w,?. (c, would now become = '/, X °/, R = 4’/, instead of 
6). However — the above calculation is certainly questionable at 
high temperatures, because then log (2 re) may certainly not 
—$ 
be expanded into a series, as at the culmination point of the impact 
x would become —/—s(s’ = s). The expansion into a series up to 
x* used by us, gives a too great value for w‚, hence also a too 
great value for w,?. Instead of rather abruptly, the damping of this 
exceedingly great velocity would take place during a much too long 
interval — so great even that s’ would lie far inside s, which ig 
of course impossible. 
At low temperatures (p great) on the other hand there can be 
no objection to applying the expansion into a series up to x*, because 
then the velocity is so small that it will be reduced to O already 
within a very short interval. Now the modulus & approaches to 1, 
Wore 
3 . 
hence L/ of cosap dp = (sin we, i.e. also to 1 ; but F will approach to 
0 
d 
Jae = log tg (45 + by) = log w—log 1,i.0. to log oo. As, however, 
Li 
at the same time 1—<4? approaches 0, we must examine what value 
(l—k*) F assumes in (7), when 4? is near 1. 
: 4 
According to a well-known theorem’) /’ approaches to log ZE 
a a 
in this case. 
Hence we get for ¢ and w,?, when & approaches 1, from (7): 
1) Cf. among others Lamp, Treatise on Hydrodynamics, p. 170; Gay.ey, Ellipt. 
funct., Art. 72; MaxweLL, Elect. and Magn. I, p. 311—3816; Durèae, p. 190 
et seq., particularly p. 213; KircnHorr, Vorl. p. 270; etc. Better than Duriax’s 
derivation, which is based on LANDEN’s transformation, is KirncHHOFF’s beautiful 
derivation. The latter is founded on the splitting up of the integral into two parts, viz. 
‘ar Uy Nar 
f = af Ee J , in which 6 is a small quantity, which is, however, supposed 
0 0 Ind 
to be great with respect to W1-k? But in both derivations only the limiting 
value of F is reached. 
It is in my opinion a better method to start from Jacosi’s_ relation 
