298 
MATHEMATICS: N. WIENER Proc. N. A. S. 
Therefore 
d 2 /t-c/(2M 2 Q 2 ) 
< 1 /t[m(o) (l-e~ Qt )/(MQ) 
+ c(S-e- Qt )(l-e- Qt )/(4,M 2 Q 3 t) 
< [w(o)/(MQ)J 2 +3c/(4M 2 Q 3 ) 
This represents the absolute departure of d"Jt from constancy. Writing 
v for m(o)/M, the initial velocity of the particle, we get 
\d?/t-c/(2M 2 Q 2 )\ v 2 /Q 2 
c/(2M 2 Q 2 ) ~ c/2M 2 Q 2 ' U 
This is a measure of the relative departure of df/t from constancy. v 
cannot exceed, on the average, the velocity given on the average to the 
particle on the basis of the equipartition of energy; actually it is much 
smaller. c/(2M 2 Q 2 ) can be found directly, as it is nearly the observed 
value of d 2 /t. Q can be readily computed from the constants of the par- 
ticles. Taking as a typical case one of Perrin's experiments on gamboge, Q 
turns out to be of the order of magnitude of 10 8 , c/2M 2 Q 2 of the order of 
magnitude of 10 -8 , and the kinetic energy velocity of the order of magni- 
tude of 10 _1 . Hence the proportionate error is of the order of magnitude 
of 1CT 8 . 
A proportionate error thus small is quite beyond the reach of our meth- 
ods of measurement, so that we are compelled to conclude that d 2 /t, un- 
der the hypotheses we have here formulated, is sensibly constant. There 
are cases, however, which seem to give a slightly different value of d 2 /t 
for small values of the time than for larger values. The explanation has 
been suggested 5 that over small periods the Einstein independence of an 
interval on previous intervals does not hold. The result of the present 
paper would be to suggest strongly, if not to demonstrate, that the source 
of the discrepancy, if, as appears, it is genuine, and not due to experimental 
error, is in the fact that Stokes' iaw itself is only a rough approxima- 
tion, and that the resistance does not vary strictly as the velocity. 
1 Paris, Bull. Soc. Math. France, 1919, pp. 47-70. 
2 "The Average of an Analytic Functional," in the last number of these 
Proceedings. 
3 Leipzig, Ann. Physik, 17, 1905 (549). 
4 Ann. Chim. Phys., Sept., 1909; tr. by F. Soddy. 
5 Cf. Kleeman, A Kinetic Theory of Gases and Liquids, §§ 56, 60. 
