1263 

 so. In connerfion with this I remind of the fact that equation (C) 

 with tlie value p^ =r. — ^ is valid tor each particle separalely, and is 



not dependent on any mean value. It may be integrated without reser- 

 vation, after which, if it is thought desirable, ecpiation (E) 

 can be taken into account. When A" is to be calculated, the 

 general mean is always taken. How and why an average over a 

 ?>group could be introduced for the calculation of this quantity, is 

 not clear to me; nor has it been demonstrated by Oknstkin and 

 Zernike ^). 



Accordingly it seems to mo that Ornstein and Zernike have not 

 succeeded in pointing out an error in our derivation. Nevertheless 

 I do not doubt but it must exist. I think I have pointed out the 

 error above in equation (6). Formely we had always thought that A 

 was the sum of a number of terms that were statistically independent 

 of each other. Equation (6) siiows that the increments setting in 

 after a moment t are dependent on the deviation already reached 

 at that moment. And it is obvious that we shall not find the 

 accurate amount for the mean value of the deviations, when we 

 leave this correlation out of account. 



It might now be imagined that this remark entailed that also the 

 l-esult A' = bt should be considered as doubtful. The thesis of the 

 calculus of piobability cited for it is namely only valid when the 

 different terms of the sum are independent of each other, which is 

 not the case here. It has been demonstrated on p. 1260 and 1261 

 that yet there is no reason to doubt the validity of this formula, 

 and that it can even be derived from equation (6). 



Above I have derived equation [C) from {B). I have done so 

 because such a derivation is also valid for other analogous cases, 

 e.g. for equation (6) on p. 1258. When we only wish to derive 

 equation {C), we can do so also in another way, as has been done 

 by Miss Snethi.age and me''). This derivation even brings us farther 

 than the al)Ove given one. It justifies us in the statement that the 



') The formulae derived by the writers for averages over a r-group might possibly 

 be of value for another question, namely this: how do the particles that at first 

 have the same velocity, spread over the different velocities? 



2) Messrs. Ornstein and Zernike stale I.e. that we give formula (C) without 

 a proof, after which they furnish a proof, which, however, does not depart from 

 ours except in this detail that we average immediately over all the particles, and 

 tliey in stages first over the particles of a ?;-group, and then over the different 

 v-groups. The result is, of course, identical. 



87* 



