( 799 ) 



and at the same time a way ■^ APr cos f dt is saved by the collisions, 

 those moleeules seem to move Avith a xeloeitv Pr cos¥d({\ -{- 4iA), 

 and so the number of collisions has increased in the same ratio. 

 KoKTKWEG, however, continues: "In order to obtain therefore the 

 same number of collisions with llie plane AB, the molecules will 

 only Iiave fo pass over a way Pecos f dt, ijistead of over a wav 

 Pvcosi{i — 'iA)dt, in other words, the number of collisions of this 

 system increases in the I'atio (1 — 4.1): A." Now between the two 



results there is onlv diflference of order — aiid iji so far as we wish 



A'' 



to neglect the (piantities of this ordei', Koktk\vk<;'s i-esult may cer- 

 taiidy be accepted. If, however, we wish to solve the proldem 

 rigorously, the fu'st result alone can be accepted. 



For KoRTEWEG makes it appear, as if — taking into account the part 

 of the way being saved — an ecpially long way is described in the 

 time (1 — 4:A)dt, as in the time dt without doing so. Now in the 

 last case the molecules pass over a way Pv cos 8 dt iji the time dt, 

 so PvcosBdt{\ — 4J) iji the time (1 — ^A)dt. hi the time dt there 

 is saved -i APi: cos t dt. in the time (i — ^A)dt therefore (1 — 4.1) 

 4: APr cos 8 dt : so the <listauce, passed over in the time (1 — 4.1)(/^ 

 by saxijig way and really moving together is somewhat slighter 

 (viz. 1() A' l*r cos 8 dt) than that passed o\ er by the i-eal motion alone 

 in the time dt ^). 



^) Perhaps Kokteweg was led when drawing up the formula nieutiuned in the 

 text by the solution of the problem in one dimension wliich be has given in 

 Nature (loc. eit.) later on. He linds there — i)erfectly accurately — Ibr Ibe time 

 passing between two collisions against the wall of a row of >? [larticles ol'tliameter 

 A wliicli can move over a total distance L witii a velocity V: 



, , _ V 



~ L^nY 

 This formula reminds us more of Korteweg's result than of ours, really however 

 it agrees with the latter, not with the former. For, if we determine the ratio of 



the number of collisions with and without saving wav, it is Q = . Now L is the 



total distance over which the molecules can move, so the path described by their 

 own motion + the path saved; nX is the path saved. So L corresponds with 

 (1 4- 4 ^i) P f cos fc dt, nx with 4 A P v C08 i dt ; so the ratio of the collisions is here 



agaiji (1+4 .4) : 1. To Kohteweg's result 1 : (1—4 A) would the formula Q = ~ ' ^ 



L-~2 «A 



correspond, which agrees with the first as to the terms of llie ordei' , but which 



is certainly not strictly accurate. 



It is true that with the formula for one dimension, with regard to its physical 

 meaning, an equation of state agrees, in which a quantity is subtracted from tlie 

 volume whirb is a function of 6 and r, not a formula of form (1); but we shall 

 see that our formula derived in the text, leads also to such a linal form. 



