J 287 



the ijz plane, as is easil\ derived from considerations of syniinetrj, 

 and will on either side of it present the same sign as the imaginary 

 agent. The excess uw, therefore, also shows these signs, which comes 

 to this that the cloud has shifted in the direction of the negative 

 .7>axis, as was to be expected. 



When we no longer assnme ti to be constant, but a = (tz, /r 

 will obtain a positive sign in the 1-' and the 3'^' (piadrant, i.e. the 

 clond will be elongated in the direction of a line that forms an 

 angle of 45° with the original axes and lies in the I""' and the 3''' 

 quadrant. 



§ 5. Distribution of the density in n floioiny 'iqiiid <it tJie critioil 

 point. When after these preparatory remarks we proceed to the 

 problem of the anomalies of density in a flowing liquid, we shall 

 (irst have to calculate 7^7/' according to equation (13^7). For this 

 purpose we first remarl< that the value given for "^^w by this 

 equation is only a consequence of the movement of (he gas relative 

 to the centre of force. When we put // = constant, and if we then 

 make the centre of force participate in the movement, it would of 

 course come to the same thing as if eveiything was at rest. We 

 shall, therefore, always have to take this relative velocity for u in 

 equation (13rt). The value of 7 ^?(; in a volume element ^/o; (/// r/2 ^ c/to 

 can now be calculated as the sum of contributions fiii-nished by 

 forces exerted by the substance in the different surrounding volume 

 elements. When we call one of these surrounding elements 

 dx' dij' dz' = dio' and the density in it n' , then the n'dio' mole- 

 cules in it can be conceived as a centre of force. When we put 

 again u = az, the velocity of the substance in d<x> relative this 

 centre will amount to a (~ — 2'). Let us further represent the potential 

 energy of* two molecules at a mutual distance /• by (f'{r), the 

 contribution to \j*iü in doi which is owing to the substance in </<'/ 

 is then : 



y 3jr n , dcf(r') x — x' 



-— ,a(z — z')ndvii' -^- ; — 



a.nl kl Or r 



in which /' represents the distance of the spacial elements dw and 

 duj'. When we turn the axes 45° round the y-axis, and when we 

 call the new axes ^, 1], $, we find for the total value of v* '" • 



V7^w z= — — I n dio . . . (10) 



^ al kTJ dr' 2r' ^ ' 



This equation gives the distribution of the imaginary agent in 

 space. We find from it for the value of w^ in a volume element 



