366 MATHEMATICAL APPENDICES 8* 



to continually change their direction of motion. Yet they move, 

 and this motion, even if it should happen to be in curved paths, 

 produces a kinetic pressure just as in gases. 



This pressure may be calculated in the same manner as gaseous 

 pressure, and the calculation leads to exactly the same result. If 

 no attractive or repulsive forces act between the molecules of the 

 liquid and of the alien body the particles of the latter move about in 

 the liquid, not in continuously curved paths, but in straight paths 

 like the particles of gas in vacuum. The only difference is this, that 

 the alien particles collide very much oftener in the liquid, and that 

 therefore the free path traversed between successive collisions is 

 very much shorter. But this difference has no influence on the 

 validity of the calculation. There is only a change in signification 

 of the function f (t, u, v, iv), which denotes the probability of the 

 path being straight for the interval t, to this extent that it has 

 values differing from only for very small values of t, and that 

 it vanishes for greater arguments ; but thereby no alteration in the 

 final result of the calculation is entailed. 



The result is similar in the other case which better corre- 

 sponds to actuality, viz. when forces do indeed act between the 

 liquid and the alien particles, but when the forces to which a 

 particle is subjected from the molecules of the liquid which surround 

 it mutually balance each other on the average. The path of a 

 particle is then continuously curved, as it is continuously under 

 the action of molecular forces ; yet we may look on the path as 

 rectilinear which is traversed during an infinitely small interval 

 of time. To this straight bit of path and to the short time needed 

 for it we have to apply the foregoing calculation, which results in 

 the same value as before for the pressure due to the motion of the 

 added alien particles, and gives the same relation between this 

 kinetic pressure p and the kinetic energy K of the particles con- 

 tained in unit of volume, viz. 



This formula remains therefore at least approximately correct 

 when the stretches of which the paths of the molecules are made 

 up are not of finite length. It would therefore be mathematically 

 stricter so to express the condition of its validity that the particles 

 whose motion causes the pressure move under the laws of inertia 

 and collision only, without being subject to external force. 



