for the Theoretical Proof of Avogadro's Law. 331 



and we can then convince ourselves directly that 



_ v ay v - 



Although I have already deduced a great variety of rela- 

 tions from fig. 1, yet it would probably furnish several 

 other equations which might be of use in particular circum- 

 stances, e. g. by denoting the magnitude and position of the 

 straight lines v, V, v', V' symmetrically by the magnitude 

 and position of the straight lines X2, P and of the line joining 

 the point P with the middle point of the straight line W. 

 Symmetrical relationships of this kind are particularly conve- 

 nient when we wish to obtain equations in which the magni- 

 tudes before and after impact play the same part as the 

 equation we have used. 



V#X . x(%, X, x') = sj x\x-\-HL— x') .%(#', .£ + X— x', x). 



Second Appendix. 



After correcting the foregoing for the press, I became ac- 

 quainted, by the kindness of the author, with Prof. Tait's 

 paper " On the Foundations of the Kinetic Theory of 

 Gases "*. While reserving for a future occasion my remarks 

 on Prof. Tait's observations on the mean path, and on the case 

 when external forces act, I will here mention only one point. 

 If in a gas on which no external forces act, and whose molecules 

 are elastic spheres, F(#, y, z) dx dy dz be the probability that 

 components of the velocity of a molecule parallel to the axes 

 of coordinates shall at the same time lie between the limits x 

 and x + dx, y and y + dy, z and z + dz, then Maxwell bases 

 the first proof which he gives f of his law of distribution of 

 velocities on the assumption that F(#, y, z) is a product of 

 these functions, of which the first contains only x, the second 

 only y, the third only z. This is the same as the assumption 

 that, for a given component of velocity at right angles to the 

 axis of abscissas, the quotient of two probabilities, viz. the pro- 

 bability that the component of the velocity of a molecule in 

 the direction of the axis of abscissas lies between x and x + dx, 

 and the probability that the same quantity lies between certain 

 other limits £ and f + d%, is altogether independent of the 

 given value of the component of the velocity of the same 

 molecule at right angles to the axis of abscissas. In a 



* Trans. Roy. Soc. Edin. xxiii. p. 65 (1886). 

 t Phil. Mag. [4] vol. xix. p. 19 (1860). 



