8o SCIENCE AND METHOD. 



mixing of cards, but to all mixing, to that of powders 

 and liquids, and even to that of the gaseous molecules 

 in the kinetic theory of gases. To return to this theory, 

 let us imagine for a moment a gas whose molecules 

 cannot collide mutually, but can be deviated by col- 

 lisions with the sides of the vessel in which the gas 

 is enclosed. If the form of the vessel is sufficiently 

 complicated, it will not be long before the distribution 

 of the molecules and that of their velocities become 

 uniform. This will not happen if the vessel is spherical, 

 or if it has the form of a rectangular parallelepiped. 

 And why not ? Because in the former case the dis- 

 tance of any particular trajectory from the centre 

 remains constant, and in the latter case we have 

 the absolute value of the angle of each trajectory 

 with the sides of the parallelepiped. 



Thus we see what we must understand by conditions 

 that are too simple. They are conditions which pre- 

 serve something of the original state as an invariable. 

 Are the differential equations of the problem too 

 simple to enable us to apply the laws of chance? 

 This question appears at first sight devoid of any pre- 

 cise meaning, but we know now what it mean.s. They 

 are too simple if something is preserved, if they 

 admit a uniform integral. If something of the initial 

 conditions remains unchanged, it is clear that the 

 final situation can no longer be independent of the 

 initial situation. 



We come, lastly, to the theory of errors. We are 

 ignorant of what accidental errors are due to, and it is 

 just because of this ignorance that we know they will 

 obey Gauss's law. Such is the paradox. It is ex- 

 plained in somewhat the same way as the preceding 



