Probabilities to the Movement of Gas-Molecules. 255 



knoTvn form, but to determine the most probable form of a 

 function, consistent with certain data, that form for which 

 a certain conditional probability becomes a maximum — 

 namely, the probability that, if the proposed function repre- 

 sented the sought ultimate distribution, the given conditions 

 would be fulfilled. One datum in the case before us is 

 summarized by the equation \ ( /(U, n) d~U du = constant (say 

 unity, or else a given number of molecules) ; U and u 

 being relative to the centre of gravity supposed at rest, the 

 integration extending over all possible values of U and u. 

 Another datum is furnished by the conservation of the energy 

 of the system. Also it is known that "H" is approximately of 

 the form j \flogfdJJdu *. Accordingly, it is proper to 

 minimise the expression 



jj flogfdJJ du+j(jjjjfdV. du- const.) 



+ h( jj/(MU 2 + ?mr) -const.), . (9) 



where j and h are undeterminate multipliers. Varying (9), 

 we have for the first term of expansion 



w 



3/(log/+l+,; + 7i(MU 8 + wM 2 )) = 0. 



Whence it is deducible (the second term being positive) 

 that 



/(U,w) = Const. exp.-/i(MU 2 + rou a ). . . (10) 



More generally, the centre of gravity moving uniformly, 

 there is a second datum, namely MU+fliu = const. But the 

 only effect of this condition is to introduce into (9) a new 

 term under the sign of integration, say k(MJJ + mil), where 

 k is a new constant. There will thus be introduced first 

 powers of U and u into the exponent of /, which may be 

 written 



-7i(MU 2 + wm 2 ) + *(MU + wim) + const., . (11) 



where h and k are to be determined from the given mean 

 energy of the molecules and the given (constant) velocity 

 of the system's centre of gravity. 



The validity of this reasoning is confirmed by observing 

 that the law of error itself, as applicable to statistics gene- 

 rally, is deducible by parity of reasoning. For instance, 



* By a generalization of the approximation whereby the exact 

 expression (7) was reduced to (8). See Boltzraann, Vorlesungen iiber 

 Gattheorie, Abschnitt iii. §§ 5, 6, & 8 ; Jeans, i Dynamical Theory of 

 Gases,' ed. 2, § 22 ; and other leading authorities. 



