250 PROFESSOR TAIT ON THE 



Probability can be, or ought to be, received if they depend upon the results of 

 a complicated mathematical analysis." To this may be added the obvious 

 remark, that the purely mathematical part of an investigation, however elegant 

 and powerful it may be, is of no value whatever in physics unless it be based 

 upon admissible assumptions. In many of the investigations, connected with 

 the present subject, alike by British and by foreign authors, the above remark 

 of De Morgan has certainly met with scant attention.] 



In my first paper I spoke of the errors in the treatment of this subject 

 which have been introduced by the taking of means before the expressions were 

 ripe for such a process. In the present paper I have endeavoured throughout 

 to keep this danger in view ; and I hope that the results now to be given will be 

 found, even where they are most imperfect, at least more approximately accurate 

 than those which have been obtained with the neglect of such precautions. 



The nature of Clerk-Maxwell's earlier investigations on the Kinetic 

 theory, in which this precaution is often neglected, still gives them a peculiar 

 value ; as it is at once obvious, from the forms of some of his results, that he 

 must have thought them out before endeavouring to obtain them, or even to 

 express them, by analysis. One most notable example of this is to be seen in 

 his Lemma {Phil. Mag. 1860, II. p. 23) to the effect that 



/ 



m + Z ax\ / 



where U and r are functions of x, not vanishing with x, and varying but slightly 

 between the limits — r and r of x; — and where the signs in the integrand 

 depend upon the character of m as an even or odd integer. This forms the 

 starting point of his investigations in Diffusion and Conductivity. It is clear 

 from the context why this curious proposition was introduced, but its accu- 

 racy, and even its exact meaning, seem doubtful. 



In all the more important questions now to be treated, the mean free path 

 of a particle plays a prominent part, and integrals involving the quantities e, or 

 e + e l (as defined in §§9, 10, 28) occur throughout. We commence, therefore, 

 with such a brief discussion of them as will serve to remove this merely 

 numerical complication from the properly physical part of the reasoning. 



X. On the Definite Integrals 



/ V JL and 

 Jo e 



33. In the following investigations I employ, throughout, the definition 

 of the mean free path for each speed as given in § 11. Thus all my results 



