Kinetic Theory of Gas. 365 



investigation, what the mathematician really investigates is 

 not the problem presented by nature, but some simplification 

 of it. The legitimacy of this process and its value depend 

 upon the important circumstance that, in dynamics, a slight 

 change in the data leads only, at least for a time, to a slight 

 change in the result. Thus in computing the mutual pertur- 

 bations of the planets, the planets may be treated as though 

 they were spheres, made up of untextured spherical shells 

 each of a uniform density throughout ; and it may be left out 

 of account that they approach to being spheroids, with moun- 

 tains on their surface, irregularities of a like kind at greater 

 depths, rocks in those mountains, minerals in those rocks, a 

 different molecular texture in each mineral^ tidal strains, heat 

 expansions by day, contractions by night, and so on — perhaps 

 seas and an atmosphere, vegetation and animals, all in con- 

 stant and complicated movement; with numberless other 

 details. Now it is legitimate to omit all these from our 

 calculation, for though every one of them produces its effect 

 in actual nature, the difference between the outcome of their 

 joint operation and that computed from the greatly simplified 

 data of the mathematician is too small to make any approach 

 to being perceived by any human agency. Hence, for any 

 purpose which is of use to man, the approximation arrived at 

 by the simpler problem is sufficient, wherever the errors are 

 of such a nature that they are not cumulative. Nevertheless, 

 it should be clearly recognized that it is a mechanism illus- 

 trating nature, and not nature itself, that has been mathemati- 

 cally investigated. So it is with all dynamical investigations : 

 the data of nature have to be simplified to bring the task 

 within the range of man's power over mathematical analysis ; 

 and the result is satisfactory because of the important funda- 

 mental principle referred to above, that in dynamics a slight 

 modification of the data furnished by nature may be safely 

 made, because it leads only to there being a small difference 

 between the calculated result and that which occurs in nature 

 — of course care being taken in each case that the approxi- 

 mation of the computed result to what occurs in nature shall 

 be sufficiently close for the object we have in view. 



These criticisms need to be pressed with special emphasis 

 when we are examining investigations into the dynamical 

 condition of gases. Here we have to substitute for the data 

 of nature others which differ from them in undesirable ways 

 and to an undesirable extent, in order to arrive at data simple 

 enough to be available as a basis for mathematical deductions. 

 Nevertheless, some of the results are not appreciably affected 

 by this too great simplification of the data : for instance, 



