246 Prof. F. Y. Edge worth on the Application of 



3. Correction of the normal formula. — There is reason to 

 believe that the conditions for the genesis of the normal law 

 are not perfectly fulfilled by a molecular chaos. With 

 regard to collisions, the elasticity can hardly be regarded as 

 perfect (Cp. Burbury, Science .Progress, vol. ii. § 7). In 

 the case of encounters, the beginning and the end of the 

 encounter being defined arbitrarily, it is to be apprehended 

 that there may be some slight interaction between the 

 molecules. Both in the case of collisions and encounters 

 the ultimate velocities have elements in common (/. c. p. 251). 

 If the independence of ' the elements aggregated and the 

 linearity of the aggregation is not perfect, the function proper 

 to represent the frequency of the aggregate would no longer 

 be the simple normal law, but the corrected form described 

 by the present writer as the generalized law of error (Cam- 

 bridge Philosophical Transactions, 1904 ; Journal of the 

 Royal Statistical Society, 1906). The first correction, 

 involving odd powers of the variables, may presumably be 

 neglected. The formula given by second correction may be 

 written in the simple case of motion in one dimension 



Fx{l + «(i-2MU 2 + fM 2 L T4 )+y5(|-2mw 2 4-|m 2 u 4 ) 



+ 7(l-2MU 2 )(l-2mK 2 )}, (9) 



where F is the normal function (1), a, /3, <y are small co- 

 efficients proportional to the excess of the respective mean 

 powers [U 4 ] [u 4 ] [U 2 u 2 ] over what each would be if the 

 normal law were perfectly satisfied (Camb. Phil. Trans. 

 he. cit. p. 118, 1904). 



The above expression, which may be written as a function of 

 U multiplied by a function of u, if 7 = (a/3 being negligible), 

 need not express correlation between the velocities. What 

 slight interdependence there may. have been among the 

 elements — the long series of velocities from which the U and 

 u in (1) resulted-— may have disappeared in the compound 

 (Camb. Phil. Trans, p.* 126, 1904). 



4. Correlation. — The correction above made may, how- 

 ever, co-exist with (norma]) correlation. Our first argument 

 is not inconsistent with Dr. Burbury 's contention that there 

 may be some interdependence between the velocities of con- 

 tiguous molecules (Science Progress, vol. ii. § 33, 1894 ; 

 Phil. Trans. 1886 : ' Dense Gases,' § 13 ; ' Kinetic Theory of 

 Grases/ p. 11 ; et passim). But there are two cases of inter- 

 dependence which should be excluded from the field to 

 which (2) applies, namely (a) between molecules in an 

 initial distribution, or on their way to the ultimate normal 



