( G84 ) 



Now, the modes of distribulion for which the vahie ot § lies 

 between | and i, -\- d c, are tliose for which |f lies l)etweeii /i' and 



di3' = ^d^ (41) 



y." 



Since «' = 0, every mode of distribution may be detined by the 

 values of |5' . . . fi', these quantities being-, like a, ii, . . . (i, capable of 

 \ery small \'ariations. 



We can therefore select, among all the modes of tlistribution, those 

 for which ^' . .41 lie between ^' and |5' -f (/,?', y' and y' + ^/y', etc. 

 The number of these may be represented by 



lid^'...dn' (42) 



where // is a coefTicient whose value need not be specified. It sufTices 

 to know that it is independent of tlie \alues chosen for /5' . . . ft'. 

 This is a consequence of the linear forui of the relations between 

 these variables and a, h, . . . tn. 



As the just mentioned modes of distribution, whose number is 

 given by (42), differ iuliuitely little from one another, the probability 

 P may be taken to be the same for each of them. Hence, the proba- 

 bility for the occurrence of one of these modes, no matter which, 

 must be 



I>Pd^'...dn' (43) 



From this we may pass to the probability for /?' lying between 

 i5' and fi' -f (/,>", whatever be the values of y' . . . ft' ; we have only 

 to integrate with respect to these last variables. Now using the funda- 

 mental property of an orthogonal substitution 



«^ + (5^ -f . . . + ft^ = «'^ + iV^ + • • • + t*'% 



and attending to (39). we write for (43) 



— — /J 7 (,'"- + • •• + y'-) 



h Pjn e dlï . . . f/ft'. 



If we integrate this expression from — x to + ao , as may be done 

 for obvious reasons, denoting by I- a coefficient tliat does not depend 

 on (3', we find for the probability in question 



1 

 ke - d^'. 



On account of (40) and (41) this is equal to 



y(-V 2x'«'''^$ (44) 



I'' being a new constant. 



