2\.„ 



and 



16 Mr. S. H. Burbury on the Law of Probability 



the r's and u% as regards dimensions. In <r Q , Q must have 



zero dimensions, that is r^ has zero dimensions, or those o£ 



u are the reciprocal of those of r. Therefore a, /3 must have 



dimensions the reciprocal of those of r 2 . If, therefore, for 



L T L 2 



Instance r has dimensions y^ , u has -=- , a x b 12 &c. have ^ > or 



if r has j , u has ^ . Also if r has ^r, D has L-^j , 



D T 2 



-—p &c, or a, ft, have p. 



Pakt II. 



26. We have thus proved that </>(r 1 ...r n ), which represents 

 the chance of the variables s 1 ...s n having the values r x ...r n , 

 or values infinitely near thereto, has the form <b(i\. . .r n ) = C6~ Q , 

 and Q is expressed as a quadratic function of r^:. .r n , con- 

 taining products as well as squares of the variables, and with 

 coefficients as above stated. Also expressing <l>(r 1 ...r n ) in 

 terms of u^.Mn, we may write 



0(ri. . .Vnjdr^ .:dr n =ylr(u 1 . . Mn)du x . . .du n X ±~r J: -r 1 



— tyiui- • Mj^dUi. . .du 7l D, 



or since D = -r- , 



^•(mi. • .u n )du L . . .du n =(j)(r 1 . . .r n )dri. . .dr n &. 



Whether we use r^.. r n or w 1 ...?/,„for independent variable 

 is a question of convenience. 



Since e~® does not contain the time explicitly, we are in 

 effect assuming that it is independent of the time, and there- 

 fore that if the system represented by ri...r n is in motion, 

 such motion is stationary. And (f>(r 1 ...r n )dr 1 ...dr n , or 

 ■Tlr(u 1 ...u,i)du 1 ...du n , represents the time during which, on 

 average of any sufficiently long time in that stationary 

 motion, the variables lie within the limits r x . . .r^ di\, &c, 

 or iti...Ui + dui, &c. 



27. Since Q contains products of the variables, the law of 

 distribution of the values of s 1 ...s n is not generally of the 

 form g-( TO i s J 2 + wl 2s 2 2 + • *nV>, w ith mx..jm n constants, and cannot 

 possibly have that form, unless firstly every 5 = 0, and 

 secondly there is no correlation, for if Q=2wis 2 , e~® is the 

 product of n factors each containing only one of the variables 

 .$!... s„. Therefore, by definition, there is no correlation. 



