( 683 ) 



u 



this coordinate for all points situated on I lie first part of the line, 3 



P 

 for all points of the second part, and soon. Then, in the distribution 



that is characterized hv a', h' in', the coordinate of the centre of 



uraxity of the <i points will hv 



or, by (34), 



|V + ;W/ + 5-' + .... + (2y> -\)m!\ -. 



r 



u 



,. + [« + 3i?-h5y+.. . + {2;>-l),il 



P 

 The positive or negative value of 



^-=[.M-;^/:?+5y + .--- + (2/'-l)Ml^. • • (36) 



P 



is thus seen to rej)resent the distance lietween the middle point of 

 the line and the centre of gravity. We have to calculate the proba- 

 bility for (his distance lying between I and 5 -|- d:. 



The problem is easily solved by means of a change of variables. 

 Instead of the quantities a, [3, .... (i, which serve to define a mode 



of distribution, we shall introduce new ones «', i>*'. n', the sui)sti- 



tution being linear and orthogonal. 



Let us take for the fii'st of the new variables 



1 1 1 



^f' = 7^ « M- --^ i^ + • • • + 77- M (3 <■) 



VP VP VP 



and for the second 



P — 1 /> — 3 p — 1 



ir = -^ — «-^ — 1?-... + ^ — fi. . . . (38) 



where the numerators form an arithmetical progression, ^vhereas y. 

 means the positive square root of the sum of the squares of the imme- 

 rators. These expressions (37) and (38) may i-eally be adopted, because 

 the peculiar conditions for an orthogonal substitution are satisfied : 

 in both expressions the sum of the squares of the coetHicients is 1, 

 and we get if we add together the coefficients of (37) after having 

 multiplied them by the corresponding coefficients in (38). As to the 

 coefficients in the expressions for y', . . . n', we may choose them as 

 we like, [)rovided the whole substitution remain orthogonal. 



The reason for the above choice of n' and ,?' will be clear ; the 

 conditioji (35) simplifies to 



«' = (39) 



and, in virtue of (35), the value (36) will be equal to 



y.n 



ê = — /3' (40) 



P 

 in all cases with which we are concerned. 



