Probable in a Correlated System of Variables. 159 



Further, 



/too S*T> 



i e-ix 2 x n - ] dx=(n-2){n — 4)(rc— 6) . . . (n-2r)\ e - | * 2 % n_2 '"~%- 

 Jo Jo 



Now n will either be even or odd, or if n be indefinitely 

 great we may take it practically either. 



Case (i.) n odd. Take r= — ^— . Hence 



_ J.W +«-»{? + A + A + • • • + nrfe^i} 



y- ix * dx ... (-) 

 p -v/|£ Vhy * 



VI - k5 (r + 1 2 ^ + T^ra + • • • + r^&ri)- w 



As soon as x is known this can be at once evaluated. 

 Case (ii.) n even. Take r=-\n — 1. Hence 



( g-fcVdY + e - ** 2 .! — + X 4 ■ ^ ■ 4- + % w ~ 2 1 



T> _ Jx XX (2 + 2.4 4 2.4.6 + --- + 2 .4.6...n^2 / 



L — TToo ~ ' 



I e'i^ X dx 

 Jo 



1+ 2 + 2.4 + 2.4.6^'-' + 2.4.6.../^2/ W 



P = 

 But 



Thus 



The series (v.) and (vi.) both admit of fairly easy calcu- 

 lation, and give sensibly the same results if n be even moder- 

 ately large. If we put P=^ in (v.) and (vi.) we have 

 equations to determine X = Xo> * ne va l ue gi ym g the "proba- 

 bility ellipsoid." This ellipsoid has already been considered 

 by Bertrand for n=2 (probability ellipse) and Czuber for 

 w = 3. The table which concludes this paper gives the 

 values of P for a series of values of % 2 in a slightly different 

 case. We can, however, adopt it for general purposes, when 

 Ave only want a rough approximation to the probability or 

 improbability of a given system of deviations. Suppose we 



