158 Prof. Karl Pearson on Deviations from the 



that denoted by % is given by 



P = 



lm-^ 2 ^^---^]: 



the numerator being an n-fold integral from the ellipsoid % to 

 the ellipsoid go , and the denominator an n-fold integral 

 from the ellipsoid to the ellipsoid go . A common constant 

 factor divides out. Now suppose a transformation of coordi- 

 nates to generalized polar coordinates, in which ^ may be 

 treated as the ray, then the numerator and denominator will 

 have common integral factors really representing the genera- 

 lized " solid angles " and having identical limits. Thus we 

 shall reduce our result to 



e X d X 



I e X d X 



This is the measure of the probability of a complex system 

 of n errors occurring with a frequency as great or greater 

 than that of the observed system. 



(2) So soon as we know the observed deviations and the 

 probable errors (or o-'s) and correlations of errors in any 

 case we can find % from (ii.), and then an evaluation of (iii.) 

 gives us what appears to be a fairly reasonable criterion of the 

 probability of such an error occurring on a random selection 

 being made. 



For the special purpose we have in view, let us evaluate the 

 numerator of P by integrating by parts ; we find 



f e-**'tf- 1 d X = [% n - 2 +(^-2)%' l - 4 + (n-2)(n-4) % ' 1 - 



+ . . . + (n— 2)(n— 4)(n— 6) . . . (n — 2r — 2)x n - 2r ~\ e -ix 2 



+ (n — 2)(ra — 4)(n— 6) . . . (n — 2r)\ e'^rf 1 - 2 ?- 1 



= (n— 2)(n — 4)(n — 6) . . . (n — 2-r) | I e'^x^^'^X 



[_n — 



i— 2r <y» l_ 2r + 2 -.n-lr + i 



+ €-**<*—=> 4 7 ETT7 ^^-ITT + " 



2r (» — 2r)(w — 2r+2) (n — 2r) + (n — 2r + 2) (» — 2r + - 



.. (n-2)J J* 



+ 



(n — 2r){n — 2r + 2) . . . (w — 2). 



