526 Laiv of Error and Correlated Averages. 



last section. If the objection is understood in a less narrow 

 sense as suggesting that antinomies arise when correlations 

 other than linear are supposed, the answer furnished by this 

 section is that correlations other than linear are not to be 

 supposed (in the case of composite organs). 



VI. In the investigation of the most general conditions of 

 the law of error, one more step has still to be taken. We 

 have so far supposed every member of a group to be a definite 

 function of a number of elements, each of which varies 

 according to some definite law of frequency. But now it is 

 to be added that the conclusion is not seriously affected if we 

 suppose these functions to be themselves variable ; provided 

 that they preserve certain characteristics approximately 

 constant, namely, the parameters of the probability-curve. 

 These are its centre and its dispersion : or 



PiX, + p 2 x 2 + &c. +p n x n ( = X) , 



/>i% 2 +^W+&c.+ i?w V(==c a ), 



where p lt p 2) &c. are " weights " (identical with the 

 Fi 1 (xi, x 2 , &c), Fg 1 (x 1? x 2 , &o.), 



employed before) ; x 1? x 2 , &c. are the average values of the 

 elements as before ; Ci, c 2 2 , &c. are each the modulus squared 

 for the corresponding elements. 



Now these parameters will not be seriously affected if we 

 suppose each of the quantities x and c 2 to be vibrating about 

 an average value. For the centre will then become changed 

 from 



which differs from X by a percentage of the order -, supposing 



the relative fluctuation of each x to be the same as that which 

 according to the usual theory is ascribed to X, namely, of the 



order —y^. For, if A# is of the order x-r V w > S/?A# (by 



the usual theory) will be of the order \/n x (x-r- \/w) or x ; 

 and x is of the order X-~n. By parity of reasoning, if 

 c i 2 ? c 2 2 > & c v each fluctuate over a relatively small range, the 

 value of C 2 will not be seriously affected. 



In fine, the result will still remain unaffected if we attri- 

 bute to each p a relatively small variation Ap; corresponding 

 to a variation of 2Ap on p 2 . 



So loose are the conditions which suffice for the fulfilment 

 of the law of error. 



