of Error and ( orreldted Averages* 519 



singularity and continuous in the sense before defined *, in 

 the neighbourhood with which' we are concerned: throughout 

 the sel of \ ;» luos formed by assigning i»> each of the variables 

 .'■,, >r. : , A < • . nil the values of' which it is cabable, 



Examples of this theory nmv be seen in all the important 

 oases in which the law of error is fulfilled i namely, errors of 

 observation (broper), gunnery, and natural groups generally ; 

 in nil which departments the fulfilment of the law is doubtless 

 due to the Independent action of numerous small agenoies. 



The apparent number of exceptions is much reduced l>\ the 

 consideration thai the rule still holds good if we substitute for 

 fi, in the above staternehc, any function of £,, say £,, which 

 does not become infinite lor any of the values assumed 



by £ i J and make similar substitutions t fpr fj, ft, &0, Pbr 



example, the law of error will be obeyed by the varying 



values oi' 



P(i, | fc,x, + &,&o.), 



where $,=#, fe*fc 8 , 5 8 «ft*tJ 



provided that Pi'(xi)> 1^. : \ \-) , 4c. neither vanish, nor beoome 

 infinite; and Pu^fo), Fj| (tf|), &c. are not very large§. 



Nor,even thus construed, are all the cases in which EY(xi), 

 Pj'(x 8 ), &c. all vanish genuine exceptions. When the first 



term of the expansion vanishes, the function in general reduces 



lo the second term, which is oi' the form 



F u "(x w v.. ,ve.)x£," ) rV'Cpta ** ApO-x-lV ' *°« 

 i 8P lf / 'txi|Xi,4o0x{ l 6 i Ac 



Which in genera] May be reduced to a sum oi' ii squares of 



the form lh 9 I '/■/' I &0. ; where each // is ;i linear function of 



the fs, Tne systems of the g'sand the t*s being thus related, 

 it is allowable to regard the v's as the independent, and the 

 £"s as the dependent, variables; Accordingly the original 

 function reduces to a sum of numerous elements independently 



Oscillating. Which is the essential condition tor the fulfilment 



of the law of error. For example, if for F we have 



(.r.-x,)-' I (# I -Xi)*+&C., 

 the first term of the expansion \anishes for the point 

 tfi-Xj, .r,= x.„ tfeo.j 



* Ante, ]». 4SL 



I By a proper change in the law oi' frequiencji for the elements, this 

 case may be reduced to tli»> preceding, 



| The elemental laws of frequency and the functions i may both 



l)t> iiiisi/iinih fried/. 



