TRANSACTIONS OF SECTION A. 621 



14. On the Law of Error in the Case of Correlated Variations. 

 By S. H. BuRBURY, F.R.S. 



If we have a great number, N, of independent magnitudes, each liable to 

 variation according to any law of its own, but remaining" always finite, the chance 

 that their sum, each divided by VN, shall lie between .r and x + dx is proportional 

 to e~'^^-dx, where h is a constant. The proof of that proposition is originally due 

 to Poisson. The first application of it was to errors of observation, each of the 

 * magnitudes ' aforesaid being such an error, and the N observations being supposed 

 independent. 



Modern writers, among others Mr. F. Galton and Mr. F. Y. Edgeworth, have 

 substituted for the single magnitude given by each independent -observation a 

 group of mutually dependent or ' correlated ' magnitudes, and for the single square 

 forming the index in e"'"- a quadratic function, i.e., sums of square and products 

 of the correlated magnitudes. If, for instance, they be denoted by x and v, the 

 expression corresponding to e-'"W.r will be e-^'"'^-+''■'^'J+'■^y\h■dl/. The coefficient 

 d expresses the fact of ' correlation ' between .r and ?/. 



The object of the following paper is (1) to extend the purely mathematical 

 investigation hitherto applied to the case of single magnitudes to the case of groups 

 of magnitudes, the members of each group, although ' correlated ' with one 

 another, being still supposed independent of any other group — in this I only 

 follow the lines of the known proof; and (2) to show how in certain cases the 

 method may be extended to groups which do not possess this property of mutual 

 independence. 



Part I. — Articles 1-15. 



1. Let/a(a'j . . . «„), or/„, be a continuous function of the variables a, . . . an. 

 Let/(,(/!ii . . . b„), or/, be the same or a different function of the variables/;, . . . bn, 

 and so on to fq(g^ . . . q„), there being N functions. Denote by Pa the integral 



. . . fada^ . . . da,„ 



and by Pft 11 J^; • /" ^^i • • • <^^"> &c. 



2. Assume the b'a to be independent of the a's, so that the variables a are not 

 contained explicitly or impUcitly in/j . . . or/g, the variables b are not contained 

 in fa, or in/,; . . . fg, and so on. Then 



[PaPs . . . P,, = . . , faf . . . fdUj . . . dqn. 



3. Of the K,j variables, «, . . . q^ form now n linear functions whose type is 



Xjrt, +Xo^i + . . . 

 /Xjff, + y^-Jji + . . . 



where the coefficients are numerical and of the order of magnitude . Let it 



be required to find the value of 



w 



. . . fafb . . . fda^ , . . dqn, 



subject to the condition that the linear functions lie respectively between limits 



•S, + . . . S, + ffoj, &c. 



