194 Prof. F. Y. Edgeworth on 



business in a column furnishing positive values of the devia- 

 tion x 2 . It is clear that, if we admitted such values indis- 

 criminately, we should obtain an indeterminate result for p. 



Of course, the more elaborate tabulation employed by Mr. 

 Galton in the paper referred to and elsewhere is useful for 

 the purpose of verif}dng that the law of error is indeed 

 fulfilled, of testing up to what distance from the mean it 

 holds good, and for other purposes. 



It may be observed that the calculation of the coefficients 

 of correlation is not affected if we take as the unit, not the 

 quartile, but the same -i-476 . . ., the modulus of each set of 

 observations. In what follows it is to be understood, when 

 the contrary is not expressed, that the variables x 1 x 2j &c. — 

 representing deviations of different organs or attributes from 

 their mean value — are thus measured. 



To extend Mr. Galton's method to the case of three vari- 

 ables x u x 2 , #3 ; first determine the coefficients of correlation 

 for each pair (x 1 x 2 ), (je x x 3 )^ (x 2 # 3 ) ; say, p 12 , p u , p u . ^ Thus 

 the probability of any particular x x and # 2 concurring is 

 'Ke~ s dx l dx 2 , where K is a properly taken coefficient, and 



1-P12 2 1-/>12 2 1-P12 2 ' 



(See above, p. 190, and Mr. Galton's work there referred to.) 

 Now the expression above written must be derived from the 

 sought expression Je"^dx 1 dx 2 dx 3} where 



R = Pl^l 2 +p 2 X 2 2 +p s W 3 Q + 22 12 #i# 2 + %qi3#l®S + 2223 V3? 



by integration with respect to x d between extreme limits + go 

 and — go . If we watch the process of integration, we shall 

 find that the coefficient of x* (in the exponent of the 

 integral) is 



Pi : 

 the coefficient of x i x 2 is 



v qw Pa— 9u 923 . 



and the coefficient of # 2 2 is 



Pi ' 

 These coefficients are to be equated respectively to 



1 — 2/o 12 t , 1 



i 9' 1 2' an( -^ 1 2 1 



