of Error and Correlated Averages. 



521 



to se between extreme limits, y being treated as constant, the, 

 result is approximately of the form K<? -A "^\ This condition is 

 fulfilled by functions of the form Je~ n , where R is a quadratic 

 function of x and y *. By parity it may be presumed that, if 

 the law of frequency for eacli element involve three or more 

 variables ,r. //, s, the law of frequency for the compound is 

 Je~ R , where R is a quantic of the second order. 



Assuming the appropriateness of this form, I have in a 

 former article f shown how to calculate the coefficients 

 from the measurements of correlated organs. I now justify 

 that assumption. 



The general theorem, besides its direct applications to the 

 measurement of organisms, has other interesting consequences. 

 We have here a theoretical ground for the practical use of the 



* The a priori demonstration that this form should prevail is con- 

 firmed, in the case of correlated organs, subject to a number of independ- 

 ent influences, by a posteriori verifications, such as that which was given 

 in a former paper (Phil. Mag. Aug. 1892, p. 107); and the following. 

 If the assumed form prevail, we should expect that p 12 , the coefficient of 

 correlation obtained by assigning values of one organ x and observing the 

 corresponding values of another organ y (op. cit. p. 191) should be equal 

 to p.,i obtained by assigning values of y and observing the corresponding 

 values of x. That this condition is very nearly fulfilled by the measure- 

 ments of three hundred men supplied to me by Mr. Galton appears from the 

 following results calculated by Mrs. Bryant, D.Sc. ; p ]2 , p 21 , &c, having 

 the meaning just explained ; and the suffixes 1, 2, 3 re 'erring respectively 

 to stature, cubit-length, and knee-height. 



Pl2 = -75 p 2:i = -81 p 31 = -93 



p 2l = '8S p.^ = 'S7 p i;! = , 80. 



The correspondence is as close as, in view of the probable error 

 (incideut to the limited number of observations), was to be expected. 



The results obtained from the positive and negative deviations from the 

 average, results which should be equal, in theory, are in fact as 

 follows : — 



From Positive 

 Deviation. 



From Negative 

 Deviation. 



i 

 Pi 2 \ "SO 



•80 



p 23 -87 



■S2 



Psi ! '80 



•87 



The analogous verifications obtained bv Mr. Gallon (Proc. Roy. Soc. 

 1886 and 1888) and Prof. Weldon (ibid. 1892) are to be added, 

 f " On Correlated Averages," Phil. Mag. Aug. 1892. 



