MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 239 



Now these must be multiplied by 2 and integrated for x l and x., from oo to + co. 

 These integrations follow at once, if we remember that 



na ''i = ll zx * dxt dx. 2 , ncr\ = \\zxi dxi dx.,, ncr^-.r^, = \\zx l x., dx : dx. it 



by definition of o^, cr.,, and r r ,. 



Thus we obtain the following system : 



n 2 - rl, n 2 - 7-f., 



if fl _ . 



11 01 o ' W 22 o-i " ., ' 



o"j 1 rjo cr; 1 ?'jo 



nr \-> 



ao \ 

 /7 2 ) 



where the a's are those of Eqn. (vii.), p. 236, obtained by taking the above differentials 

 of the logarithms of z in order. 



We can now write down the correlation surface, giving the frequencies of errors in 

 the constants : 



n 



, L j 



" ^ IT? a-4) " ^(i - "/-fD " h ^a 



f it, (2 rr,) . /; 2 rr. 



x expt . _ i L J - . 







| i ,, 



cr, 1 / 12 a.,, i i } ., i r ,) 



2m-,. 2/M-,., 2/n-;, 1 ... 



-- Ao-., A?',., ----; ACT, Ar,., - - Ao"! Ao-., ^ . (XL). 



o- 2 (1 - rl,) ff, (1 - r|,) cr^^l - r la ) 



Now several important conclusions follow at once from this result : 

 (a) For the case of normal frequency (but in general only for this case) the errors 

 in the means are uncorrelated with the errors in the variations and correlations. 

 The error correlation surface breaks up into two parts, of which the first part we 

 have written down involves only the means, and would coincide exactly with the cor- 

 relation surface (ix.), with which we started, if we write in (ix.) A/i, for x lt A/;, for 

 x.,, and o-i/^/n, cr.>/</n, for o-j and cr, respectively. 



It follows accordingly that the standard deviations of the errors in the means are 



....... (xii.), 



and the correlation of the errors made in the means is 



