64 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



But the mean of the whole population of offspring is at 1 + fw from our origin. 

 Thus we have the final results : 



Mean number of allogenic couplets in offspring of fathers with s allogenic couplets 



= L H _ J (/,. _ s) allogeuic couplets. 

 Deviation from mean of general population of this array of offspring 



Deviation of fathers from mean of population 



s -L-/6 = i'(4s n). 

 Thus 



Deviation of offspring from mean of jxtpulation j_ 

 Deviation of fathers from mean of population 



We have then the following results, which could certainly not have been fore- 

 seen : 



(<(.) The regression is constant for all arrays, or the regression curve is a straight 



line. 



(b.) The slope of this straight line is 3, or, since we have seen that the population 

 is stable, the parental correlation is -$ also. 



Now these results are of very singular importance. A very general theory of the 

 pure gamete type leads to linearity of the regression curve, a result amply verified 

 by observations t>n inheritance in populations ;* and this result is quite independent 

 of the number of couplets supposed to form the total character of the parent, or of 

 the fact that in this case the arrays of offspring are skew and do not obey the normal 

 law.t Further, the value of the correlation reached is numerically identical with the 

 value obtained by FBANCIS GALTON in his original investigations on the inheritance 

 of stature ! The generalised theory of the pure gamete is thus shown, whatever the 

 number of couplets taken, to lead to precisely the chief results already obtained by those 

 who have studied heredity statistically. So far then it might appear that a 

 generalised theory of the pure gamete was capable of being brought into accordance 

 with the chief results of biometric experience in heredity. This would undoubtedly 

 be a great step forward, as linking up perfectly definite inheritance results with a 

 physiological theory of heredity. Unfortunately the whole drift of recent biometrie 

 observations on heredity emphasises three points ; 



First. That the parental correlation appears to be markedly greater than -j, nearef 

 to "45 to '5, 



* GALTON, ' Natural Inheritance,' p. 96 ; ' Biometrika,' w>l. 2, pp. 216 and 362-3. 

 t This is further demonstration that linearity of regression has nothing whatever to do with the Gauss- 

 Lapkcian law of errors, i.e., normal curves or surfaces, 



