1904.] serve to Test various Theories of Inheritance. 



273 



u- Ci = cr C2 = <r c = 2"-710, T\f = 2/y c = 1*0280, iii = no = Jw, and 

 r 2 / = 0. We have 



(^/o- c ) 2 = 0-4716 + 0-2642 (a/oy) 2 , 



where cry = 2"*568. 



It will be seen at once that the variability of the array is too small 

 for the mean father and far too large for the exceptional father. I 

 think we may safely conclude that for man in the case of the three 

 characters under investigation no such theory of alternative inheritance 

 applies. 



(9) I now turn to the third or Mendelian hypothesis. Are the 

 variabilitjr distributions better represented by parabolas of horizontal 

 axes than by horizontal straight lines ? We have a series of points in 

 each case and the question is : What is the best fitting parabola ? 

 The equation to the Mendelian parabola, as given in my memoir 

 already referred to,* is in the notation of the present paper 



9 J(3q) <r, 



(xvi), 



where q is the number of Mendelian couplets, and x is measured from 

 the mean.f Writing ^x 2 / ' 2 = £ ^/°7 = Vi this is of the form, a and 

 b being numerics : 



£ = a + h]. 



Applying the method of least squares, since a is an absolute 

 constant, we find for the best value of b, fx x being the total in the 

 x array : 



o = S W (^x 2 - f q-c 2 ) %} 



°"c 2 S (fl x x 2 ) 



or, q the number of Mendelian couplets is to be found from 

 1 _ 9 V(3)oy S (M^ 2 -fo-c 2 K 



J(q) 4 ay 2 



(xvii). 



Working this out for the case of stature ^first, I found, after some 

 rather laborious arithmetic, that q = 48. Thus the best fitting 

 Mendelian parabola needs no less than 48 couplets. We may write 

 the parabola in the form 



a- f J 



* ' Phil. Trans.,' A, vol. 203, pp. 66-67. 



f To reduce to the result of the above memoir we have 2 iC = tu s , crc = erf — 

 {\o) n x = e ( s ~ 1 = n > where £ is an undetermined constant depending on 

 the relation between actual scale and number of Mendelian couplets. 



