54 PROFESSOR K. PEARSON ON A GENERALISED THEORY OF ALTERNATIVE 



identity. The analytical expression of this is represented by the fundamental 

 formula : 



(AA') x (') } = { 



' Aa' 



,, OO. 



A'a 



j L AV 



where (AA') and (') are the parental zygotes, and the right-hand side of the 

 equation represents the four possible constitutions of the offspring. Such a formula 

 as the above may be accepted without any hypothesis as to dominant and recessive 

 characters, but these terms were certainly essential to Mendelian theory as propounded 

 by MENDEL himself, and it becomes very doubtful whether we ought to attach his 

 name to any theory which discards these " recognition marks." It is very convenient, 

 however, to have names for the alternative elements expressed by capital and small 

 letters respectively. I propose for the purpose of this paper to term an A-element a 

 protogene, and an a-element an allogene. Two protogenic elements will give rise to a 

 protogenic zygote AA, two allogenic elements to an allogenic zygote aa, and a 

 protogenic and allogenic element to what Mr. BATESON has termed a heterozygote 

 Aa. We may thus class his homozygotes into protozygotes and allozygotes. We 

 reach pure Mendelianism by making our protozygotes " dominants," our allozygotes, 

 " recessives," and our heterozygotes " hybrids of dominant character." In so far as 

 our theory of pure gametes replaces protozygote, allozygote, and heterozygote by 

 " dominant," " recessive," and " hybrid with dominant character," it becomes a 

 generalised Mendelian theory, but only in this case. Otherwise we must look upon 

 it as an attempt in one direction only of course to give a consistent mathematical 

 basis to the various formula? which have been propounded for describing statistical 

 data classed under Mendelian categories ; shortly we shall endeavour to develop a 

 general pure gamete theory. 



The results were worked out in a purely impartial frame of mind ; indeed, once 

 state the hypotheses, and the analysis is far too complex to allow us to predict rf 

 priori what can possibly result from it, nor does the investigation admit of any but 

 one solution. If the hypotheses are admissible, then any narrower pure gamete 

 formula must lead to results embraced under our general conclusions. 



What we have to admit at the present time are the following conditions : 



(i.) The existence of a vast bulk of evidence that heredity, as far as measurable 

 characters are concerned, follows within a population perfectly definite laws. 



(ii.) The existence of another mass of experiments, in which simple and pure 

 Mendelianism is certainly inadmissible, but in which certain ratios undoubtedly 

 approach the values they would have on such a simple and pure Mendelian theory. 



It is possible, therefore, that a generalised theory of the pure gamete would account 



