RELATIVES ON THE SUPPOSITION OF MENDEL] AN INHERITANCE. 
415 
14. Multiple Allelomorphism . — The possibility that each factor contains more 
than two allelomorphs makes it necessary to extend our analysis to cover the 
inheritance of features influenced by such polymorphic factors. In doing this we 
abandon the strictly Mendelian mode of inheritance, and treat of Galton’s “ par- 
ticulate inheritance” in almost its full generality. Since, however, well-authenticated 
cases of multiple allelomorphism have been brought to light by the Mendelian method 
of research, this generalised conception of inheritance may well be treated as an 
extension of the classical Mendelism, which we have so far investigated. 
If a factor have a large number, n, of allelomorphs, there will be n homozygous 
phases, each of which is associated with a certain deviation of the measurement 
under consideration from its mean value. These deviations will be written 
i x , i 2 , . . . i n , and the deviations of the heterozygous phases, of which there are 
\n{n — l), will be written j 12 , j 13 , j 23 , and so on. Let the n kinds of gametes exist 
with frequencies proportional to p, q, r, s, and so on, then when the mating is 
random the homozygous phases must occur with frequencies proportional to p 2 , 
q 2 , r 2 , . . . , and the heterozygous phases to 2 pq, 2pr, 2qr, . . . 
Hence, our measurements being from the mean, 
p\ + q\ + rH 3+ . . . + 2pqj n + 2prj is + ... =0 . . . (XII*) 
As before, we define a 2 by the equation 
p 2 i 2 + q 2 i 2 +rH 2 + • • ■ + 2pqj 1 2 + 2prj n 2 + . . . = a 2 . . . • (I*) 
and choosing l, m, n, . . . , so that 
p\2l - h) 2 + <? 2 (2m - i 2 ) 2 + . . • 2pq(l> + m-j 12 ) 2 + 2pr(l + n-j 13 ) 2 + . . . 
is a minimum, we define /3 2 by 
4 l 2 p 2 + 4 m 2 q 2 + . . . 2 pq{l + m) 2 + 2 pr(l + n ) 2 . . . = /I 2 , 
the condition being fulfilled if 
l=pi 1 +qj n + rj ls + . . . , 
m=pj 12 + qi 2 + rj 23 + . . . , 
and so on. 
Now 
y3 2 = S(4Z*p a ) + S(-2pql + m 2 ), 
= S(2p(l + p)l 2 ) + S(4:pqlm), 
and since 
pl + qm + rn + . . . = 0, 
f3 2 = S(2pl 2 ), 
which may now be written as a quadratic in i and j, represented by the typical 
terms 
2 P h 2 + 4p 2 2Vi 2 + 2 iMp + ?)ii2 2 + tpgviJ is- 
Now we can construct an association table for parent and child as in Article 6, 
though it is now more complicated, since the /s cannot be eliminated by equation (XIP), 
and its true representation lies in four dimensions ; the quadratic in i and j derived 
TRANS. ROY. SOC. EDIN., VOL. LII, PART II (NO. 15). 64 
