HOMOTYPOSIS IN THE VEGETABLE KINGDOM. 
289 
characters in the spermatozoon and certain characters iu the ovum from which the 
individual has developed. These characters cannot of course be determined, still less 
measured, but we have no reason to doubt their existence. In the paa-ticular sper¬ 
matozoon from which the individual has developed, let them have deviations x^, x.-,, 
. . . from their mean values for all the spermatozoa of the race, and let y^, y.-,, y^ . . . 
be the corresponding deviations for the ovum characters. Then 
2 =/(»!, Xg . . . y^, y., 2/3 • • • ) 
where / is a cpiite unknown function. 
The mean of the z-character will, however, correspond to the mean values of the 
spermatozoon and ovum characters, and if we suppose the variation of these 
characters small as compared with their mean value, we assume as usual for such 
deviations : 
z = UyX^ + ao.Xo -f agXg + . . . + /3^2 /i + + • • • ('•)> 
where the as and /3’s are independent of the xs and ys, and define the male and 
female inheritance. 
Now let (T be the standard deviation of the character 2 in the population ; cr^ the 
standard deviation of x,j. of y,^. Let be the correlation of Xj, and x,^, of y^ 
and y^. Then we Avill suppose that there is no selection of particular ova by })arti- 
cular spermatozoa, or that and y,^ are not correlated. Then if n — number of 
individuals in the ])opulation : 
O" = 
o >S(.r/) 
n 
n \ j I 
)' 
71 
' n 
^ {ypi/q) 
•11 
where S is the sum for all individuals of any x or y for constant subscript, and S is 
the sum of a and ^ for every possible subscrij)t. This follows by simple squaring and 
remembering that S(.r^ 2 /y) = 0 . We thus reach : 
a- — S (a/cr/) + 5 ) + 2 S ( 
apa,jO-pCr,fyj 
) -b 2 ^ 
/ / ^ 
yCT p(T (j) 2iq 
)• • (ii-)- 
Now let us considei’ the correlation of two individuals due to the spermatozoa and 
ova p\it forth l)y the same two individuals. Let 2 , and 2 ,j be the values of their 
characters, and x\ x'\ y', y" represent the fundamental characters in the two sper¬ 
matozoa, and two ova on which they depend. 
Then we have 
2i = S S {jipy'j) 
23 = s s d^py'p) • 
Now let us multiply 2 , by 2.3 and sum for every fraternal pair; then if E, be the 
2 p 
VOL. CXCVII.—A. 
