242 
GENETICS: H. H. LA UGHLIN 
Proc. N. a. S. 
ancestor is not in the rjc-trail or j-trail, and does not, due to inbreeding, 
appear elsewhere in the ancestral pedigree. 
n = Value as in formula 1. 
p = Value as in formula 1. 
No. 10. 1 : C-tx = or 1 : C-ty = The probability that no member of any- 
equivalent chromosome-pair, nor of the xx or xy chromosome-pair, from 
the given ancestor will enter the Fi zygote, i. e., the probability (barring 
crossing-over and other special phenomena, and the appearance of the 
gi^en ancestor elsewhere in the pedigree), of the complete elimination 
of the hereditary influence of the given ancestor and all of his or her lineal 
ancestors, on the propositus, i. e., the Fi zygote. 
This formula is applicable only to those ancestors in the x-trail or y-traiL 
1 : C-,, = (1 : Q.(l : C-J 
1 : C,_y = (1 : C_,).(l : C-y) 
Values of 1 : C-^ as in formula 9 
r" — 
^-x - — 
r — 
^-y - 
Summary. — The present problem of measuring ancestral influence by 
tracing chromosomes or gene-s-radical is demonstrated to be a special case 
under the following general situation: 
Given : First. A single unit factor (a gene) in a known situation (in the 
Pi generation) and the quotient measuring the probability of its reacting in 
a definite manner or reaching a definite new situation at the end of a single 
system of moves or transformations (in the Fi generation). 
Second. The demonstrated synchronization or sequence in the systems 
of moves or transformations of two or more such unit factors. 
By applying the correct principles of combination and chance, the prob- 
ability that a given complex situation will result from a given set of con- 
stituent conditions at the end of a given number of definite kinds of trans- 
formations may be mathematically formulated. 
In case of measuring ancestral influence by tracing chromosomal descent, 
the situations and transformations concern definite chromosome-complexes 
and their breaking-up, their re-combination and their transmission to off- 
spring. 
The formulas here set forth are foundational. They are stated in general 
terms whose validity depends upon their presenting correct mathematical 
pictures of the chromosomal processes which work out in the germ-cell 
cycle. 
