MATHEMATICS: G. A. MILLER 
61 
From this table we get, for the first time, a definite quantitative 
conception of the average degree of inbreeding existing in any animal. 
In the complete publication which will shortly follow this note additional 
details will be given, including probable errors and other statistical 
constants. 
From this analysis of Jersey inbreeding records two results, among 
others, stand out with particular clearness and significance. These are : 
1. That American Jersey cattle, at the present time, may be said in 
general and on the average to be about one-half as intensely inbred, when 
account is taken of the eighth ancestral generation, as would be the case 
if continued brother X sister breeding had been followed. The form 
of the inbreeding curve is, however, very different in the two cases, 
the brother X sister curve being concave to the base line throughout, 
while the actual Jersey curves tend to have their principal curvature 
convex to the base. 
2. That, in general and on the average, Register of Merit animals 
are less intensely inbred than the general population of Jersey cattle. 
A detailed report of the work on inbreeding in Jerseys will be pub- 
lished shortly. The above notes are intended to be merely of a pre- 
liminary character. 
* A list of references to these papers will be found in Amer. Nat., 48, 513 (1914). 
UPPER LIMIT OF THE DEGREE OF TRANSITIVITY OF A 
SUBSTITUTION GROUP 
By G. A. Miller 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF ILLINOIS 
Received by the Academy. December 24. 1915 
In a recent article published in the Bulletin of the American Mathe- 
matical Society (22, 68, 1915) I proved the following theorem: A sub- 
stitution group of degree n = kp -\- r,p > k and r > k, which is neither alter- 
nating nor symmetric cannot he more than r times transitive unless k = \ 
and r = 2. From this theorem and the well known theorem that there 
is at least one prime number in the interval m (exclusive) and 2m — 2 
(inclusive), whenever m^i,it results directly that the degree of transi- 
tivity of a substitution group of degree n which does not include the al- 
ternating group of this degree must be less than 3^/n — 2, In the article 
noted above the condition n>12 was added in stating this result but 
this condition can clearly be omitted. 
The general expression 3 — 2 for an upper limit of the degree of 
transitivity of a substitution group of degree n which does not include 
