62 
GEOLOGY: ADAMS AND DICK 
the alternating group of this degree gives clearly a much smaller value 
for this upper limit, when n becomes large, then the expression- + 1, 
3 
which is found in the standard works of reference, for instance, in 
PascaPs Repertorium der hdheren Mathematik, volume 1, (1910), page 211. 
It may therefore be of interest to note that it is possible to deduce from 
the italicized theorem noted above a general expression which gives 
still smaller values for the upper limit in question. 
In fact, if is a prime number in the interval ^/ n (exclusive) and 
2 \/w — 2 (inclusive) then r can always be so selected that it does not 
exceed p-\-n/p. Since this expression is an increasing function of p 
when p>'\/n and has the value | when p = 2\^ n,\t results that the 
prime number p can be so chosen that r is always less than ^y/n— 1. 
That is, the degree of transitivity of a substitution group of degree n which 
does not include the alternating group of this degree is always less than 
^y/n — 1. For all integral values of w>43 this theorem gives a smaller 
upper limit for the degree of transitivity than the formula \n-\-\., and 
for all values of #>4 it gives a smaller upper limit than SVw — 2. 
THE EXTENSION OF THE MONTANA PHOSPHATE DEPOSITS 
NORTHWARD INTO CANADA 
By Frank D. Adams and W. J. Dick 
COMMISSION OF CONSERVATION OF CANADA 
Received by the Academy, January 10. 1916 
With the development of agriculture in Western Canada there will 
arise in the not distant future an insistent and ever increasing demand 
for fertilizers and manures. Phosphoric acid is one of the most impor- 
tant constituents of such fertilizers. While no deposits of phosphates 
have hitherto been found in Western Canada, great beds of phosphate 
of lime have in recent years been discovered in Utah, Idaho, Wyoming, 
and Montana. 
It was for the purpose of ascertaining whether these phosphate- 
bearing rocks extend northward into Canada that the present investi-^ 
gation was undertaken by the Commission of Conservation of Canada. 
The phosphate beds from Utah to Montana occur at a certain definite 
horizon, usually known as the Phosphoria Formation, which is found 
near the summit of the Pennsylvanian subdivision of the Carboniferous 
system. This Pennsylvanian is usually composed of four formations, 
the phosphate lying at the base of the highest of these members and being 
usually associated with bands of chert. 
