238 MATHEMATICS: G. A. MILLER Proc. N. A. S. 
which is the condition that the equations of geodesies admit the quadratic 
integral Aij = const. These results are due to Levi-Civita 
ds ds 
{Annali, 24, 1896, pp. 273, 276; also Annalen, 54, 1901, p. 188), who has 
found all the Riemann spaces which admit of geodesic representation. 
When in particular it is assumed that 5„ and possess corresponding 
orthogonal systems, and these are taken as parametric we have 
V' = — ^ w = _i_ - - ^ ^^^i - n 
2gu 2gii bxj 2gjj dxj 
and the equations (3.4) and (3.6) are readily solved; giving 
i = 1 uih -\- C) xpi -\- C 
j = 1 
where / 5^ i in the first product, xpi is an arbitrary function of x* alone, and 
c and C are arbitrary constants (Pirro, Rend. Palermo, 9, 1895, p. 169; 
also, Levi-Civita, I.e., p. 287). 
NUMBER OF SUBSTITUTIONS OMITTING AT LEAST ONE 
LETTER IN A TRANSITIVE GROUP 
By G. A. Miller 
Department of Mathematics, University of Illinois 
Communicated June 10, 1922 
In an article pubhshed in Amer. J. Math., 26, 1904, p. 1, H. L. Rietz 
proved the theorem that in "a primitive group G of degrees of composite 
order g there are more than g/(x + 1) substitutions of degree less than n, 
where x is the number of systems of intransitivity of the sub-group which 
leaves a given letter fixed." If 6" is a transitive group which is not neces- 
sarily primitive this subgroup Gi is of degree n — a, where a is not neces- 
sarily unity, and may be supposed to have a transitive constituents of 
degree 1 in addition to its transitive constituents whose degrees exceed 1. 
If ^i; "h 1 represents the total number of these transitive constituents the 
theorem quoted above remains true for every possible transitive group 
except the regular groups, as may be inferred from an abstract which 
appeared in Bull. Amer. Math. Soc., 8, 1902, p. 17, where the exception 
noted above is, however, not stated. 
The object of the present note is to furnish a very elementary proof of 
this general theorem, as follows: If each of the substitutions of Gi involves 
