Observations upon Complement Fixation in Infections with Endamoeba Histolytica

special interest. Let G1 represent one such subgroup and suppose that the degree of G is n, while the number of its transitive constituents is k. It will be convenient to think of G1 as having as many conjugates under G as the degree of the transitive constituent of G which involves the letter omitted by all the substitutions of G1. Two conjugates of G need therefore not be distinct. If a substitution s of G1 omits I letters of G it appears in I of the n subgroups composed of all the substitutions which omit a given letter of G. Hence, such a substitution is counted I times if the omitted letters in all of these subgroups are counted and the number of letters omitted in all these substitutions is 12. Since the number of letters omitted in all the substitutions of G is kg, g being the order of G, the number of letters omitted by all the substitutions of the conjugates of G1 is g times the number of transitive constituents of G when it is regarded as a group of degree n. This constitutes a proof of the following theorem: The sum of the squares of the letters omitted by all the substitutions of any substitution group G which has k transitive constituents is equal to the product of the order of G and the sum of the transitive constituents in the set of k subgroups obtained by taking successively all the substitutions of G which omit one letter of each of its transitive constituents. 1 G. A. Miller, Annals Math., 14, 95 (1912-13).