A quantitative analysis of the toxicity of drugs or poisons applied jointly requires that they be administered at several dosages in mixtures containing fixed proportions of the ingredients. From a study of the dosage‐mortality curves for several such mixtures, preferably in comparison with equivalent curves for the isolated active ingredients, most cases of combined action can be classified into one of three types:

(1) The first type is that in which the constituents act independently and diversely, so that the toxicity of any combination can be predicted from that of the isolated components and from the association of susceptibilities to the two components. The coefficient of association can be measured experimentally and should be constant at all proportions of the ingredients. When high, the toxicity of the mixture is reduced. The form of the dosage‐mortality curve has been examined for several hypothetical mixtures. Whenever the curves for the two constituents were assumed to differ in slope, there was a relatively abrupt bend in the curve for the mixture, the rectilinear segments above and below the break approaching in slope the values for the original constituents. This observation indicates that in homogeneous populations the slope of a dosage‐mortality curve is of toxicological significance. Since the same numerical relations would be expected if a single poison were to have two independent lethal effects within the animal, there is theoretical basis for fitting the linear segments of a dosage‐mortality curve separately when a break occurs after transformation to probits and logarithms. This argument has been extended to time‐mortality experiments to explain the smoothly concave curves characteristic of natural mortality.

(2) The second type of joint action is that in which the constituents act independently but similarly, so that one ingredient can be substituted at a constant ratio for any proportion of a second without altering the toxicity of the mixture. With homogeneous populations, dosage‐mortality curves for the separate ingredients and for all mixtures should be parallel. Although by hypothesis the susceptibility to one ingredient is completely correlated with that to the other, mixtures in this category are more toxic than in the preceding class where association may vary from 0 to 1. The numerical relations have been illustrated by an experiment on the toxicity to the house‐fly of solutions containing pyrethrin and rotenone. A mixture with a little less than four equitoxic units of pyrethrin to one of rotenone agreed closely with the definition but one in which the ingredients were about equally balanced showed a significantly greater toxicity than expected on the hypothesis of independent action, indicating the presence of synergism.

(3) Synergism forms the third type of joint action, characterized by a toxicity greater than that predicted from studies on the isolated constituents. It is the reverse of antagonism, which has not been considered directly. Two methods are proposed for the analysis of synergism. The more direct is to relate equitoxic dosages of mixture to its percentage composition in terms of the more active ingredient. When both are in logarithms the relation is linear over a useful range of compositions. This procedure preserves the original structure of the experiment, can be extended readily to three or more ingredients and leads to a convenient practical result. Theoretically it is less satisfactory than a second method in which for equitoxic dosages of each mixture the content of one ingredient (A) is related to the content of the other (B.) The equation which satisfies this relation most completely is (1 +k₁A) Bⁱ=k₂, where the three constants are computed …