[[p. 481]] I have received from Mr. Thos. Blakiston, of Tokio, Japan, a communication to the Japan Mail by himself and Prof. Alexander, [[p. 482]] commenting on my article in Nature, vol. xxvi. p. 86, and pointing out some errors as to the estimated advantage derived by the mimicking butterflies. On referring to my article, I find that I have, by an oversight, misstated the mathematical solution of the problem as given by Dr. Fritz Müller and confirmed by Mr. Meldola, and have thus given rise to some confusion to persons who have not the original article in the Proceedings of the Entomological Society to refer to. Your readers will remember that the question at issue was the advantage gained by a distasteful, and therefore protected, species of butterfly, which resembled another distasteful species, owing to a certain number being annually destroyed by young insectivorous birds in gaining experience of their distastefulness. Dr. Müller says: "If both species are equally common, then both will derive the same benefit from their resemblance--each will save half the number of victims which it has to furnish to the inexperience of its foes. But if one species is commoner than the other, then the benefit is unequally divided, and the proportional advantage for each of the two species which arises from their resemblance is as the square of their relative numbers." This is undoubtedly correct, but in my article I stated it in other words, and incorrectly, thus: "If two species, both equally distasteful, resemble each other, then the number of individuals sacrificed is divided between them in the proportion of the square of their respective numbers; so that if one species (a) is twice as numerous as another (b), then (b) will lose only one-fourth as many individuals as it would do if it were quite unlike (a); and if it is only one-tenth as numerous, then it will benefit in the proportion of 100 to 1."

    This statement is shown by Messrs. Blakiston and Alexander to be untrue; but as some of your readers may not quite see how, if so, Dr. Müller's statement can be correct, it will be well to give some illustrative cases. Using small and easy figures, let us first suppose one species to be twice as numerous as the other, a having 2000 and b 1000 individuals, while the number required to be sacrificed to the birds is 30. Then, if b were unlike a it would lose 30 out of 1000, but when they become so like each other as to be mistaken, they would lose only 30 between them, a losing 20, and b 10. Thus b would be 20 better off than before, and a only 10 better off; but the 20 gained by b is a gain on 1000, equal to a gain of 40 on 2000, or four times as much in proportion as the gain of a. In another case let us suppose c to consist of 10,000 individuals, d of 1000 only, and the number required to be sacrificed in order to teach the young birds to be 110 for each species. Then, when both became alike, they would lose 110 between them, c losing 100, d only 10. Thus c will gain only 10 on its total of 10,000, while d will gain 100 on its total of 1000, equal to 1000 on 10,000, or 100 times as much proportional gain as c. Thus, while the gain in actual numbers is inversely proportional to the numbers of the two species, the proportional gain of each is inversely as the square of the two numbers.

    I am, however, not quite sure that this way of estimating the proportionate gain has any bearing on the problem. When the numbers are very unequal, the species having the smaller number of individuals will presumably be less flourishing, and perhaps on the road to extinction. By coming to be mistaken for a flourishing species it will gain an amount of advantage which may long preserve it as a species; but the advantage will be measured solely by the fraction of its own numbers saved from destruction, not by the proportion this saving bears to that of the other species. I am inclined to think, therefore, that the benefit derived by a species resembling another more numerous in individuals is really in inverse proportion to their respective numbers, and that the proportion of the squares adduced by Dr. Müller, although it undoubtedly exists, has no bearing on the difficulty to be explained.