In August, 1899, I presented a memoir to the Royal Society on the inheritance of coat-colour in the horse and of eye-colour in man, which was read November, 1899, and ultimately ordered to be published in the 'Phil. Trans.’ Before that memoir was printed, Mr. Yule’s valuable memoir on Association was read, and, further, Mr. Leslie Bramley-Moore showed me that the theory of my memoir as given in § 6 of the present memoir led to somewhat divergent results according to the methods of proportioning adopted. We therefore undertook a new investigation of the theory of the whole subject, which is embodied in the present memoir. The data involved in the paper on coat-colour in horses and eye-colour in man have all been recalculated, and that paper is nearly ready for presentation. But it seemed best to separate the purely theoretical considerations from their application to special cases of inheritance, and accordingly the old memoir now reappears in two sections. The theory discussed in this paper was, further, the basis of a paper on the Law of Reversion with special reference to the Inheritance of Coat-colour in Basset Hounds recently communicated to the Society, and about to appear in the ‘ Proceedings. While I am responsible for the general outlines of the present paper, the rough draft of it was taken up and carried on in leisure moments by Mr. Leslie Bramley-Moore, Mr. L. N. G. Filon, M. A., and Miss Alice Lee, D. Sc. Mr. Bramley-Moore discovered the u -functions ; Mr. Filon proved most of their general properties and the convergency of the series; I alone am responsible for sections 4, 5, and 6. Mr. Leslie Bramley-Moore sent me, without proof, on the eve of his departure for the Cape, the general expansion for z on p. 26. I am responsible for the present proof and its applications. To Dr. Alice Lee we owe most of the illustrations and the table on p. 17. Thus the work is essentially a joint memoir in which we have equal part, and the use of the first personal pronoun is due to the fact that the material had to be put together and thrown into form by one of our number.—K. P.