13. The main object of this paper is to supply simple algebraical proofs of the general formulae for the number of particles, at any time t, of the rth product in a series of successive radioactive transformations. This is done both for ‘Case 1’, where the matter is initially all of one kind (paragraphs 3 and 4), including a stable end-product (paragraph 5), and for ‘Case 2’, where the successive products are initially all in existence and in equilibrium (paragraph 6). In paragraph 7 it is remarked that there exists a relation between the amounts of the end-product in ‘Case 1’, supposed to be the (r + 1)th product, and of the preceding rth product in ‘Case 2’, for the same time t, and the necessity of this relation is explained on general grounds. In paragraph 9 numerical values are given for the particular example (paragraph 8) considered by Rutherford in his Newer alchemy (pp. 11, 12), whilst paragraphs 10 and 11 deal with the degree of accuracy attained, and show the need for close accuracy in the experimental data.