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Abstract In offering to the Royal Society the ensuing Supplement to my two former papers on the Law of Mortality, with subsequent remarks on invalidism, I am anxious to acknowledge that I have derived great advantage from the encouragement and persuasion of my esteemed brother-in-law, Sir Moses Montefiore, Bart., given me to endeavour to compile and publish some of my later observations on the subject; knowing that, though I felt flattered by the attention originally shown by scientific gentlemen to these papers, they appeared to me capable of advantageous illustrations. Therefore I may venture to hope that if this Supplement merit the attention of those interested in this branch of science, I may consider that he has added a mite further to entitle him to the good wishes of those who applaud him for his constant endeavours to promote the general interest of mankind—endeavours which he has shown to extend through Europe and Asia in the cause of humanity, and to be exercised at home in various ways, among which I notice his attention to the practice of Life, Fire, and Marine Assurance; he being the President of the Alliance British and Foreign Life and Fire Assurance Company; of which I was the founding Actuary, and in which Institution, though retired from it, I feel greatly interested; it having been established about the year 1824 by the late N. M. de Rothschild, Esq., the late John Irving, Esq., the late Samuel Gurney, Esq., and Francis Baring, Esq., and himself conjointly with other gentlemen, and he being also President of the Alliance Marine Assurance Society, founded at the same time by them with him. Art. 1. In the year 1820 the Royal Society did me the honour to publish in their Transactions a paper of mine on the Analysis and Notation applicable to the valuation of Life Contingencies, in which I introduced a new and general notation, which appealed to me far more extensively useful, and more explanatory of its object, than any other notation I had met with; and in that paper I think I introduced a new manner of dealing with the subject, by offering an analysis, with examples of the extensive use of it, applicable to some of the most intricate questions which had up to that period met with anything like a proper solution; and showed, by selections from the treatise of Life Annuities of my late learned and much-respected friend, Francis Baily, Esq., a mode of solution of all the problems in chapter 8 of that work, depending on a particular order of survivorship; problems previously considered many years before, and presented by my late friend William Morgan, Esq., of the Equitable Society, to the Royal Society, and published in their valuable Transactions; and which had been since considered, in a learned work on Life Annuities, by my late respected friend Joshua Milne, Esq., with some ingenious notation with respect to those contingencies. But still, the solutions given to many of the problems, though there were but three lives concerned, were of such an intricate practical form, as to be in my opinion perfectly useless; especially on considering that it was necessary to obtain, by Tables of single and joint lives, by necessary interpolations, the required data; as the differences to be used for the interpolations, in consequence of the great irregularity of the numbers of those Tables, are so irregular as to throw great doubt on the necessary accuracy of the results. And I think the examples I gave of my method could leave no doubt as to the comparative simplicity which resulted from it, and consequently comparative utility of my analysis; an analysis which applies where there are more than three lives concerned, and, in fact, where there are any number of lives to be considered. And I may refer the reader to my solutions in that tract, to enable him to make the comparison.