^{page 307 note *} If we multiply the numerator and denominator of the above expression by the same quantity, or by l _x+1, l _y+1, its value will not be altered, though its form will be changed; thus,

which is the expression given in Tables and Formula for the same function, found above.

^{page 309 note *} As the factors d_x and l_x+ ½ consist only offrom two to five digits their product, might perhaps be as expeditiously obtained without the intervention of logarithms, by placing those two slips together according to the difference of age, with a blank slip at the right, and a weight on the top. The factors being then opposite one another may be multiplied together on scroll paper, and the results set down on the blank slip as they are found. This table, When completed, could then be used in connection with Table XL. of Mr. Jones' work.

^{page 311 note *} If the numerator and denominator of this fraction be multiplied by D _x+1.y+1, its form will be changed into

therefore,

which is Mr. Gray's formula, as given in his Tables and Formula.

This form of expression, from its compactness, possesses more of the character of an algebraical formula than the one found above, and has been accurately and extensively exemplified by the publication of Assurance and Annuity Tables, the offspring of this new and ingenious method of construction; but it must be borne in mind that Mr. Sang's object was to adapt his formula to the commutation form, which had already worked so successfully in the computation of his Single Life Tables. It is very probable, however, from his entire acquaintance with the subject, that he was quite aware of the connection thus shown to exist between any given value and the next younger, though such a formula would evidently not suit his object.

^{page 313 note *} The present value of a benefit payable at death is evidently greater than a similar benefit payable six months after death. And as the payments in the former case are, on an average, made six months earlier in each year, the expression for payable at death, which we may denote by will become

and this is the formula from which Mr. Sang's tables were computed. But if this fraction be multiplied by it will become

that is, or, which is the same thing, the assurance payable six months after death.

The same relation might be shown to subsist between Mr. Sang's single life assurance values, and the value of the same benefit due six months later. And although it is unfortunate that he made such an assumption when computing his Single Life Tables, there is not on that account such inconvenience in using them as might be supposed, where a desire is felt to obtain the smaller value. For since =0·0064186, the assurance values there tabulated may be easily converted into similar values payable six months later, by subtracting this quantity, or to suit his five place logs. 0·00642 from each column of logarithms, to which the corresponding natural numbers will be the values required: or the value for a single case may be found very nearly by dividing by 1·0149, a number easily remembered.