In my relation of the late expedition to the north-west, if I recollect right, I have observed, that all the changes and variety of weather, that happen in the temperate zone throughout the year, may be experienced at the Hudson’s Bay settlements in 24 hours. But I may now extend this observation; for in my cellar the thermometer stands at 81, in the next story at 102, and in the upper one at 105; and yet these heats, violent as they are, would be tolerable, but for the sudden changes that succeed them. On the 10th of December last the mercury was at 86; on the 11th it was so low as 38 of the same instrument. What havock must this make with an European constitution? Nevertheless, but few people die here out of the ordinary course; tho’ indeed one can scarce call it living, merely to breathe, and trail about a vigorless body; yet such is generally our condition from the middle of June to the middle of September. Dear Sir,
Yours most affectionately,
Henry Ellis.
CIII. The Invention of a General Method for determining the Sum of every 2d, 3d, 4th, or 5th, &c. Term of a Series, taken in order; the Sum of the whole Series being known. By Thomas Simpson, F.R.S.
Read Nov. 16, 1758.
AS the doctrine of Series’ is of very great use in the higher branches of the mathematics, and their application to nature, every attempt tending to extend that doctrine may justly merit some degree of regard. The subject of the paper, which I have now the honour to lay before the Society, will be found an improvement of some consequence in that part of science. And how far the business of finding fluents may, in some cases, be facilitated thereby, will appear from the examples subjoined, in illustration of the general method here delivered.
The series propounded, whose sum (S) is supposed to be given (either in algebraic terms, or by the measures of angles and ratio’s, &c.) I shall here represent by a + bx + cx² + dx³ + ex⁴, &c. and shall first give the solution of that case, where every third term is required to be taken, or where the series to be summed is a + dx³ + gx⁶ + kx⁶, &c. By means whereof, the general method of proceeding, and the resolution of every other case, will appear evident.
Here, then, every third term being required to be taken, let the series (a + dx³ + gx⁶, &c.), whose value is sought, be conceived to be composed of three others.
⅓ × (a + b × (px) + c × (px)² + d × (px)³ + e × (px)⁴, &c.)
⅓ × (a + b × (qx) + c × (qx)² + d × (qx)³ + e × (qx)⁴, &c.)
⅓ × (a + b × (rx) + c × (rx)² + d × (rx)³ + e × (rx)⁴, &c.)
having all the same form, and the same coefficients with the series first proposed, and wherein the converging quantities px, qx, rx, are also in a determinate (tho’ yet unknown) ratio to the original converging quantity x. Now, in order to determine the quantities of these ratios, or the values of p, q, and r, let the terms containing the same powers of x, in the two equal values, be equated in the common way:
So shall,