The whole day and night of the 24th of last month it seemed as if Mount Vesuvius would again have swallowed up this country. On that day it suffered two internal fractures, which intirely changed its appearance within the crater, destroying the little mountain, that had been forming within it for some years, and was risen above the sides; and throwing up, by violent explosions, immense quantities of stones, lava, ashes, and fire. At night the flames burst out with greater vehemence, the explosions were more frequent and horrible, and our houses shook continually. Many fled to Naples, and the boldest persons trembled. For my own part, I resolved to abide the event here at Portici, on account of my family, consisting of eight children, and a very weak and aged mother, whose life must have been lost by a removal in such circumstances, and so rigorous a season. But it pleased God to preserve us; for the mountain having vented itself that night and the succeeding day, is since become calm, and throws out only a few ashes.
LXXXV. A further Attempt to facilitate the Resolution of Isoperimetrical Problems. By Mr. Thomas Simpson, F.R.S.
Read April 13, 1758.
ABOUT three years ago I had the honour to lay before the Royal Society the investigation of a general rule for the resolution of isoperimetrical problems of that kind, wherein one, only, of the two indeterminate quantities enters along with the fluxions, into the equations expressing the conditions of the problem. Under which kind are included the determination of the greatest figures under given bounds, lines of the swiftest descent, solids of the least resistance, with innumerable other cases. But altho’ cases of this sort do, indeed, most frequently occur, and have therefore been chiefly attended to by mathematicians, others may nevertheless be proposed, such as actually arise in inquiries into nature, wherein both the flowing quantities, together with their fluxions, are jointly concerned. The investigation of a rule for the resolution of these, is what I shall in this paper attempt, by means of the following
General Proposition.
Let Q, R, S, T, &c. represent any variable quantities, expressed in terms of x and y (with given coefficients), and let q, r, s, t, &c. denote as many other quantities, expressed in terms of ẋ and ẏ; It is proposed to find an equation for the relation of x and y, so that the fluent of Qq + Rr + Ss + Tt, &c. corresponding to a given value of x (or y), may be a maximum or minimum.
Let A E, A F, and A G, denote any three values of the quantity x, having indefinitely small equi-differences E F, F G; and let E L, F M, and G N, (perpendicular to A G) be the respective values of y, corresponding thereto; and, supposing EF (= FG = ẋ) to be denoted by e, let c M and d N (the successive values of ẏ) be represented by u and w. Moreover, supposing P´p´ and P´´p´´ to be ordinates at the middle points P´ P´´, between E, F and F, G, let the former (P´p´) be denoted α, and the latter (P´´p´´) by β; putting A P´ = a and A P´´ = b. Then, if a and α (the mean values of x and y, between the ordinates E L and F M) be supposed to be substituted for x and y, in the given quantity Qq + Rr + Ss + Tt, &c. and if, instead of ẋ and ẏ, their equals e and u be also substituted, and the said (given) quantity, after such substitution, be denoted by Q´q´ + R´r´ + S´s´ + T´t´, &c. it is then evident, that this quantity Q´q´ + R´r´ + S´s´ + T´t´, &c. will express so much of the whole required fluent, as is comprehended between the ordinates E L and F M, or as answers to an increase of E F in the value of x. And thus, if b and β be conceived to be wrote for x and y, e for ẋ, and w for ẏ, and the quantity resulting be denoted by Q´´q´´ + R´´r´´ + S´´s´´ + T´´t´´, &c. this quantity will, in like manner, express the part of the required fluent corresponding to the interval F G. Whence that part answering to the interval E G will consequently be equal to Q´q´ + R´r´ &c. + Q´´q´´ + R´´r´´ &c. But it is manifest, that the whole required fluent cannot be a maximum or minimum, unless this part, supposing the bounding ordinates E L, G N to remain the same, is also a maximum or minimum. Hence, in order to determine the fluxion of this expression (Q´q´ + R´r´ &c. Q´´q´´ + R´´r´´ &c.) which must, of consequence, be equal to nothing, let the fluxions of Q´ and q´ (taking α and u as variable) be denoted by Q ̇α and qu⋅; also let Rȧ and ru denote the respective fluxions of R´ and r´; and let, in like manner, the fluxions of Q´´, q´´, R´´, r´´, &c. be represented by Q ̇β, qẇ, R ̇β rẇ, &c. respectively. Then, by the common rule for finding the fluxion of a rectangle, the fluxion of our whole expression (Q´q´ + R´r´ &c. + Q´´q´´ + R´´r´´ &c.) will be given equal to Q´ qu⋅ + q´Q ̇α + R´ru⋅ + r´R ̇α &c. + Q´´qẇ + q´´Q ̇β + R´´rẇ + r´´R ̇β &c. = 0.
But u + w being = GN - EL, and β - α = GN - EL ⁄ 2 (a constant quantity), we therefore have ẇ = -u⋅, and ̇β = ̇α: also u being (= 2rp´) = 2α - 2EL, thence will u⋅ = 2 ̇α: which values being substituted above, our equation, after the whole is divided by ̇α, will become