To exemplify, now, by a particular case, the method of operation above pointed out, let there be proposed the fluxionary quantity xⁿ ym ẏp ⁄ ẋp ⁻ ¹; wherein the relation of x and y is so required, that the fluent, corresponding to given values of x and y, shall be a maximum, or minimum. Here, by taking the fluxion, making ẏ alone variable (according to the rule) and dividing by ÿ, we shall have pxⁿ ym ẏp⁻ ¹ ⁄ ẋp ⁻ ¹ = υ. And, by taking the fluxion a second time, making y alone variable, and dividing by ẏ, will be had mxⁿ ym ⁻ ¹ ẏp ⁄ ẋp ⁻ ¹ = ̇υ. Now from these equations to exterminate υ, let the latter be divided by the former; so shall mẏ ⁄ py = ̇υ ⁄ υ; and therefore aym ⁄ p = υ (a being a constant quantity). From whence ym ⁄ pẏ = (a ⁄ p)¹ ⁄ p ⁻ ¹ × ẋx⁻ⁿ ⁄ p ⁻ ¹; and consequently p ⁄ m + p × ym ⁺ p ⁄ p = (a ⁄ p) ¹ ⁄ p ⁻ ¹ × p - 1 ⁄ p - n - 1 × xp ⁻ ⁿ ⁻ ¹ ⁄ p ⁻ ¹.
Let there be now proposed the two fluxions xⁿymẋ and xpyqẏ, the fluent of the former being required to be a maximum, or minimum, and that of the latter, at the same time, equal to a given quantity. Then the latter, with the general coefficient b prefixed, being joined to the former, we shall here have xⁿymẋ + bxpyqẏ. From whence, by proceeding as before, bxpyq = υ, and mxⁿ ym ⁻ ¹ ẋ + qbxpyp ⁻¹ ẏ = ̇υ. From the former of which equations, by taking the fluxions on both sides, will be had pbxp ⁻¹ yqẋ + qbx py q ⁻ ¹ ẏ (= ̇υ) = mxⁿ ym ⁻ ¹ ẋ + qbxp yq ⁻ ¹ ẏ. Whence pbx p ⁻ ¹ yq = mxⁿ ym ⁻ ¹; and therefore pbyq ⁻ m ⁺ ¹ = mxⁿ ⁻ p ⁺ ¹. And in the same manner proper equations, to express the relation of x and y, may be derived, in any other case, and under any number of limitations.
LXXXVI. Observations on the Alga Marina latifolia; The Sea Alga with broad Leaves. By John Andrew Peyssonel, M.D. F.R.S. Translated from the French.
Read April 13, 1758.
HAVING cast anchor at Verdun, the road at the entrance of the river of Bourdeaux, I was fishing with a kind of drag-net upon a bank of sand, which was very fine and muddy. We collected a number of sea-plants, and among them the great broad-leaved Alga, which I did not know: and as the root or pedicle of this plant appeared to be very particular, I observed it with attention. The following is its description, and the detail of my observations.
From a pedicle, which is sometimes flat, and sometimes round (for they vary in these plants, and might be about three lines in diameter, and an inch high, of a blackish colour, and coriaceous substance, approaching to the nature of the bodies of lithophyta), a single flat leaf arises, about an inch or an inch and half broad, thick in its middle to about three lines, ending at the sides in a kind of edge, like a two-edged sabre, almost like the common Alga, formed of longitudinal fibres interlaced with other very delicates ones, and the whole filled with a thick juice, like the parenchyma of succulent plants, such as the Sedum, Aloes, and the like, of a clear yellowish green, and transparent. This first leaf is always single, and serves instead of a trunk or stem to the whole plant.
When it rises to about a foot high, more or less, it throws out at the sides other leaves formed of a continuation of the longitudinal fibres; and these second leaves are of the same thickness and substance with the first: they are two or three feet long, and the whole plant is five or six, or more (for one can hardly tell the length); and is not capable of supporting itself, but is sustained by the strength of the waters, in which it floats.
The substance of the plant is not so solid as that of the common Alga, which is capable of drying as it fades, and of being kept: whereas the leaves of this great Alga shrink and wither in the air, become of a blackish colour, and very friable, or indeed soon fall into putrifaction. I never observed, that they bore any fruit: perhaps this was not the season.
But what we find particular in this plant is its root or foot: First, this pedicle extends in ribs, like what we call the thighs of certain trees: these thighs are in right lines: perhaps they run in the same direction or situation, that is, placed north and south, or east and west; but this I could not observe. They are about three or four lines high towards the pedicle, and, ending, are lost. They flourish and spread at the bottom, forming an elliptical bladder, like an egg, flattened above and below, and rounded at the sides, being intirely empty: it is rough without, and very smooth within. This egg, or oval bladder, is exactly round at the ends of the great diameter, but varies a little in the lesser diameter, and forms itself like the body of a fiddle. The under part is a little flattened; and there is a hole, which is very considerable, in the center of the two diameters. This hole is about an inch wide, and is quite round: it gives passage to the root, or pivot, which I shall by and by mention: the edges appear to turn a little inward: and it is by this hole that the egg fills with sea-water. The whole substance of this bladder or egg is of a coriaceous matter, firm and transparent, and of a clear green; nor can there be any fibres, either longitudinal or transverse, observed upon it.