Let there be now proposed the two fluxions _xⁿyᵐẋ_ and _xᵖy⒬ẏ_, 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_ + _bxᵖy⒬ẏ_. From whence, by proceeding as before, _bxᵖy⒬_ = υ, and _mxⁿy⁽ᵐ ⁻ ¹⁾ẋ_ + _qbxᵖy⁽ᵖ ⁻¹⁾ẏ_ = ̇υ. From the former of which equations, by taking the fluxions on both sides, will be had _pbx⁽ᵖ ⁻¹⁾y⒬ẋ_ + _qbxᵖy⁽⒬ ⁻ ¹⁾ẏ_ (= ̇υ) = _mxⁿy⁽ᵐ ⁻ ¹⁾ẋ_ + _qbxᵖy⁽⒬ ⁻ ¹⁾ẏ_. Whence _pbx⁽ᵖ ⁻ ¹⁾y⒬_ = _mxⁿy⁽ᵐ ⁻ ¹⁾_; and therefore _pby⁽⒬ ⁻ ᵐ ⁺ ¹⁾_ = _mx⁽ⁿ ⁻ 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.