_Take the fluxion of the given expression_ (_whose fluent is required
to be a_ maximum _or_ minimum) _making_ ẏ _alone variable; and, having
divided by_ ÿ, _let the quotient be denoted by_ υ: _Then take, again,
the fluxion of the same expression, making_ y _alone variable, which
divide by_ ẏ; _and then this last quotient will be_ = ̇υ.
When _ẏ_ is not found in the quantity given, υ will then be = 0; and,
consequently, the expression for ̇υ, equal to nothing also. But if
_y_ be absent, then will ̇υ = 0, and consequently the value of υ = a
constant quantity. It is also easy to comprehend, that, instead of _ẏ_
and _y, ẋ_ and _x_ may be made successively variable. Moreover, should
the case to be resolved be confined to other restrictions, besides that
of the _maximum_ or _minimum_, such as, having a certain number of
other fluents, at the same time, equal to given quantities, still the
same method of solution may be applied, and that with equal advantage,
if from the particular expressions exhibiting all the several
conditions, one general expression composed of them all, with unknown
(but determinate) coefficients, be made use of.
In order to render this matter quite clear, let _A, B, C, D, &c._ be
supposed to represent any quantities expressed in terms of _x, y_, and
their fluxions, and let it be required to determine the relation of _x_
and _y_, so that the fluent of _Aẋ_ shall be a _maximum_, or _minimum_,
when the cotemporary fluents of _Bẋ, Cẋ, Dẋ, &c._ are, all of them,
equal to given quantities.
It is evident, in the first place, that the fluent of _Aẋ_ + _bBẋ_
+ _cCẋ_ + _dDẋ, &c._ (_b, c, d, &c._ being any constant quantities
whatever) must be a _maximum_, or _minimum_, in the proposed
circumstance: and, if the relation of _x_ and _y_ be determined (_by
the rule_), so as to answer this single condition (under all possible
values of _b, c, d, &c._) it will also appear evident, that such
relation will likewise answer and include all the other conditions
propounded. For, there being in the general expression, thus derived,
as many unknown quantities _b, c, d, &c._ (to be determined) as there
are equations, by making the fluents of _Bẋ, Cẋ, Dẋ, &c._ equal to the
values given; those quantities may be so assigned, or conceived to be
such, as to answer all the conditions of the said equations. And then,
to see clearly that the fluent of the first expression, _Aẋ_, cannot
be greater than arises from hence (other things remaining the same)
let there be supposed some other different relation of _x_ and _y_,
whereby the conditions of all the other fluents of _Bẋ, Cẋ, Dẋ, &c._
can be fulfilled; and let, _if possible_, this new relation give a
greater fluent of _Aẋ_ than the relation above assigned. Then, because
the fluents _bBẋ, cCẋ, dDẋ, &c._ are given, and the same in both cases,
it follows, according to this supposition, that this new relation must
give a greater fluent of _Aẋ_ + _bBẋ_ + _cCẋ_ + _dDẋ, &c._ (under all
possible values of _b, c, d, &c._) than the former relation gives:
_which is impossible_; because (whatever values are assigned to _b,
c, d, &c._) _that_ fluent will, it is demonstrated, be the greatest
possible, when the relation of _x_ and _y_ is that above determined, by
the General Rule.
To exemplify, now, by a particular case, the method of operation above
pointed out, let there be proposed the fluxionary quantity (_xⁿ yᵐ
ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ 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 ẏ⁽ᵖ ⁻ ¹⁾_) ⁄ _ẋ⁽ᵖ ⁻ ¹⁾_ = υ. And, by taking the fluxion
a second time, making _y_ alone variable, and dividing by _ẏ_, will be
had (_mxⁿ y⁽ᵐ ⁻ ¹⁾ ẏᵖ_) ⁄ _ẋ⁽ᵖ ⁻ 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 _ay⁽ᵐ ⁄ ᵖ⁾_ = υ (_a_ being a constant
quantity). From whence _y⁽ᵐ ⁄ ᵖ⁾ẏ_ = _(a ⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × _ẋx⁽⁻⁽ⁿ ⁄
⁽ᵖ ⁻ ¹⁾⁾⁾_; and consequently (_p_ ⁄ (_m + p_)) × _y⁽⁽ᵐ + ᵖ⁾ ⁄ ᵖ⁾_ = _(a
⁄ p)⁽¹ ⁄ ⁽ᵖ ⁻ ¹⁾⁾_ × (_p_ - 1) ⁄ (_p_ - _n_ - 1) × _x_⁽⁽_ᵖ_ ⁻ _ⁿ_ ⁻ ¹⁾
⁄ ⁽_ᵖ_ ⁻ ¹⁾⁾.
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.
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.
The vault at the top, surmounted by the thighs, is as it were
granulated; but at the rounding of the egg it produces a kind of
_mammæ_, or little elevations, very round and cylindrical, intirely
full; of the same nature and substance with the egg.
In examining the under part of the egg, we found a second rank of these
_mamellæ_, somewhat longer than the first, and at equal distances
from one another, in a circular line; then a third yet longer; then a
fourth, which at the extremities were bifurcated; and at last a fifth
rank, which divided into three, and sometimes into five, branches:
these last, placed round the hole, were wreathed inwards, and several
were joined together, and only formed a small body; and in wreathing
themselves thus they close and embrace the pivot mentioned below.
None of these _mamellæ_ have any apparent opening: their substance is
compact, of the same nature with the bladder or egg, that produces them.
Below the trunk and thighs the plant protrudes a pivot, of a like
substance with that of the bladder. This pivot, which is large at its
origin, proceeding thus from the trunk and thighs, forms something like
the knot of the sea-tree: it descends perpendicularly to the trunk,
diminishing as it lengthens, and as it grows round; and then divides
into a number of _mamellæ_, branched and wreathed inwards so firmly, as
not to be retracted; of a coriaceous nature, blackish, forming a bunch
like what we call the Rose of Jericho. I cannot recollect the name of
this plant or flower.
This bunch, or wreathed rose, incloses a heap of gravel, as if
petrified or hardened, and ends upon a level with the hole of the egg,
exactly as high as the last rank of _mamellæ_, which wreath upon,
embrace, and sustain it, leaving always an empty space to let the
sea-water pass in, which should fill the inside of the egg or bladder,
and even to let in little fishes and shells.
I was surprised to find in one little living muscles, as they always
are attached to some solid body by their beards. Now by what means
could they enter into this egg? I conjectured, that they had their
beginning there, by the seminal matter of muscles carried in by the
sea-water. I also found some small star-fish, whose rays might be about
four or five lines long.
If my stay here had been longer, I had continued my observations;
and perhaps should have made some discoveries. It belongs to the
academicians of Bourdeaux to push these observations further, if they
think proper.
From the Entrance of the river of Bourdeaux, the 4th of August, 1756.
Peyssonel.
LXXXVII. _An Account of the distilling Water fresh from Sea-water
by Wood-ashes. By Capt._ William Chapman: _In a Letter to_ John
Fothergill, _M. D._
[Read April 13, 1758.]
Whitby, 10th 2d mo. Feb. 1758.
THY kind acceptance of my last emboldens me to inform thee, how, on
my return from a voyage to the north part of Russia, I procured a
sufficient quantity of fresh water from sea-water, without taking with
me either instruments or ingredients expressly for the purpose.
Some time in September last, when I had been ten days at sea, by an
accident (off the north cape of Finland) we lost the greatest part of
our water. We had a hard gale of wind at south-west, which continued
three weeks, and drove us into 73° lat. During this time I was very
uneasy, as knowing, if our passage should hold out long, we must be
reduced to great straits; for we had no rains, but frequent fogs, which
yielded water in very small quantities. I now blamed myself for not
having a still along with me (as I had often thought no ship should be
without one). But it was now too late; and there was a necessity to
contrive some means for our preservation.
I was not a stranger to Appleby’s method: I had also a pamphlet wrote
by Dr. Butler, intituled, _An easy Method of procuring of fresh Water
at Sea_. And I imagined, that soap might supply the place of capital
lees, mentioned by him. I now set myself at work, to contrive a still;
and ordered an old pitch-pot, that held about ten quarts, to be made
clean: my carpenter, by my direction, fitted to it a cover of fir deal,
about two inches thick, very close; so that it was easily made tight by
luting it with paste. We had a hole thro’ the cover, in which was fixed
a wooden pipe nearly perpendicular. This I call the still-head: it was
bored with an augre of 1½ inch diameter, to within three inches of the
top or extremity, where it was left solid. We made a hole in this,
towards the upper part of its cavity (with a proper angle) to receive
a long wooden pipe, which we fixed therein, to descend to the tub in
which the worm should be placed. Here again I was at a loss; for we
had no lead pipe, nor any sheet-lead, on board. I thought, if I could
contrive a strait pipe to go thro’ a large cask of cold water, it might
answer the end of a worm. We then cut a pewter dish, and made a pipe
two feet long; and at three or four trials (for we did not let a little
discourage us) we made it quite tight. We bored a hole thro’ a cask,
with a proper descent, in which we fixed the pewter pipe, and made
both holes in the cask tight, and filled it with sea-water: the pipe
stuck without the cask three inches on each side. Having now got my
apparatus in readiness, I put seven quarts of sea-water, and an ounce
of soap, into my pot, and set it on the fire. The cover was kept from
rising by a prop of wood to the bow. We fixed on the head, and into it
the long wooden pipe above-mentioned, which was wide enough to receive
the end of the pewter one into its cavity. We easily made the joint
tight.
I need not tell thee with what anxiety I waited for success: but I
was soon relieved; for, as soon as the pot boiled, the water began
to run; and in twenty-eight minutes I got a quart of fresh water. I
tried it with an hydrometer I had on board, and found it as light as
river-water; but it had a rank oily taste, which I imagine was given it
by the soap. This taste diminished considerably in two or three days,
but not so much as to make it quite palateable. Our sheep and fowls
drank this water very greedily without any ill effects. We constantly
kept our still at work, and got a gallon of water every two hours;
which, if there had been a necessity to drink it, would have been
sufficient for our ship’s crew.
I now thought of trying to get water more palateable; and often perused
the pamphlet above-mentioned, especially the quotation from Sir R.
Hawkins’s voyage, who “with four billets distilled a hogshead of water
wholsome and nourishing.” I concluded he had delivered this account
under a veil, lest his method should be discovered: for it is plain,
that by four billets he could not mean the fuel, as they would scarce
_warm_ a hogshead of water. When, ruminating on this, it came into my
head, that he burnt his four billets to ashes, and with the mixture
of those ashes with sea-water he distilled a hogshead of fresh water
wholsome and nourishing. Pleased with this discovery, I cut a billet
small, and burnt it to ashes; and after cleaning my pot, I put into
it a spoonful of those ashes, with the usual quantity of sea-water.
The result answered my expectations: the water came off bright and
transparent, with an agreeable pungent taste, which at first I thought
was occasioned by the ashes, but afterwards was convinced it received
it from the resin or turpentine in the pot, or pipes annexed to it.
I was now relieved from my fears of being distressed thro’ want of
water; yet thought it necessary to advise my people not to be too free
in the use of this, whilst we had any of our old stock remaining; and
told them, I would make the experiment first myself; which I did, by
drinking a few glasses every day without any ill effect whatever. This
water was equally light with the other, and lathered very well with
soap. We had expended our old stock of water before we reached England;
but had reserved a good quantity of that which we distilled. After my
arrival at Shields, I invited several of my acquaintance on board to
taste the water: they drank several glasses, and thought it nothing
inferior to spring-water. I made them a bowl of punch of it, which was
highly commended.
I have not the convenience of a still here, or should have repeated the
experiment for the conviction of some of my friends: for as to myself,
I am firmly persuaded, that wood-ashes mixed with sea-water will yield,
when distilled, as good fresh water as can be wished for. And I think,
if every ship bound a long voyage was to take a small still with Dr.
Hales’s improvements, they need never want fresh water. Wood-ashes
may easily be made, whilst there is any wood in the ship; and the
extraordinary expence of fuel will be trifling, if they contrive so
that the still may stand on the fire along with the ship’s boiler.
I shall think myself sufficiently recompensed, if any hints here may
tend to the relief of my brother sailors from the dismal extremity of
want of water; an extremity too little regarded by those, who have
never experienced it.
_P. S._ During my passage from Russia we very rarely had any _aurora
borealis_; and those few we saw were faint, and of short continuance:
at which I was much surprised; for about ten years ago, being in a
high north latitude, we had very beautiful ones almost every night in
the month of September; which exceeded any I have seen described in
the _Philosophical Transactions_, or _Memoires de l’Academie Royale_.
Wm. Chapman.
LXXXVIII. _Observatio Eclipsis Lunaris facta_ Matriti _a Pª._ Joanne
Wendlingen, _Societatis_ Jesu, _in Regali Observatorio Collegii
Imperialis ejusdem Societatis, Die_ 30 Julii 1757.
_Quælibet observatio bis instituta fuit, semel interjecto oculum
inter lentemque ocularem vitri clari, cærulei, plani, ac bene tersi,
fragmento. Hæ observationes notantur hac voce_ cerul. _Telescopium,
quo usus sum, est_ Gregorianum _trium pedum_ Anglicanorum, _omnino
præclarum_.
_Communicated by_ Matthew Maty, _M. D. F.R.S._
[Read April 20, 1758.]
IMMERSIONES. |Tempus verum| Differentia
| h ´ ´´ |
PRincipium eclipsis, _clar._ | 9 47 34 |
Mare Humorum, _clar._ | -- 52 47 |
{ _cær._ | -- 54 28 | ´´
Grimaldus { _clar._| -- 55 27 | ---- 59
{_cær._ | 10 1 21 | ---- 13
Bullialdus {---- | 10 1 34 |
{---- | -- 9 35 | ---- 10
Keplerus {---- | -- 9 45 |
{---- | -- 16 15 | ---- 12
Copernicus {---- | -- 16 28 |
{---- | -- 18 14 | ---- 10
Heraclides {---- | -- 18 24 |
Manilius {_clar._ | 10 30 43 | ´´
{_cær._ | 10 30 54 | ---- 11
Menelaus {---- | -- 34 11 | ---- 9
{---- | -- 34 20 |
Promontorium {---- | -- 44 49 | ---- 9
{---- | -- 44 58 |
Mare Crisium {---- | -- 45 33 | ---- 11
{---- | -- 45 44 |
Proclus, _clar._ | -- 46 54 |
{---- | -- 54 48 | ---- 14
Plato {---- | -- 55 2 |
{---- | 11 7 3 |
Langrenus {---- | 11 7 23 | ---- 20
EMERSIONES. |Tempus verum| Differentia
| h ´ ´´ |
Plato {_clar._ | 11 40 34 | ´´
{_cær._ | 11 40 48 | ---- 14
Heraclides {---- | -- 41 27 | ---- 12
{---- | -- 41 39 |
Grimaldus {---- | -- 47 57 | ---- 7
{---- | -- 48 4 |
Keplerus {---- | -- 52 58 | ---- 10
{---- | -- 53 8 |
Copernicus {---- | 12 1 36 | ---- 12
{---- | 12 1 48 |
Menelaus {---- | -- 17 18 | ---- 10
{---- | -- 17 28 |
Finis eclipseos, _clar._ | -- 52 15 |
_Observatio Eclipsis Lunaris, facta ab eodem, eodem modo, eodem loco,
iisdemque instrumentis Die_ 24 Januar. _Anni_ 1758.
_Ab hora 5ᵗᵃ usque ad finem observationis tantum commovebatur
imprægnata plurimum vaporibus athmosphæra, ut tota lunæ illuminatæ
portio præter morem undulare videretur. Flabat boreas, indicante
thermometro Reaumuriano. 1. grandem infra aquæ congelationem._
IMMERSIONES. |Tempus verum| Differentia
| h ´ ´´ |
Principium | 4 7 42 | ´´
Grimaldus {_cær._ | -- 9 55 | ---- 38
{_clar._ | -- 10 33 |
Aristarchus {---- | -- 15 37 | ---- 34
{---- | -- 16 11 |
Mare Humorum {---- | -- 20 48 | ---- 24
{---- | -- 21 12 |
Copernicus {---- | -- 26 45 | ---- 13
{---- | -- 26 58 |
Plato {---- | -- 35 41 | ---- 41
{---- | -- 36 22 |
Tycho {---- | -- 39 31 | ---- 38
{---- | -- 40 9 |
Menelaus {---- | -- 45 43 | ---- 39
{---- | -- 46 22 |
Plinius {---- | -- 50 17 | ---- 27
{---- | -- 50 44 |
Promontorium Somni {---- | -- 58 15 | ---- 27
{---- | -- 58 42 |
Cleomedes {---- | 5 0 22 | ---- 14
{---- | 5 0 36 |
Proclus {_cær._ | 5 2 9 |
{_clar._ | 5 2 15 | ---- 6
Princip. Maris Crisii {---- | -- 3 6 | ---- 25
{---- | -- 3 31 |
Langrenus {---- | -- 7 40 | ---- 14
{---- | -- 7 54 |
Immers. tot. Maris Crisii {---- | -- 8 19 | ---- 11
{---- | -- 8 30 |
Immersio totalis Lunæ {---- | -- 12 50 | ---- 40
{---- | -- 13 30 |
In fine cœlum serenum, & athmosphæra quieta.