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ried over other curves, the dimensions of the locus of p will be equal to the
triple product of the number of dimensions of all the curves employed in the
description.

5. If the invariable angles pnr, pmg, (fig. 11 ) move so, that while the
sides PN, PM, pass always through the poles c and s, the angular points n and
M describe the curves an and bm; and at the same time, the invariable angle
RDQ, revolve about the third pole d, so that the intersections r and a describe
the curves er and go. ; then the dimensions of the locus of p, when highest,
shall be equal to the quadruple product of the numbers that express the dimen-



48 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735.

sions of the given curves an, er, oa and bm, multiplied continually into each
other. If more poles are assumed, about which angles be supposed to move,
as RDQ moves about d in this description, and the intersections of the sides be
still carried over curves, as in this example ; the dimensions, of the locus of p,
when highest, shall still be found equal to the quadruple product of all
the numbers that express the dimensions of the curves employed in this
description.

6. Suppose that the three invariable angles pqk, klr, rnp, (fig. 12) move
over the curves gq, el, an, so that the sides pq, kl, pn produced, pass al-
ways through the poles c, d, s, and that the intersections of their sides k and
R, at the same time move over the curves fk and br ; then the dimensions of
the locus of p, when highest, shall be equal to the product of the numbers
that express the dimensions of the given curves multiplied by 6. If more poles,
with the necessary angles and curves, are assumed between c and d, as here
D is assumed between c and s, and the motions be in other respects like to
what they are in this example; then in order to find the dimensions of the
locus p, when highest, raise the number 2 to a power whose index is less than
the number of poles by a unit; add 2 to this power, and multiply the sum by
the product of the numbers that express the dimensions of the curves employed
in the description ; then this last product shall show the dimensions of the
locus of p when highest.

The author is able to continue these theorems much further : but it is not
worth while, especially since there is not any considerable advantage obtained
by increasing the number of poles, above the method delivered in the above-
mentioned treatise, of the description of curve lines. On the contrary, the
descriptions there given, by means of 2 poles, will produce a locus of higher
dimensions by the same number of curves and angles, than these that require
3 or more poles; and are therefore preferable, unless perhaps in some parti-
cular cases.

7. However, he has also found how to draw tangents to the curves that
arise in all these descriptions : of which he gives one instance, where 3 right
lines are supposed to revolve about 3 poles, and 2 of their intersections are
supposed to be carried over given curve lines, and the third describes the locus
required.

Let the right lines ca, sn, dn, (fig. 13) revolve about the poles c, s, d ;
where that which revolves about d, serves to guide the motion of the other
two ; its intersection with ca moving oyer the curve oa, while its intersection
with SN moves over the curve fn. Suppose that the right line ah touches
the curve ca in a, and that the right line Aa touches the curve fn in n. In



VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 4^

order to draw a tangent to the locus of p ; join dc, ds and cs, and constitute
the ; ngle dqr, equal to cqb, so that gr lie the contrary way from gd that qb
lies from gc, and let qr meet dc in r. Constitute also the angle dnt, equal
SNA, with the like precaution, and let nt meet ds in t. Join kt, and produce
it till it meet cS in h ; then join ph, and make the angle cpl equal to sph, so
that PL and ph may lie contrary ways from cp and sp; then pl shall be a tan-
gent at p, to the locus described by p, the intersection of cq and sn.

Mr. Maclaurin has also applied this doctrine to the description of lines through
given points. But he supposes he has said enough at present on this subject ;
and concludes, after observing that in the abovementioned treatise, he has
given an easy theorem, for calculating the resistance of the medium, when a
given curve is described with a given centripetal force in a resisting medium,
which he here repeats, because it has been misrepresented in a foreign Journal.

Let V express the centripetal force with which the body that is supposed to
describe the curve, is acted on the medium ; let v express the centripetal force
with which the same curve could be described in a void ; suppose z = -, then
the resistance shall be proportional to the fluxion of z multiplied by the fluxion
of the curve ; supposing the area, described by a ray, drawn from the body to
the centre of the forces, to flow uniformly. Let this theorem be compared
with what the celebrated ma^iematician mentioned by that Journalist has given
on the same subject, and it will easily appear what judgment is to be made of
his assertion ; and since several persons, and particularly the gentleman men-
tioned above in this paper, testify that Mr. Maclaurin communicated to them
this theorem, before any thing was published on this subject by the learned
mathematician he nauies, his observation on this occasion must appear the
more groundless.

From this theorem, the author draws this very general corollary; that if the
curve is such as could be described in a void by a centripetal force, varying ac-
cording to any power of the distance, then the density of the medium in any
place, is reciprocally proportional to the tangent of the curve at that place,
bounded at one extremity by the point of contact, and, at the other, by its
intersection with a perpendicular raised at the centre of the forces to the ray
drawn from that centre to the point of contact. Let al be the curve described
by a force directed to the point s (fig. 14) ; let lt touch the curve at l, and
raise st perpendicular to sl, meeting lt in t ; then the density in l shall be
inversely as lt, if the resistance be supposed to observe the compound propor-
tion of the density, and of the square of the velocity.

Besides what is here observed, he proposes to illustrate and improve several

VOL. veil. H



50 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735.

Other parts of the treatise concerning the description of curve lines in this
supplement.

That treatise requires these additions and illustrations the more, that though
the whole almost was new, it was published in a hurry, when the author was
very young, before he had time to consider sufficiently which were the best
ways of demonstrating the theorems, or resolving the problems, for which this
supplement he hopes will make some apology.

The following paper, dated at Nancy, Nov. 27, 1722, is that which the
author mentions in his letter.

Section I.— Pkop. 1. Which respects the Description of Lines. — About
the poles c, b, d, (fig. 15) let the lines cd, Bm, nr be moved; and let the
concourse of the legs sm, nr be drawn along the given line pg, and the con-
course of the legs cd, nr along the given line pa; then the concourse of the
legs cd, sd will describe a conic section.

Draw rt parallel to the line bd given in position, meeting Bd in t; joint
pt, producing it to meet bd in f ; and it will give the point f. For as the
ratio of ru to rt is given, being the same as that of dg to db, because of the
similar triangles omBG and rmta ; and since ru is to rt, as og to qf, the
ratio of qf to ug will be also given ; so that, because of the given line qg,
there will be given qp, and hence the point f and the line pf. Since there-
fore Btand cr cut off the parts pt, pr, from the ITnes pf, pa, given in position,
their intersection d will always be in a given ratio in a conic section, by
Lem. 20, lib. \, Newton's Principia.

If the point d be taken anywhere in the right line bf; and if dg be always
to aG, as bd to qf ; the conic section will be the same as d describes.

The conic section passes through c, p, b and a, by completing the parallelo-
gram psav. It also passes through l, where the line bg produced meets pv, as
also through k, where cd cuts the given line pg. Hence the pentagon pkclb
is inscribed in the section. And if the 5 points ckpbl be given, through
which the conic section is to be drawn, or if the conic section is to be cir-
cumscribed about the given pentagon clbpk, let any 2 sides, ck, lb, be pro-
duced to their intersection d ; then join the rest pl, pk, and let the intersec-
tions of cd, Dr, and sd, dk, be silways drawn along those lines pl, pr ; then
the intersection d will describe the section.

Prop. 2. About the given points f, c, g, s, (fig. l6) as poles, let the lines
FQ, cn, gq, sl be moved ; and let the intersections of the lines fq and cn,
FQ and Ga, Ga and sl, viz. the points m, a, l, always touch the lines ae,
BB, HL, given in position ; then the intersection of the lines cn, sl will describe
a conic section.



VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 51

Let the lines am, hr meet bq in e and h. Join cf and gs, meeting in
D ; join Da, meeting cm and sl in n and r : then if en and hr be joined,
these will be lines given in position, by Lemma 1 . For since the points f, c, d,
are in the same right line ; and the intersections of the lines fm, cm, and Fa,
Da, run over given lines, the intersection of the legs cm, do will also touch
the given one. And, for the like reason, since s, d, g are in the same right
line, the intersection of do and sl will also touch the given one.

Therefore, omitting the poles f, g, there is to be found the curve which the
intersection p, of cn, sl, will describe, while by the lines cn, dn, sr, revolv-
ing about the poles c, d, s, the concourse of the two cn, dn touches the
given line en, and the concourse of the two sr, dn touches the given one hr;
and that this is a conic section is plain from the foregoing proposition.

Some Experiments relating to Electricity. By Mr. Stephen Gray, F. R. S.

N°439, p. 166.

Feb. 18, Mr. Gray tried what effect would be produced on several sorts of
wood with respect to the luminous part of electricity : the wood was made
into rods of the same form with those iron ones mentioned in a former letter
on this subject ; the woods were fir, ash and holly ; these being successively
disposed on electric bodies, after the same manner as the iron rods had been,
the tube being applied to one end, there appeared a light on it, but not with
so great a force, nor did the light extend to so great a length ; neither was the
form of it conical, but rather cylindrical ; but its extremity seemed to consist
of a short fringe of light ; when the light, that was given to the rod by the
application of the tube, ceased, on a motion of the hand towards the point
of the rod, the light came out again, as has been mentioned of the iron rods ;
but when the hand or finger was held near the point of these wooden rods,
there was no pricking or pushing of the finger felt, as when the iron rods were
used. He had some of these rods made much larger at one end than the
other, and now applying his finger to the larger end, there not only appeared
a light, but the finger was pushed, especially when the holly rod was used, and
the cheek was a little pricked, but the smart was not near so great as when
the iron rods were used ; the large end of the rod was pointed with a much
greater angle than the smaller one, yet there was very little, if any difference,
in the form or size of the light from either end.

Having procured two pair of lines made of worsted yarn, one of them of
a mazareen blue, the other of a scarlet colour ; on the 3d of April, he sus-
pended the boy first on the blue lines, and found that all the effects were the

H 1



52 PHILOSOPHICAL TRANSACTIONS. ' [aNNO 1735.

same, as when he was suspended on lines of blue silk. He then suspended
him on the scarlet lines ; but now, though the tube was as well excited, and
the experiment often repeated, yet there was no effect produced on him, either
of attraction of a pendulous thread, nor of pricking nor burning, by apply-
ing a hand near him : one of the iron rods being then first on the blue lines,
all the same effects were exhibited, as when the same rod had been laid on
silk lines of that colour; but on laying the same rod upon the scarlet lines, no
manner of attraction, &c. was perceived.

In the Philos. Trans. N° 422, Mr. Gray gave an account of the experiments
made on the communicative electricity of water, and found that water is at-
tracted by the tube ; with several remarkable circumstances with which this
attraction is attended ; but he has now found, that when the stand with those
little ivory cups there mentioned, are set on any electric body, the same phae-
nomena are produced, not only by holding the tube near the water, but when
that is removed, and the tip of the finger placed over the water, viz. there is
a little hill, or protuberance of water of a conical form, from the vertex of
which proceeded a light and a small snapping.

May the 6th, was made the following experiment. The boy being sus-
pended on the silk lines; and the tube being applied near his feet, as usual;
on holding the end of his finger near a gentleman's hand, that stood on a cake
made of shell lack and black rosin ; at the same time another gentleman stand-
ing at the other side of the boy with the pendulous thread; the boy was then
bid to hold his finger near the first gentleman's hand, on which it was pricked,
and the snapping noise was heard; and at the same time the thread, which
was by its attraction going towards the boy, fell back, the boy having lost a
great part of his attraction; on a second moving his finger to the gentleman's
hand, the attraction ceased: then the thread being held near that gentleman,
he was found to attract very strongly; but having since repeated this experi-
ment, though the attraction of the boy is much diminished, yet he does not
quite lose it, till 2, 3, and sometimes 4 applications of his finger to the hand
of him that stands on the electric body, but without touching him. At another
time three persons stood, one of them on a cake of shell lack, &c. the other
on one of sulphur, the third on a cake of bees-wax and rosin ; the persons all
holding hands, the boy applying his finger near the first man's hand, they all
three became electrical, as appeared by the attraction of the thread, when
held near to any of them.

June the 10th in the morning, were repeated the experiments with the
wooden rods, the most material of which were made with the holly rod : this
being laid on the glass cylinder, and a fir board, about a foot square and ,v of



/



VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 53

an inch thick, being placed erect on a stand, set on another glass cylinder, so
that the centre of the board was placed near the point of the rod, but not to
touch it by near i- inch ; then the tube being held near the thick end of the
rod, there issued out a light from the small end, which was that next the board;
and it came along with a hissing noise, and struck against the board : when the
boy touched the board, there was a light ; and at the same time another on
the end of the rod, but he heard no snapping nor pricking of his finger, as
when the brass plate and iron rod were used.

Experiments with the Scarlet and Blue Worsted Yarn repeated. — When the
boy was suspended on the scarlet lines, he attracted the white thread at a very
small distance ; but the attraction ceased in about 6 or 7 seconds of time.
Then the boy being taken off, an iron rod was laid on the lines, but there
was no attraction of the thread by the body of the rod; but when the thread
was held near either of its pointed ends, it showed a small repulsion, and in
the dark a very small light was seen at each end of the rod.

When the boy was suspended on the blue lines, he attracted the thread to
him when it was held at least a foot from him, and he continued his attrac-
tion to near 75 seconds, the iron rod continued its attraction not more than
36 seconds.

When he was suspended on the blue lines, he continued his attraction 50
minutes, on the scarlet lines 25 minutes, on the orange coloured lines 21
minutes.

By these experiments we see the efficacy of electricity on bodies sus-
pended on lines of the same substance, but of different colours, and also that
the attraction continues much longer on silk than on yarn, and consequently
silk is the properest body we can make use of, to suspend those bodies on, to
which we would communicate an electricity.

An Account of the Births and Burials, with the Number of the Inhabitants
at Stoke- Damerel, Devonshire. Communicated by the Rev. Mr. fVm. Barlow..
]S° 439, p. J7I.

On taking a survey, about Michaelmas 1733, of the inhabitants of Stoke-
Damerel, in the county of Devon, the number of persons, men, women,
and children, residing in the parish, amounted to 336l. By the register, it
appears that in the same year, 28 couples were married, 61 males and 61 fe-
males baptized, and 61 people buried.

Whence it appears that the number of persons who died, is only one more
than half the number of children born ; and that about 1 in 54 died.



54 VHILOSOPHICAL TRANSACTIONS. [aNNO 1736.

An extraordinary Case of the Foramen Ovale of the Heart, being found open
in an Adult ; communicated by Claudius Amyand, F. R. S. N° 439, p. 1 72.

A person dying at the age of 22, of an illness that had perplexed his phy-
sicians, was opened, to discover an imposthume, which was apprehended in
the belly. As nothing was observed there worth notice, excepting a great re-
laxation of the viscera, the cause of his death was looked for in the thorax ;
there the lungs were strongly attached to the pleura on each side, and a
large collection of water in each cavity, especially on the left, where the pos-
terior lobe was inflamed, and tending to suppuration; the quantity of water
in the pericardium was greater than usual, and the heart much larger than
could be expected in so great an atrophy as the patient was reduced to ; in
it the foramen ovale was found open, so as to give passage to a large finger,
when a fungous substance, which grew from the circumference of the foramen,
and stopped up the same, was removed. The valve was hardly perceptible, it
being callous and furled up. The ductus arteriosus was found close as usual.
This patient had enjoyed great health till lately, and had given no sign of this
opening of the foramen ovale, which is preternatural in adults.

A Catalogue of the Fifty Plants from Chelsea Garden, presented to the Royal
Society hy the Company of Apothecaries, for the Year 1734, pursuant to
the Direction of Sir Hans Sloane, Bart. P.R.S. By Isaac Rand, F.R.S.
N''440, p. 173.

This is the 13th catalogue of this collection, completing 650 plants.

The Apparent Times of the Immersions and Emersions of Jupiter's Satellites,
which will happen in the Year 1737. Computed to the Meridian of the
Royal Observatory at Greenwich. By James Hodgson, F. R. S. N" 440,
p. 177.

Another repetition of the catalogue of these eclipses, pre-computed, and
published for the accommodation of gentlemen intending to make observations
on them.

A Proposition relating to the Combination of Transparent Lenses with Rejecting
Planes. By J. Hadley, Esq. V. Pr. R. S. Communicated Jan. 9, 1734.
N°440, p. 185.

Having proposed the use of a telescope with the instrument for taking angles



vol.. XXXIX.3 l-HILOSOPHICAL TRANSACTIONS. 55.

which Mr. Hadley formerly laid before this Society, (See N° 420) it gave oc-
casion to consider the effects of combining several kinds of telescopes with re-
flecting planes, and, among others, led to the following proposition :

That if two lenses, of equal focal length, be put together in the form of a
telescope, and a plane speculum be placed before one of them, so that the axis
of the telescope make any angle with its surface, and a ray of light, the line of
whose direction lies in a plane perpendicular to that surface, and passing through
the axis of the telescope, fall on it, and be reflected from it, so as to pass
through the telescope ; then the line of its last direction, after passing the
telescope, will make an angle with that of its first direction, before its incidence
on the speculum, very nearly equal to double the angle made between the axis
of the telescope, and the surface of the speculum.

Lemma. — Let the line fg be the common axis of the two lenses id and ke,
of equal focal lengths, fig. 6, pi. 1 ; to which let the lines ad, db and be, be
each equal ; and let a ray of light, issuing from a point in the axis p, fall on
the lens id at i, and be there refracted into the line ig, cutting the axis in g,
and meeting the lens ke in k ; where let the ray be again refracted into the line
KH, cutting the aforesaid axis in h : the angles ipd and khe are very nearly
equal,

Demonstr. — It is known from dioptrics, that the lines pi, I6, kh, and fg, are
all in the same plane ; and by the construction the lines ad, db, and be are
equal ; and by prop. 20 of Huygens's dioptrics, the lines fa, pd, and fg are
continually proportional ; consequently fa is to ad as pd to dg ; and dividing,
PA is to AD as PD — pa (= ad) is to DG — AD (= bg). Therefore ad is to
bg as FD to dg. By the same prop, the lines bg, eg, hg are also continually
proportional, and be (=: ad) is to bg, as eh is to eg. Hence it follows, that
the lines pd, dg, and eh, eg, are proportionals. But pd is to dg, as the tangent
of the angle igd or kge, to the tangent of the angle ipd ; and eh is to eg, as
the tangent of the angle kge to the tangent of the angle khe. The tangent
of the angle kge therefore has the same ratio to the tangents of each of the
angles ifd and khe, and consequently those angles are equal, a. e. d.

In the demonstration of the above-cited proposition of Hiiygens, the thick-
ness of the lenses are neglected, and the distance of the points i and k, from
the line fg, supposed very small ; so that if either of those are too great, there
may arise a sensible difference between the angles ifd and khe.

Let DP and cg, fig. 7> represent the two lenses, put together as before,
having their common axis in the line bl ; and bn a plane speculum, to which
that line is inclined in the angle ghn ; and let ab be a ray of light falling on
the speculum at b, as is before expressed, and let it be there reflected towards



56 PHILOSOPHICAL TRANSACTIONS. [aNNO 1736.

the point c of the lens cg, where it is refracted towards the point d of the lens
DF, and there again refracted into the line de, cutting the axis in k. The angle
AOP contained between this last line de, continued backward, and the first line
of incidence of the ray ab, will be very nearly equal to double the angle of in-
clination of the axis of the lenses el, to the plane of the speculum bn ; i. e.
double the angle ghn.

Demonslr. — Produce the lines of incidence and reflection of the ray ab and
BC, till they meet the axis of the two lenses in i and l ; and through the point
B draw BK perpendicular to the plane of the speculum, and cutting the same
axis in k; then the angles kbl and kbi are equal. The angle klb is the differ-
ence of the angles ikb and kbl ; and the angle hib is the sum of the angles ikb
and KBI = to KBL : therefore the angle ikb is equal to half the sum of the angles
hib and klb. But by the foregoing lemma, the angles klb and fed are very
nearly equal. Therefore the angle ikb is nearly equal to half the sum of the
angles hib and fed ; that is, to half the angle pob ; and its complement bhi or
GHN is nearly equal to half the angle aop, the complement of pob to a semi-
circle, a. E. D.

If the first incidence of the ray be supposed to be in the line ed, it will pro-
ceed in the same track as before, but with the contrary directions ; so that the
angle eob made between the first incident ray, and the last reflected, will still
be equal to the double of ghn, as before.

It is evident that on this principle an instrument might be constructed, the
effects of which would in a great measure resemble those of that before-men-
tioned (N° 420) : but it would be liable to the errors arising both from the
spherical figure of the lenses, and also the different refrangibility of the rays of
light, when the object is seen at a distance from the axis of the telescope ;
though those errors, by a proper disposition of the parts of the instrument, may



Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 8 of 85)