XXXVI. A Disquisition concerning certain Fluents, which are Assignable by the Arcs of the Conic Sections; where are Investigated some New and Useful Theorems for Computing such Fluents. By John Landen, F. R. S. p. 298. Mr. Mac Laurin, in his Treatise of Fluxions, has given sundry very elegant theorems for computing the fluents of certain fluxions by means of elliptic and hyperbolic arcs; and Mr. D'Alembert, in the Memoirs of the Berlin Academy, has made some improvement on what had been before written on that subject. But some of the theorems given by those gentlemen being in part expressed by the difference between. an arc of an hyperbola and its tangent, and such difference being not directly attainable, when such arc and its tangent both become infinite, as they will do when the whole fluent is wanted, though such fluent be at the same time finite; those theorems therefore in that case fail, a computation thereby being then impracticable, without some further help.

The supplying that defect Mr. L. considered as a point of some importance in geometry, and therefore he earnestly wished, and endcavoured, to accomplish that business; his aim being to ascertain, by means of such arcs as aboveinentioned, the limit of the difference between the hyperbolic arc and its tangent, while the point of contact is supposed to be carried to an infinite distance from the vertex of the curve, seeing that, by the help of that limit, the computation would be rendered practicable in the case wherein, without such help, the before-mentioned theorems fail. And having succeeded to his satisfaction, he presumes the result of his endeavours, which this paper contains, will not be unacceptable to the Royal Society.

1. Suppose the curve ADEF, pl. 4, fig. 7, to be a conic hyperbola, whose semi-transverse axis AC is = m, and semi-conjugate = N, Let CP, perpendi

cular to the tangent DP, be called p; and putƒ:

=

m2 - n2

2 m

z = 2. Then, as is well

m

p and z being each =

2. Suppose the curve adefg fig. 8, to be a quadrant of an cllipsis, whose semitransverse axis cg is = √ m2 + n2, and semi-conjugate ac = n. Let ct be per

pendicular to the tangent dt, and let the abscissa cp ben√. Then will the

= m

; and its fluxion will be found = ÷ mn2z - 1ż X

be supposed = 2. Then will bey, and the

-c

dbz

(az

, q = &, a = − d = 12, b = 1,

4. Taking, in the last article, r and s each = 4, q = 4, a =

Consequently, taking the fluents by art. 1, and correcting them properly, we

AF the fluent of

and L the limit to which

2 √ mz

√n2 + 2fz →
-j√my

n2 + 2fy — y2

the difference DP-AD, or FR - AF, approaches, on

carrying the point D, or F, from the vertex a ad infinitum.

5. Suppose y equal to z, and that the points D and F then coincide in E, the points d and p being at the same time in e and q respectively. Then cv being perpendicular to the tangent ev, that tangent will be a maximum and equal to cg -ac= √ m2 + n2 -n; the tangent Ea, in the hyperbola, will be = ✅m2 + n2; √(1 the abscissa BC = m (1+); the ordinate BE = n ×

n

√ m2 + n2

n

and it appears that Lis=2EQ-2AE ev=n+ √ m2 + n2 — 2AEL. Thus the limit proposed to be ascertained, is investigated, m and n being any right lines whatever!

Jż √ mz √n2 + 2fz

6. The whole fluent of generated while z from O becomes = m, being equal to L; and the fluent of the same fluxion (supposing it to begin. dt; it when z begins) being in general equal to L + AD DP = FR — AF — appears, that, k being the value of z corresponding to the fluent L + AD DP, will be the value of z corresponding to the fluent L + AF FR, and

follows therefore that the tangent dt, together with the fluent of

It.

generated while z from O becomes equal to any quantity k, is equal to the fluent

DP, the fluent of
√n2 + 2fz- z2
3x √ mz

+ DP — AD — L will be 0. Therefore, the fluent of

--

n2 + mz m- 2

being the fluent of z

√ n2 + Qfz — z2

√ n2 + Qƒz — z3 ̧n2 + mz

+ the

it is obvious,

is = DP — AD — L + the fluent of

= DP—AD — L + the elliptic arc dg (fig. 8) whose abscissa cp is = n√ Consequently, putting E for of the periphery of that ellipsis, it ÷ appears that generated while z from 0 becomes = m, is

8. By taking, in art. 3, 9, r, and s, each; and a=d= mn n2z 2√3/1/

b = 1, and c = n2; we find, that, if y be = n2 + mz' √ n2 + 2ƒz— z2

will be = 0.

It is obvious therefore, that the fluent of from o becomes equal to any quantity k, is equal to the fluent of the same fluxion, generated while z from

to half the fluent of the same fluxion, generated while z from O becomes = m; which half fluent is known by the preceding article.

9. It appears, by art. 4, that of the tang, dt; and it appears, by the last article, that

Consequently, by taking the correct fluents, we find the tang. dt (= the tang. fw) = the arc ad the arc fg! the abscissa cp being =

n, the abscissa crn, and their relation expressed by the equation n°

n1u2 - n1v2 m2u3v2 = 0, u and v being put for cp and cr respectively. Moreover, the tangents dt, fw, will each be = ; and ct X cw = cv2 = ac

x cg.

m2uv

If for the semi-transverse axis cg we substitute h instead of ✔m2 + n2, the relation of u to v will be expressed by the equation n° — n'u2 — n1v2 — (h2 — n2) × u2v2 = 0, and dt (= fw) will be =

- n2 n3

x uv.

If u and v be respectively put for fr and dp, their relation will be expressed by the equation ho - hˆu2 — h*v2 + (h2 — n2) × u2v2 = 0, and dt (= fw) will be

h2 n2

h3

x uv.

10. Suppose y = to z, that is, v=u, and that the points d and f coincide in e. In which case the tangent dt will be appears then that the arc ae - the arc eg is is E for the quadrantal arc ag, we find that the

a maximum,
cg — ac.
cg ac.

arc ae is =

and = cg Consequently, putting B+ h − n !

--- ac. It

2

+ !

2

There are, Mr. L. is aware, some other parts of the arc ag, whose lengths may be assigned by means of the whole length ag, with right lines; but to investigate such other parts is not to his present purpose.

11. Taking m and n each = 1; that is, ac = AC = 1, and cg= 2; let the arc ag be then expressed by e: put c for of the periphery of the circle whose radius is 1; and let the whole fluents of

and
W

generated while z from O becomes 1, be denoted by F and G respectively. Then, by what is said above, F G will be e; and, by his theorem for comparing curvilineal areas, or fluents, published in the Philos. Trans. for the year 1768, it appears that FX G is c. From which equations we find F = e Ve2c, and Gee2 - 2c.

But m and n being each = 1, L is = F; therefore 1 + √2-2AE, the value of L, from art. 5, is in this case = ↓e — ¿√ e2 — 2c. Consequently, in the equilateral hyperbola, the arc AE, whose abscissa BC is = √(1 + √), will be

= ÷ + √ ↓ − ↓e + √ e2 - 2c, by what is said in the article last mentioned. Hence the rectification of that arc may be effected by means of the circle and ellipsis !

The application of these improvements will be easily made by the intelligent reader, who is acquainted with what has been before written on the subject. But there is a theorem, demonstrable by what is proved in art. 8, so remarkable, that he cannot conclude this disquisition without taking notice of it.

12. Let Ipqn, fig. 9, be a circle perpendicular to the horizon, whose highest point is 1, lowest n, and centre m; let p and q be any points in the semicircumference pqn; draw ps, qt parallel to the horizon, intersecting lmn in s and t; and, having joined lp, pt, make the angle lpu equal to lip, and draw ru parallel to qt, intersecting the circle in r, and the diameter Imn in v. Let a pendulum, or other heavy body, descend by its gravity from p along the arc pqrn: the body so descending will pass over the arc pq exactly in the same time as it will pass over the arc rn; and therefore, qt and ru coinciding when it is equal to lp, it is evident that the time of descent from p to q will then be precisely equal to half the time of descent from p to n!

And it is further observable, that, if pqn be a quadrant, the whole time of descent will be = √ 1 × (ie + ÷ √ e2 2c); the radius lm, or mn, being a; and b being put for (16 feet) the space a heavy body descending freely from rest falls through in one second of time.

Jax √

x j;

In general ns being denoted by d, and the distance of the body from the line ps, in its descent, by x, the fluxion of the time of descent will be expressed by √ [2ad — d3 — (2a — 2d). — ]; the fluent of which, corresponding to any value of x, may be obtained by art. 7. By which article it appears, that the whole time of descent from any point p will be = √ bdx (2a - d) X (E+2AE-pn-ps). The semi-transverse AC (fig. 7) being = ns;

a

the semi-transverse cg (fig. 8)
and the semi-conjugate in each figure

= np;

= ps.

After writing the above, Mr. L. discovered a general theorem for the rectification of the hyperbola, by means of two ellipsis; the investigation of which he purposed to make the subject of another paper.

XXXVII. On the Management of Carp in Polish Prussia. By J. Reinhold Forster, F. A. S. p. 310.

Though the carp is now commonly found in ponds and rivers, and generally thought to be a fresh water fish, the ancient zoologists ranged the same among

* I have great reason to think, that many other fish, which, it is commonly conceived, can only live in the sea, may also exist, at least for several years, and perhaps breed, in fresh water. The smelt or sparling (salmo eperlanus Linnæi) never comes up our rivers, but for a short time; and then does not penetrate much farther than where the water continues to be brackish. I have however been informed by Sir Francis Barnard, the late governor of New England, that in a large pool