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III. Again, sineДA: sine BC:: sine c: sine AB. (G. 176.) :: sine B sine AC.

That is, the sines of the variable angles have a constant ratio to the sines of their opposite sides, consequently their fluxions will be in the same ratio; but the rectangle of radius and the fluxion of the sine of any arc the rectangle of the fluxion of that arc and its cosine (1st equation page 127.) Hence, flux. sine of

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sine A sine BC:: cos Lcx LC: cos AB X AB,

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But it has been shown in the first part of this proposition, that

AB COS C¦¦ AC: COS ▲ B.

:: Zc: Ac:: sine ▲ A × COS AB : sine BC × Cos ▲ B.

(L) Hence, as in the former propositions, if we denote the sides of the triangle ABC, by a, b, c, and their opposite angles by A, B, C, we derive the following proportions, wherein a and a are invariable quantities.

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5. c: b:: sine a X cos c: sine a x cos B.

These proportions may be varied in their form, thus from

the third.

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viz. cc::tang c: tang c. (N. 104.)

And exactly in the same manner, from the fourth proportion,

we derive Bb::tang B: tang b.

PROPOSITION VII. (Plate IV. Fig. 6.)

(M) In any oblique-angled spherical triangle ABC, suppose the two sides AB and AC to remain constant*, it is required to find the fluxions of the other parts.

Let BC and AC change their positions to вn and an, an being equal to AC; produce ac and An to the points o and p, so that AO and Ap may be quadrants, and join cn.

Also, produce BC and вn to the points q and r, making вq and Br quadrants, and through c draw cm parallel to qr.

Then op will be the measure of the increment of the A, qr the measure of the decrement of the B, and mn will be the increment of the side BC.

I. Sine ap sine an::sine op: sine cn,

viz. rad sine ac:: LÀ: sine cn=

Also, sine Bq : sine BC:: sine qr: sine cm,

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sine. AC

rad

XLA.

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II. In the small straight-lined triangle cmn.
1. rad: cn::sine mcn: mn. (Y. 34.)
sine AC
sine AC X sine c
rad

Viz.rad:

BC

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× ZA::sine/C:BC=

rad2

But sinec sine AB:: sine B sine Ac. (G. 176.) •. sine cx sine AC sine BX sine AB, consequently sine B X sine AB

rad2

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Again, by substituting AB for AC, and c for B, we obtain

* La Lande's Astronomy, vol. iii. art. 4015 to 4034; Cagnoli, page 321, &c.

A A

(N.353.) ¿C COS / BX sine AB

rad x sine BC

× ▲ ▲, and

(0.353.) BC=

sine BC

cot / B

(P) By denoting the three sides of the triangle by a, b, c, and their opposite angles by A, B, C, the following proportions are deduced, where c and b are constant quantities.

1. A a::rad2: sine bx sine c.

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a::cosecb sine c:: cosec c: sine b,

and A:a:: cosec c: sine B:: cosec B: sine c.

3. Å: B::radx sine a : cos cx sine b.

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PROPOSITION VIII. (Plate IV. Fig. 6.)

(Q) In any oblique-angled spherical triangle ABC, suppose the two angles B and c to remain constant*, it is required to find the fluxions of the other parts.

Take the supplemental triangle DEF (S. 136.) then DE, and DF will be constant, therefore by Proposition VII.

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1. D: EF:: cosec DF: sine/F:: cosec / F: sine DF,
Viz. BC ZA:: cosec Zc: sine AC:: cosec Ac: sine c.
2. DEF:: cosec DE: sine E::cosec E: sine DE,
Viz. BCA:: cosec / B: sine AB:: cosec AB sine __ B.

*La Lande's Astronomy, art. 4034 to 4045; Cagnoli, page 325, &c.

3. LD: LE::rad x sine EF: cos / FX sine FD,

Viz. BC AB::rad x sine ▲A: COS AC X sine c. :

4. LD: LF:: rad x sine EF: COS E X sine DE, AC:: rad x sine A: COS AB X sine B.

Viz. BC

5. LF

EF::cot ZE: sine EF,

and E EF:: cot/F: sine EF. Viz. AC: ZA:: cot AB : sine A, and ABA::cot AC: sine LA.

6. AC: AB:: cot AB: cot Ac,

and AC AB:: tang AC: tang AB (C. 103.)

(R) By denoting the three sides of the triangle by a, b, and c, and their opposite angles by A, B, and c, we derive the following proportions, wherein в and c are constant quantities.

1. a: A::cosec c: sine b:: cosec b: sine c.

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(S) In all the preceding propositions if the sides of the triangle be diminished without limit, the triangle may be considered as rectilinear, and instead of the sines and tangents of the sides, we may substitute the sides themselves (Z. 193.) Hence the variations of plane triangles are readily deduced from those of spherical triangles, in every case where the fluxions are proportional to the sines or tangents of the sides. Thus, by the 4th proportion (P. 354.) we have shewn that Ac::rad x sine a cos BX sine c, that is, (supposing the triangle straight-lined) a:c::rad xa; cos B×c.

Again, by the 8th proportion (R. 355.) bc::tang b: tang c, that is, when the triangle is straight-lined, bc::b:c, and in the same manner the rest may be deduced.

The variations of rectilinear triangles may be deduced from the triangles themselves, without reference to spherical triangles, in a manner exactly similar to those deduced from the - spherical triangles. Vide Traité de Trigonométrie, par M. Cagnoli, chapitre X.

THE USE OF THE FLUXIONAL ANALOGIES.*

PROPOSITION IX.

(T) To find when that part of the equation of time dependent on the obliquity of the ecliptic is the greatest possible.†

Here the sun's longitude will form the hypothenuse of a right-angled spherical triangle, his right ascension will be the base, and the obliquity of the ecliptic will be a constant angle. Let the hypothenuse be denoted by b, and the base by c. Then bc:: sine 26: sine 2c (A. 346.)

when bc, then sine 2b-sine 2c; but when two arcs have equal sines, the one must be the supplement of the other. (K. 31.) Consequently b+c=90°, therefore when b-c-o, that is when b-c is a maximum, b+c=90°.

The equation of time dependent on the obliquity of the ecliptic is therefore the greatest possible, when the sun's longitude and his right ascension together are equal to 90°. The sun being in the first quadrant of the ecliptic.

PROPOSITION X.

(U) Given the parallax in altitude of a planet, to find its parallax in latitude and longitude.

Let в represent the pole of the ecliptic, a the zenith, and c the place of the planet.

Then b will represent the parallax in altitude, B the parallax in longitude, and a the parallax in latitude.

*A variety of examples will be met with in the perusal of La Lande's Astronomy, vol. iii.

+ Simpson's Fluxions, page 550.

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