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Thus, from the fourth set of equations cx cos ab × sin ≤ c. that is, bc::cos a: sin c.

cos b sin c

::

(O. 169.)

sin b

COS C

::cos b x sin b; cos c × sin c.
::sin 26: sin 2c* (0.108.)

PROPOSITION III.

(Plate IV. Fig. 3.)

(B) In any right-angled spherical triangle CGF, right-angled at G, suppose one of the sides as FG to remain constant, it is required to find the fluxions of the other parts.

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FC; Bm the decrement of the Z F; co the decrement of
CG; and
Ds the increment of the Z. c, also FC is the complement of BC.
Therefore,

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(C) Hence, if the fixed side FG be represented by c, the hypothenuse Fc by b, the side ca by a, and the angles opposite to these sides by C, B, and a respectively (as at Z. 346.) we derive the following general equations, or formulæ, which comprehend all the different cases that can possibly happen wherein the angle B is 90° and one of its adjacent sides (viz. c.) constant.

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* Vince's Trigonometry, 2d edition, page 159; Traité de Trigonométrie, par

M. Cagnoli, art. 677, page 329.

+ Simpson's Fluxions, page 280.

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(D) The preceding formulæ may be varied in the expression by reference to pages 104, 105, &c. or they may be turned into proportions thus, from the first set of equations × sine b sine cxa, that is,

a: A:: sine b: sine c.*

PROPOSITION IV. (Plate IV. Fig. 3.)

(E) In any right-angled spherical triangle FDE, right-angled at D, suppose the hypothenuse EF to remain constant, it is required to find the fluxions of the other parts.

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But no is the increment of BC, or of its=FD; Bm is the measure of the BFm, or of its equal SFD, and consequently it is the decrement of the EFD; also BC FD, and the Zc is measured by the arc DI which is the complement of ED.

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Again, co is the decrement of CG, and co is the complement of GI the measure of the E, therefore co is the increment of the E.

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(F) Hence, if the hypothenuse EF be represented by b, the side FD by c, ED by a, and their opposite angles by B, C, and a

* Vince's Trigonometry, 2d edition, page 140.

A

respectively, (as at Z. 346.) we derive the following general formulæ, which include all the varieties that can possibly happen, wherein the angle в is 90°, and the hypothenuse (viz. b) con

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(G) These formulæ may be varied in the expression or turned into proportions, &c. as before observed. Thus from the third set of equations

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PROPOSITION V. (Plate IV. Fig. 4.)

(H) In any oblique-angled spherical triangle ABC, suppose the angle A and its adjacent side AB to remain constant †, it is required to find the fluxions of the other parts.

Let BC, by its revolution about B, be changed to вm, then AC will become Am, and the ABC will be reduced to the ABM. Now, mc will be the decrement of AC, the CBM will be the decrement of the B, and if cn be drawn perpendicular to вn, mn will be the decrement of BC; produce Bn and BC so that Bp and Bo may be quadrants, then op will be the measure of the CBM. Hence,

I. cn op: sine BC: sine BO,

and in the small straight-lined triangle cnm right-angled at n mn: cn: sine mcn cos mcn. (Y. 34.)

mn op::sine BC X sine mcn sine BO × cos ▲ mcn.

* Vince's Trigonometry, Prop. 49th, page 140, 2d edition.

Traité de Trigonométrie, par Cagnoli, page 311. La Lande's Astron. vol. iii. art. 3998 to 4003.

But men will be the complement of the angle c, because Ben is a right angle, and Bo=90° by the construction, hence BCB:sine BC X cos c: rad x sine c.

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II. Again, rad mc:: sine

that is, rad: AC:: cos

c. (N. 104.)
mcn mn. (Y. 34.)

C: BC.

III. Also, mcnc::rad: cos mcn. (Y. 34.) and nc opsine BC: sine BO=rad.

mc

that is, AC

op sine BC:: sine c,

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IV. By taking the supplemental* triangle to ABC, viz. dfe, (S. 136.) the E and the side FE will be constant quantities, and the rest will be variable; now it is shown in the first case of this proposition, that flux. side opp. a constant: flux. Z adj. to the constant side:: sine of the side opp. to the constant: tang opp. to the constant side.+

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Hence, DFF:: sine DF: tang / D, that is, (S. 136.)

C: AC: sine c: tang BC,

and B AC:: sine c: sine BC. (III. case above.)

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(I) Let a perpendicular BD be drawn from the end of the fixed side AB and opposite to the fixed. A, then the several parts of the right-angled triangle BDA will be invariable; and in the right-angled triangle BDC, the perpendicular BD will be a fixed quantity, and all the other parts will be variable; hence, by the assistance of the formulæ already given, (C. 347.) all the variations of which this proposition is susceptible may easily be derived; some of the principal will here be inserted, denoting the sides of the triangle by a, b, c, and their opposite angles by A, B, C, as in the figure, where a is the constant angle, and c the constant side.

* Called by the French writers the polar triangle, from the manner in which it is described.

+ Cagnoli, page 314, art. 555.

b: B::sine a
b a::rad

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c: tanga

a Bsinea

BC rad

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sine c. Ist equation C. 347.
cos c. 4th equation C. 347.
sine c. 3d equation C. 347.
tang c. 1st equation C. 347.

cos a. 1st equation C. 347.

catangc tanga. 2d equation C. 347.

PROPOSITION VI. (Plate IV. Fig. 5.)

(K) In any oblique-angled spherical triangle ABC, suppose the angle A and its opposite side BC to remain constant*, it is required to find the fluxions of the other parts.

Let BC change its position to mo, and let these circles intersect each other in an indefinitely small angle at r; make ŕn— rв, and rc=rp; then, nm will be the decrement of rв, and po will be the increment of rc; and because np BC, by construction, and mo=BC, by hypothesis; if the common part mp be taken from each, there will remain nm op; then,

I. Considering the indefinitely small triangles вnm, cрo as rectilinear, and right-angled at n and p, we have

In the right-angled triangle cpo, rad: co::cos poc: op, and In the right-angled triangle Bnm, rad: вm:: sine nвm : nm= op. (Z. 34.)

co Bm: sine nвm: cos poc.

that is, AC AB:: COSB: cos c.

For, co is the increment of AC, Bm is the decrement of AB; and the three angles pco, pcr, and Acr are together equal to two right angles (M. 134.), of which pcr is a right angle, therefore Zpco+Acrone right angle=▲ Pco+ poc, hence poc

ACB.

II. Take the supplemental triangle to ABC, viz. DFE (S. 136.), then, since A and BC are constant quantities in the triangle ABC, EF and the D will be constant quantities in the triangle DFE; and it has been shown in the first part of this proposition, that the fluxions of the variable sides are as the cosines of their opposite angles, we have

DE DF::COS F: cos E, but DE is the measure of the ZB, and DF is the measure of the c, also cos F=COS AC, and cos ECOS AB'.' ≤B: ≤c::cos AC; cos ab.

*La Lande's Astronomy, vol. iii art. 4003 to 4010. Cagnoli, page 316, &c.

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