D's semi-diameter at noon, 26th April (Naut. Alm.)=16′.5", and at midnight=15.58"; horizontal parallax at noon=59′.1", and at midnight 58'.36"; also the 's semi-diameter = 15'.55".

Answer. The reduced time is 2.16; 's true semi-diameter 16'.11"; horizontal parallax=58'.57"; apparent altitude of the D's centre 23°.56'.24"; and the correction of her altitude 51.46".

The apparent altitude of the O's centre 32°.0'.8"; correction 1.23"; apparent distance of the and D's centres 68.41.36"; true distance 68°.18'.47", and the true longitude 105°.37' East.

(Naut. Alm. distance at noon being = 67°.6'.15", and at III 68°.44'.1".)

EXAMPLE III.

Suppose on the 10th of July 1822, in longitude 116° W., by account, the following observations to be taken:

Required the true longitude, the eye being 21 feet above the level of the sea.

D's semi-diameter at noon, July 10th (Naut. Alm.) — 15'.44′′, and at midnight 15'.49"; horizontal parallax at noon 57.46", at midnight 58'.4"; also the O's semi-diameter = 15'.46".

Answer. The reduced time is 5h.15'.33" July 10th; D's true semi-diameter 15'.52"; horizontal parallax 57'.53"; apparent altitude of her centre 19°.42′.52"; and correction of her altitude 51'.51".

The apparent altitude of the sun's centre is 45°.32′.29"; correction 50"; apparent distance of centres 106°.47'.13"; true distance 105.51'.29", and the true longitude 116°.0'.2" W. (Naut. Alm. distance on the 10th at IIIh=107°.2.33", and at VI-105°.28'.18".)

EXAMPLE IV.

Suppose on the 6th of May 1822, in longitude 72°.10^ E by account, the following observations to be taken :

Alt. D's lower limb. 19°.10.20

Required the true longitude? the eye being 18 feet above the level of the sea.

D's semi-diameter at noon, May 6th (Naut. Alm.) 14'.42′′, and at midnight the same; horizontal parallax at noon 53′.58", and at midnight 53'.57".

Answer. The reduced time is 7b.52'.39", D's true semidiameter=14'.48"; horizontal parallax=53.58", apparent altitude of the 's centre 18°.56.30", and her correction 48'.19".

App. altitude of the star 20°.10',56′′, correction 2′.35′′;
App. distance of centres 31°.18'.12"; true distance 31°.12.20
The true longitude 72°.9′ East.

(Naut. Alm. distance at VI 30°.16′.54′′, and at IX1= 310.45'.25").

(W) In all the preceding examples the watch is supposed to have been previously regulated; when that is not the case, the error of the watch must be found from observations of the altitudes of the sun or of a star, taken either before or after that of the distance, as directed in Problems XIV. or XV., pages 316 and 319. Or if the sun or star be sufficiently distant from the meridian, the mean of the sun's or star's alti-tudes, taken at the same time as the distance is taken, together with the latitude of the place and the sun's declination, &c. may be used to correct the watch; with this corrected time proceed as before.

EXAMPLE V.

At sea, April 26th 1822, in latitude 47°.39′ N. and longitude 178°.20.30" West from Greenwich, by account, at 4b.23′ P.M.

per watch not previously regulated; suppose the observed altitude of the sun's lower limb to be 25°.57′.10", and that of the moon's lower limb 46°.22′.32", and at the same time the distance of the sun's and moon's nearest limbs to be 75°.33'.37"; required the longitude? The eye being 18 feet above the level of the sea.

D's semi-diameter at midnight, April 26th (Naut. Alm.) = 15.58", and at noon April 27th-15.51"; horizontal parallax at midnight 58'.36"; and at noon April 27th=58'.11"; also the 's semi-diameter=15′.55′′.

Answer. The reduced time is 16h.16'.22"; 's true semidiameter 16.7"; horizontal parallax 58'.28"; apparent altitude of her centre 46°.34'.36'', and correction of her altitude 39'.16".

The apparent altitude of the sun's centre is 26°.9.2"; correction 1.48"; apparent distance of centres 76°.5'.39′′; true distance 75°51'.20".

The sun's declination (Naut. Alm.) on the 26th=13°.25'.19" N. on the 27th-13°.44′.36" N. which reduced to the time and place of observation (B. 270.)=13° 38′.23′′ N.; with this declination, the O's true altitude 26°.7.14"; and the co-latitude 42°.21, find the correct apparent time at ship (F. 316.)— 4h.24.6"; and hence the true longitude is 177°.51.45" west. (Naut. Alm. distance at XV 75°.11'16", and at XVIII =76°.47′.7":)

CHAP. XIII.

OF THE FLUXIONAL ANALOGIES OF SPHERICAL TRIANGLES.*

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

(X) A preparatory Proposition.

CONSTRUCT a general figure as at Prop. XIX. Book III. Chap. I. (L. 151.) Thus, let A be the pole of the circle HGFE; F the pole of ABH; Ethe pole of CGI; and Cthe pole of EDI.

* See Cote's "De Estimatione Errorum in mixtà Mathesi." Cambridge, 1722. Simpson's Fluxions, vol. ii. page 278, et seq.-Crakelt's translation of Mauduit's Trigonometry, Chap. v. page 164, &c.-Traité de Trigonométrie, par M. Cagnoli, Chap. xix. page 310, &c -La Lande's Astronomy, Paris, 1792, vol. iii. page 588, et seq; or art. 3997 to 4051, &c. &c.

Suppose these circles to be invariable whilst another great circle DFCB revolves about the pole F, and let cn be at right angles to the great circle mnorsd; then the three triangles ABC, CGF, and EDF will be variable, viz.

I. In the right-angled triangle ABC, the A will be a fixed quantity, and the other parts will be variable; viz. вm will be the increment of AB; no the increment of BC; co the increment of AC; and DS the increment of the arc ID which measures the LC.

II. In the right-angled triangle CGF, the side FG will be a fixed quantity, and the other parts will be variable; viz. co will be the decrement of CG; no the decrement of FC; Bm, the decrement of the CFG; and Ds the increment of the ▲ C.

III. In the right-angled triangle FDE, the hypothenuse EF will be a fixed quantity, and the other parts will be variable; viz. sd =no, will be the increment of FD=BC; SD the decrement of ED; and Bm, the decrement of the EFD CFG (N. 135.)

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

(Y) In any right-angled spherical triangle ABC, right-angled at B, suppose one of the angles as a to remain constant, it is required to find the ratios of the fluxions of the other parts.

1. In the triangles Fâm, Fcn, having the same acute at F, sine FB sine FC::tang Bm tang cn (M. 167.)

But FB 90°, FC is the complement of BC, and вâm and cn, being very small arcs, have the same ratio to each other as their tangents.

2. In the triangles DCI, con, where the DCI may be supposed equal to the con.

Tang DI sine cr::tang cn But DI is the measure of the

sine no (M.167.) c, ci=90°, and the tangent

of cn and the sine of no, have the same ratio to each other as

But DIC, DC=90°, and the very small arcs cn and co, have the same ratio to each other as their sines.

3. In the triangles BFM, DFS, where FSFD extremely near. sine FB sine BM:: sine FS: sine DS. (M. 167.)

But FB 90°, FS

FD=BC, and the small arcs are as their sines.

But no will represent the fluxion of BC; co the fluxion of ac; Ds the fluxion of DIC; and Bm the fluxion of AB. (B. 126.) Therefore,

(Z) Hence, in any right-angled spherical triangle ABC, right-angled at B; by denoting the sides opposite to the angles A, B, C, by a, b, c, respectively, we derive by substitution and reduction, the following general equations, which include all the varieties that can possibly happen wherein the LB is 90° and one of the other angles (viz. a) constant.

COS C
X

tang c

rad

tang c

(A) Any of the foregoing equations may be turned into proportions, or varied in the expression, by reference to pages 104, 105, &c.

* Simpson's Fluxions, page 280.