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Also, since the arcs Ee, Pp, are confounded with the portions of their respective tangents which meet at T, and the parallels EP, ep, divide TE, TP, proportionally, we have ET : PT :: Ee: Pp; and consequently,

ET : PT :: v : V’.

Put se = a, SP = b, and the required angle SEP of elongation = E. Then, since SP: SE:: sin E: sin EPs, we shall have

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31. A still simpler and more convenient expression for sin E may be obtained by introducing the radii of the respective orbits for the velocities of the bodies. Thus, by the theory of central forces (Gregory’s Mechanics, book ii. art. 282) v : v':: Vb : Va;

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Here the positive sign obtains in the case of the inferior planets, the negative sign in that of the superior planets.

Example. -

The mean distance of Jupiter from the sun being 52028 times that of the earth; required the elongation of the planet from the sun at the time of its apparent station : The logarithmic operation for tan E indicated by equa. (4) is as follows: From 2 log b .... 52028.... 14324742 Take log (5 + 1) 62028.... 0-7925878

Diff. -- 20... . . . . . . . . . . . . . . 20-6398864

Its half is tan E. 115° 35'.... 10:3199232

Here, the tangent being negative, it belongs to the second quadrant (chap. iv. art. 5). - o

Note. By like operations the mean elongations of the other planets at the time of their apparent stations may be found as below.

Mercury........ 18° 12.
Venus... . . . . . . . . 28° 51’
Mars ... . . . . . . . 136°12'

Ceres .......... 126° 7'
Saturn ......... 108°47'
Uranus......... 103°13'.

ScholIUM.

32. The inversion of our 4th equa. furnishes in certain cases a convenient expression by which to approximate to the mean distance of a planet from the sun.

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b = 4 tan *E:Etan E (1 + a tan-e)".

an equation which gives the radius of the planet's orbit in terms of that of the earth, from the elongation at the instant when the motion of the planet became direct after having been retrograde.

Thus, in the case oCeres it was found by observation that E at the time of apparent station was 122° 37' 40”; whence b was found = 3-2018. The slight difference between this elongation and the one just given, arises from this, that the latter was actually observed, the planet moving in an elliptical orbit, while the former is what would be observed if the orbit were circular.

The following is the logarithmic operation, for finding the mean value of b in Ceres:

Log O-25.......... - - - - - -.... I-3979400 2 log tan E ................... 20-3873552

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Log (1 + 4 tan” e) = log ... 160995.... 202068.125

Its half... . . . . . . . . . . . . . . . . . . 10-1034062

Log tan E. ....... . . . . . . . . . . . 10-1936776 Sum — 20 ........ log -E 1.9819...... 0-2970838

# tan”. E ........ 1-2199

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The negative value of 1.9819 cannot here obtain, because b would in that case be negative, which is impossible.

CHAPTER XI.

On the Investigation of Differential Equations for estimating the minute Variations in the Sides and Angles of Triangles.

1. By reason of the imperfection of instruments, the unavoidable though generally slight inaccuracies of observers, the effects of parallax, refraction, the precession of the equinoxes, the varying obliquity of the ecliptic (see note, p. 119), and other causes, the sides and angles of either plane or spherical triangles, whether observed on the earth or in the heavens, can never be taken with perfect exactness. It, therefore, becomes necessary in cases where great accuracy is required, to strike out some means of estimating the extent of error which may be occasioned in certain sides and angles of triangles, by any assignable or supposable errors in the other parts. This is an interesting department of research, in which the celebrated Cotes in his treatise De aestimatione errorum in mixtá mathesi, and many subsequent mathematicians have laboured with considerable ingenuity and success. 2. The inquiry before us is one (of very few in my estimation) in which the contemplation of magnitudes as augmenting and diminishing by differences leads to a more natural and satisfactory explication, than that in which magnitudes are regarded as varying in conse, quence of motion. Hence I shall in this chapter employ the notation of finite and infinitesimal *. instead of that of fluxions: although I am fully persuaded that in a great majority of mathematical inquiries the fluxional notation and metaphysics are preferable to those of differentials. 3. When variable quantities augment or diminish by portions which are finite or susceptible of mensuration, the portions which constitute the augmentation or dimimution are called differences. If the variations, instead of being finite, are indefinitely small, they are called differentials. The former are aptly denoted by the capital Greek letter A placed before the letter which reresents the variable quantity, as Air, Ay, Az, &c. the atter by the small or lower case Greek letter 2, as or, *y, oz. The processes of differentiation and integration being similar to those of finding fluxions and fluents, as taught in our standard works”, need not here be explained otherwise than in connexion with the present investigation. 4. To determine the difference, or finite variation of the sine of an arc or angle, let us take from formula (u), chap. iv. the equation, sin A — sin B = 2sin # (A - B) cosł (A + B). Here, supposing. A greater than B we may denote by AB, the augmentation which A must receive to become equal to B; and, in like manner, by A sin B, the difference which must be added to sin B to make it equal to sin A. Thus, we shall have A = B + AB, and sin A = sin B + A sin B. Substituting these values in the preceding equation, it becomes A sin B = 2 sin #AB cos (B + #AB).... (1.) 5. To obtain the difference of the cosine, take from the same class of formulae the equation, cos B – cos A = 2 sin à (A - B) sin à (A + B); * See the treatises on Fluxions by Maclaurin, Simpson, and - “Dealtry.

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