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value by an acquired term: and when, according to the notation that has been adopted,

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We may now easily deduce à (sin. 2 w), and d (cos. 2 w), or those variations of sin. 2w and cos. 2 w, which are respectively due to the preceding variations in the values of u and u'. Thus

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If therefore the variation of sin. 2w, arising from du' =

cos.dmv,

be required, we have only to substitute, instead of dr', the third line of the preceding value of '; and still more simple will be the process, if the terms involving e é are to be excluded by reason of their smallness: for then it will be sufficient to make dv′ = 2 é sin. c'mv; and in this case (see Trig. p. 26. [b]),

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Similarly, . cos. 22 é. cos. (2 v2mv - ć' mv)

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If it is necessary to express the terms that involve ee, then, instead of ', we must substitute the terms of the third line of the value of v', and we must also, in the process, take account of the three first terms of sin. 2, and cos. 2 w, as given in pp. 151, &c. there will then result by the common trigonometrical formulæ, (see Trig. pp. 24, &c.).

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and a similar expression may easily be obtained for d. cos. 2 w.

If we wish to deduce d. sin. 2 w and d. cos. 2 w arising from

the variation du

4a

cos. 2g v, we have merely to substitute,

m

instead of do', the first term sin. 2 gv) of the second line

4.

of the value for v; or, we may immediately obtain the varia

my2
4

tion, by writing, in the expression, instead of 2e, and 2g v

instead of cm v: there will then result

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And, after the manner of deducing the terms that involve e é, be deduced the terms that involve y2 e.

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If we now, for the purpose of exhibiting at one view, sin. 2 and cos. 2w, complete their values, by adding, that what have been already (see pp. 151, 164.) exhibited, as so many corrections, the terms arising from the variation du and from the two variations of u, we shall have

sin. 2 sin. (2 v — 2mv) -2me.sin. (2v-2 m v cv)

+2 me. sin. (2 v − 2 m v + cv)

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It is easy, after the manner of p. 164. to increase the preceding values by terms that involve e é, &c.

We will now proceed to deduce the corrections that are due to the terms composing II, and that arise from the eccentricity (de) of the Solar orbit; excluding, however, from the formula of corrections, those terms that involve the square of é or any rectangle such as e é, é' y2, &c.

du

dv

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involves e, the simplest term therefore of the correction of the above term must involve ee': of such terms, however, it is not, at the present, intended to take account,

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where cos. (2 v 2m v), its first term, is put for cos. 2 w, since terms involving e é are to be excluded. The last line is composed of the terms in the fifth and sixth lines of cos. 2 w (see p. 166.).

Hence, if we expand (see Trig. p. 26. [d]) the first line of the preceding expression, and then combine the resulting terms with the terms in the last line, we shall have the required correction,

or

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cos. 2) = 3*, Co. (2v-2 mu-¿'mv)).

K

cos.

2h'

1

cos. (2v − 2mv+cmv).

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, (see pp. 130, 143, 154.),

u3

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d (u3 sin. 2 w) = 3 u. du'. sin. 2 + u3, ò. sin. 2 w

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e. Cos. c'mv x sin. (2 v2 m v)

[sin. (2v-2 m v-c' mv)-sin. (2 v2mv+cmv)}

2mv+cmv).

(sin. (2v - 2 mv – cm v) - 1 sin. 2v−2mv+c'mv

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=

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2.(2-2m+c'm)

Since c' is very nearly 1 (it.9999907779) the denominators in the preceding expression are nearly 2. (2—m), and 2(3—2m): and, in all the preceding cases, since no account is to be made of terms that involve e2 é, ye, &c. may be written instead of :

a

and for the same reason, we shall have the correction due to

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1

Hence the whole correction due to ПI, on account of the eccentricity of the Solar orbit and confined to terms that involve merely, is of the form

Ae.cos. dmv+Be'. cos. (2v-2mv-d'mv)+Ce'. cos. (2v-2mv+ć'mv),

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