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If we take the differential of the first equation (1), and sup

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which value being substituted in the second equation (2), and the coefficients of the terms affected with de equated, there results,

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a term much less than the other terms in the expression for d 7,

when, the inclination of the planes, is a small quantity.

By equating the terms affected with d v,

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v, is the body's longitude measured on the plane of the body's orbit; therefore (see pp. 388, 389.)

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and (see p. 393.) the variation dy is now expressed by a formula possessing all the characteristics of the former variations.

d R
d Ꮎ

enters: but

d R
do

in the two

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In this and in the former expression for dy (see p. 399.) the partial differential coefficient cases, is of different values, since R in the first case is a function of v, 0, 4, &c. and in the second of v, 0, 4, &c.

The expression may still farther be varied, by transforming R into a function of new quantities, the relations between these latter and the quantities v, e, &c. being determined by certain equations for instance, if instead of, we assume a quantity u and determine the relation between v, and u by this equation,

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then, since R, instead of being a function of v, 0, &c. is now to become a function of u, 0, &c.

* This is the same expression as that which Laplace, by a different method, has deduced in the supplement au X. Liv. de Mec. Cel.

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therefore, by a comparison like the preceding one of p. 399.

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*This certainly is a very simple expression, and it is the same which Lagrange, Mem. Inst. 1808. pp. 62, 64, &c. and Poisson, Ecole Polytechnique, tom. IX. have, by methods differing both from the preceding and from each other, deduced. The simplification, however, obtained by the last step is more apparent than real, since it is obtained by introducing a quantity u the relation of which to v, &c. is determined only by a differential equation.

A simpler form may, in like manner, be given to some of the other variations, by transforming R into a function of different quantities: thus, we have, very nearly,

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Substitute for dq its assumed value and equate the coefficients of like terms, and then

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The only variation that remains to be invested with the properties which the other variations possess, is de: now, since

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But (see p. 395.) the equation of condition is

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Substitute for the value of dy in p. 397, and

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transformation similar to the preceding one of p. 398. In the

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present transformation, however, since a different expression for must be supposed constant, and y, v, and v to

0 is required,

vary in the equation,

tan. (v

in which supposition,

e) cos. tan. (v — 0),

dvdp.cos.2 tan. (v,-) cos2. (v.-0)+

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=

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dv cos.2 (v,-0)

cos. p. cos. (v-A)

d R

dv

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nearly, (since y. sin2. ø is a very small quantity),

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The variations of the six elements may now be exhibited under one view,

* Since is very small, dy = d (tan. 4) is written instead of dø. d R

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is here used, after a manner similar to that in p. 398, and
d R
dr

to distinguish it from the partial differential coefficient of R,

when R is transformed into a function of new quantities.

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