By Garrett P.

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**Example text**

In astronomical language, M is the sun, m a planet, ξ η ζ are the heliocentric rectangular coordinates, r is the radius vector, θ the longitude in the orbit, ω the longitude of the perihelion, ν of the node, θ − ω is the true anomaly, θ − ν the argument of latitude, µ the constant part of the half square of undisturbed heliocentric velocity, diminished in the ratio of the sun’s mass (M ) to the sum (M + m) of masses of sun and planet, κ is the double of the areal velocity diminished in the λ same ratio, is the versed sine of the inclination of the orbit, q the perihelion distance, and κ τ the time of perihelion passage.

In general, it appears to√be a useful rule, though not always followed by analysts, to employ the real radical symbol R only for r positive quantities, unless the negative sign be expressly prefixed; and then √ will denote r2 positive or negative unity, according as r is positive or negative. The arc given by its sine, in the expression of the element ω, is supposed to be so chosen as to increase continually with the time. 35. ). ) r = {ξ 2 + η 2 + ζ 2 }, it is easy to perceive that the six combinations of the 4 first elements are as follows: {κ, λ} = 0, {κ, µ} = 0, {κ, ν} = 0, {λ, µ} = 0, 45 {λ, ν} = 1, {µ, ν} = 0.

For the function of elements C, we obtain the following approximate expression C1 for that function, of the form supposed by our theory: 1 (λ1 − p1 )2 + (λ2 − p2 )2 (λ3 − p3 )2 C1 = − + 2t µ2 ν2 t 1 − {(λ1 − p1 )p1 + (λ2 − p2 )p2 + (λ3 − p3 )(p3 − 3 gt)} 2 3 4 5 t t t 2 2 2 2 2 2 2 2 + {µ (p1 + p2 ) + ν p3 } − ν gp3 + ν g . ) 2 δλ3 2 2 2 δC δC δC 1 t 2 2 2 2 2 2 − +µ +ν µ dt. ) 2 2 2 = − {(λ1 − p1 ) + (λ2 − p2 ) + (λ3 − p3 ) } 3 t3 2 2 2 + {µ p1 (λ1 − p1 ) + µ p2 (λ2 − p2 ) + ν p3 (λ3 − p3 )} 24 4 5 t 2 t 4 2 4 2 4 2 − ν g(λ3 − p3 ) + (µ p1 + µ p2 + ν p3 ) 45 240 6 7 t 4 t 4 2 − ν gp3 + ν g .