perihelion to the ascending node; and the perihelion will lie on the northern or southern side of the ecliptic according as ω is less or greater than 180°.
As π and Ω are counted from the vernal equinox, and i is measured from the plane of the ecliptic, the quantities necessarily refer to a particular equinox, and this is always specified.
It was long customary to measure longitudes in the orbits of comets in the direction of the Earth's motion; to limit i to the first quadrant; and to specify the direction of the comet's motion, whether "direct" or "retrograde"; but many astronomers, especially on the Continent, now follow Gauss in regarding retrograde motion as a result of the inclination passing into the second quadrant; and in accordance with that view they measure a comet's longitude always in the direction of its own motion, and permit i to take any value between 0° and 180°. The circumstance that i is measured at the ascending node limits its range to the first and second quadrants, for if it were to pass into the third or fourth quadrants the ascending node would be converted into a descending one. For a comet having direct motion the numerical values of the elements are the same in Gauss's system as in the old system, but for a comet having retrograde motion they are different, and in that case, if their values according to the old system are indicated by a subscript ∘, the equations requisite for passing from the old to the Gaussian system are:—
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i = 180° − i∘
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ω = 360° − ω∘ = −ω∘
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There is frequently much confusion respecting the angles π and ω, and it is important to have a clear understanding of the relations of ω to π and Ω. In the old system of elements π is measured from the vernal equinox, along the ecliptic in the direction of the Earth's motion, to the ascending node of the comet, and thence along the comet's orbit, still in the direction of the Earth's motion, to the comet's perihelion. In Gauss's system π is measured from the vernal equinox, along the ecliptic in the direction of the Earth's motion, to the
