occur to the reader is this: "If the disturbances of the planets, supposing their orbits to have no independent excentricities, amount only to a few seconds, how is it likely that the small alterations of place, which are produced by the trifling excentricities and inclinations of their orbits, will so far alter their forces upon each other as to produce any sensible difference in the magnitude of the irregularities?" In answer to this we must say, "It is true that these forces, or alterations of forces, are exceedingly small, and those parts of them which act in the same direction for a short time only (as for a fraction of the periodic time of a planet) do not produce any sensible effect. But we can find some parts of them which act in the same manner during many revolutions: the effects of these may grow up in time to be sensible; and those in particular which alter the mean distance and the periodic time may produce in time an effect on the longitude of the planet (49.), very much more conspicuous than that in the alteration of the orbit's dimensions."

(162.) In this consideration is contained the whole general theory of those inequalities known by the name of inequalities of long period. They are the only ones depending on the excentricities (besides those similar to the moon's evection) which ever become important.

(163.) To enter more minutely into the explanation, let us take the instance of the long inequality of Jupiter and Saturn: the most remarkable for its magnitude, and for the length of time in which the forces act in the same manner, as well as for the difficulty

which it had given to astronomers before it was explained by theory, that has been noticed since the first explanation of the Moon's irregularities.

(164.) The periodic times of Jupiter and Saturn are very nearly in the proportion of 2 to 5 (the periodic times being 4332 days, 17 hours, and 10,759 days, 5 hours), or the number of degrees of longitude that they will describe in the same time, omitting all notice of their excentricities, will be in the proportion of 5 to 2 nearly. Suppose, now, that they were exactly in the proportion of 2 to 5; and suppose that Jupiter and Saturn started from conjunction; when Saturn has described 240 degrees, Jupiter will have described 600 degrees (as these numbers are in the proportion of 2 to 5): but as 360 degrees are the circumference, Jupiter will have gone once round, and will besides have described 240 degrees. It will, therefore, again be in conjunction with Saturn. When Saturn has again described 240 degrees, that is, when Saturn has described in all 480 degrees, or has gone once round and has described 120 degrees more, Jupiter will have described 1200 degrees, or will have gone three times round and described 120 degrees more, and, therefore, will again be in conjunction with Saturn. When Saturn has again described 240 degrees, that is, when it has gone exactly twice round, Jupiter will have gone exactly five times round, and they will again be in conjunction. So that, if the periodic times were exactly in the proportion of 2 to 5, there would be a continual succession of conjunctions at the points

whose longitudes exceeded the longitude of the first place of conjunction by 240°, 120°, 0, 240°, 120°, 0°, &c. Thus, in fig. 38, if B1 is the place of Jupiter at

first, and C1 that of Saturn, Jupiter will have gone quite round, and also as far in the next revolution as B2, while Saturn has described part of a revolution only to C2: then Jupiter will again have gone quite round, and also as far in the next revolution as B, while Saturn has described part of a revolution to C3: then Jupiter will have performed a whole revolution, and part of another to B1, while Saturn has performed part of a revolution to C1: and then the same order of conjunctions will go on again. If, then, the periodic times were exactly in the proportion of 2 to 5, the conjunctions would continually take place in the same three points of the orbits. This conclusion will not be altered by supposing the orbits excentric: for though the places of conjunction may then be somewhat altered, the conjunctions, after the third (when. Saturn has gone round exactly twice, and Jupiter exactly five times), will go on in the same order, and happen at the same places as before.

(165.) But the periodic times are not exactly in the

proportion of 2 to 5, but much more nearly in the proportion of 29:72. This alters the distance of the places of conjunction. We must now suppose Saturn to move through 242°-79, and Jupiter (by the proportion just mentioned) will then have moved through 602°-79, or through a whole circumference and 242°-79, and they will be in conjunction again. The next conjunction will take place when Saturn has moved through double this angle, or 485° 58, or when Saturn has performed a whole revolution, and 125°58 of the next revolution: and the following conjunction will take place when Saturn has moved through 728°37, or when Saturn has gone twice round, and has described 8°.37 more. Now, then, the same order of conjunctions will not go on again at the same places as before, but the next three after this will be shifted 8°37 before the former places, the three following the last-mentioned three will be again shifted 8°.37, and so on. The places of successive conjunction, in fig. 38, will be at B1, C1, b2 Cz, bз C3, b1 C1, b5 C5, bε ce, &c. The shifting of the places of conjunction will take place in nearly the same manner, whether the orbits are excentric or not.

(166.) From this the following points are evident:First. In consequence of the periodic times being nearly in the proportion of 2 to 5, many successive conjunctions happen near to three equidistant points on the orbits.

Secondly. In consequence of the proportion being not exactly that of 2 : 5, but one of rather less inequality, the points of conjunction shift forward, so that each suc

cessive set of conjunctions is at points of the orbits more advanced, by 8°.37, than the preceding one.

(167.) Let us now inquire how long it will be before the conjunctions happen at the same parts of the orbits as at first.

This will be when the series of points b, br, b10, &c., extends to B. For then the series b5, bg, b1, &c., will extend to B1, and the series bз, bé, bg, &c., will extend to B2. The time necessary for this will be gathered from the consideration, that in three conjunctions the points are shifted 8°.37; and that the points must shift 120° from B,, before they reach B: and that we may, therefore, use the proportion, As 80°-37 is to 3, so is 120° to 43 nearly, the number of conjunctions that must have passed before the points of conjunction are again the same. And as Saturn advances 242°-79 between any conjunction and the next, he will, at the forty-third conjunction from the first, have described 10440°, or 29 circumferences; and Jupiter, therefore (by the proportion of their periodic times), will have described 72 circumferences. The time, then, in which the conjunctions return to the same points is twenty-nine times Saturn's periodic time, or seventy-two times Jupiter's periodic time, or about 855 years.

*

(168.) Now let us examine into the effects of this slow motion of the points of conjunction upon the forces which one body exerts to disturb the other.

(169.) If the orbits had no independent excentricity,

* These numbers are not quite exact: the proportion of 29: 72 not being quite accurate.