114

HISTORY OF PHYSICAL ASTRONOMY.

function. Before any progress can be made in computing these pertur-bations, it is necessary that the algebraic expression on which they dependshould be decomposed into individual terms. This object is generallyeffected by expanding the function into a series of sines and cosines ofangles increasing proportionally with the time, and arranged accordingto ascending powers of the eccentricities and inclinations. If no otheroperation was necessary, and the inequalities were represented by the terms,simply as they appear in the expanded function, the computation of those thatare of sensible magnitudes would not be very troublesome; for, as the eccen-tricities and inclinations of the planets are generally very inconsiderable,at least if we except the smaller planets, the series converges with greatrapidity; and for all purposes of comparison between theory and observation,it would not be necessary to take into account any terms beyond those ofthe second order with respect to these quantities. But this is by no meansthe real state of the question. After the perturbing function has beenexpanded in the form just mentioned, it undergoes two successive inte-grations before the terms composing it can he made subservient inrepresenting the planet’s inequalities in longitude. By this processthe circumstances of the problem are totally altered, and its difficultiesin a vast degree augmented, for it may happen that a term which isquite insignificant in the original expansion will acquire by doubleintegration a divisor which will render it very considerable *. Thisdivisor is a multiplier of the time in the original terms of the expandedfunction, and it is generally formed by combining together in endless ways,by addition and subtraction, the mean motions of the disturbing and dis-turbed planets. The geometer, therefore, cannot safely reject any termsin the expanded function from a mere regard to their order in the series,unless he has assured himself at the same time that they cannot be affectedto a sensible extent by any of the divisors depending on the various com-binations of the mean motions. It is this circumstance which occasionsthe necessity of calculating the terms of the third and even the fifth ordersin the theory of Jupiter and Saturn; and indeed a similar course is in-dispensable when the question refers to any of those inequalities of longduration which extend over several revolutions of the disturbing and dis-

The expression for the longitude contains a term of the form k

d.R.

t d t

in which

i? denotes the perturbing function, s the epoch of the disturbed planet, and It a constantquantity. Now R, when expanded, is found to contain a series of terms of the form

P cos

^ (in — i? n!) t -J-

n — it 4 * Q

)

in which n n' denote the mean motions of the

disturbed and disturbing planets, « I the epochs, i i'any integers whatever, P a functionof the eccentricities and inclinations of the two planets, and Q a function of the

perihelia and the nodes. Hence 'SJ- will contain a series of terms of the form

k i

(in — i’ri)‘-

P sin ( (in — tV) t + it—”t + Q

and it is clear that if in any case i n

be very nearly equal to i r n f , the corresponding term of the longitude may be considerable,even although P be of a high order relative to the eccentricities and inclinations. Forexample, in ithe theory of Jupiter and Saturn, 5 n is very nearly equal to 2 n 1 , and hencearises the famous inequality in the longitudes of those planets which Laplace was the firstto trace to its true physical source.