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PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 129 
where a; 6, c are constants ; such variations obviously do not in the long 
tun produce any marked change in the solar system ; while other variations 
are secular—that is to say, are expressed by terms such as aé¢ + b0? + ...; 
these variations of course have the effect of continually altering the 
orbits, leading ultimately to a completely different configuration. 
The method of the classical planetary theory is to express the solution 
in this way: differential equations are found for the variations of the 
elliptic elements, and from them is found an approximate solution, which 
in the earlier memoirs was of the kind just described. 
The question naturally arose, What would be found to be the true 
nature of the secular inequalities if the equations were solved rigorously 
instead of approximately? The first approximation can be represented 
by terms like c¢, where c is a constant ; but it is possible that this is only 
the first term in the expansion of (say) - sin mt, where m is a very small 
number. If this were the case, the secular terms would really be periodic, 
though of a very long period. In researches relating to the stability of 
the solar system, and the expression of the coordinates after long intervals 
of time, the settlement of this question is of fundamental importance. 
Although the founders of the planetary theory succeeded to some 
extent in their approximation in thus replacing secular terms by trigono- 
metric terms of long period, the most important contribution to the sub- 
ject previous to the period under review was the method by which 
Delaunay | discussed the motion of the moon, the essence of which may 
be described as follows. 
Let 8, J, P be the three bodies, and let the mass of P be zero ; then 
the motion of S and J, being elliptic, may be supposed known, and to 
determine the motion of P we have a system of the sixth order. This 
can be brought to the form 
Bi OM Bish ON popes ina} 
dt cq, dt op, 
where H is a function of ¢ and of the generalised coordinates p,, ¢,. 
H may be called the disturbing function, and can be expanded as an 
infinite series, each term of which consists of 4 function of p,, p., ps; 
multiplied by the cosine of a linear function of 9¢,, q, g;, & Delaunay then 
fixes the attention on some particular one of these terms, and shows how 
to find a transformation from the variables p,, g, to new variables p’,, q’, 
such that the equations become 
ap, CHA ity bao bloat rigs 
wae ee Seater red tee a 
where H’ is a function of p’,, q’,, ¢ of the same kind as H ; but H’ does not 
contain any term corresponding to the term in H which is under considera- 
tion. This transformation has therefore robbed the disturbing function 
of one of its terms ; by a fresh transformation we can deprive H’ of any 
other term, and so on. In this way all the important periodic terms are 
abolished from the disturbing function, and when the residue has become 
negligeable, the equations are integrated ; and the coordinates are thus 
expressed in terms of six arbitrary constants, and the time by means of 
series in which the time occurs only in the arguments of periodic terms. 
1 «Théorie du mouvement de la Lune.’ Paris. Vol. i. 1860; vol. ii. 1867. 
K 
