PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 137 
In 1889, Eginitis! gave the analytical expression for those secular 
inequalities of the mean distances which are of the third order in the dis- 
turbing forces. After showing that they can arise only from a term 
1 /(3B" ,\* 
nai ( “Ot at) j 
ut 
where = denotes the aggregate of terms of the first and second order, 
he finds [ae by substituting the ordinary development of the disturb- 
ing function, squares it, and shows that secular inequalities arise from the 
multiplication of terms with the same arguments. He further shows that 
these secular inequalities are periodic, though their period is very long. 
The transition from the old planetary theory, with its secular and 
periodic inequalities, to the new Dynamical Astronomy, in which all terms 
are periodic, had its effect on theorems such as that now under considera- 
tion. Tisserand ? in 1888 gave the new formulation of the theorem of the 
invariability of the mean distances, when the solution of the problem of 
three bodies is expressed as in Delaunay’s lunar theory. He shows that 
the theorem does not hold when terms of the order of the fourth power of 
the ratio of the mean motions are taken into account, and for the general 
problem of three bodies confirms Haretu’s result that the theorem does 
not hold for terms of the order of the cube of the disturbing forces. 
In 1878 Adams? published some curious results relating to one of the 
expansions in the lunar theory. Let e¢ be the eccentricity of the lunar 
orbit, and let y be the sine of half the inclination of the moon’s orbit to 
the ellipse ; these quantities are supposed defined as in Delaunay’s 
theory : let n be the moon’s mean motion, (1—c)n the mean motion of the 
lunar perigee, (I—g)n the mean motion of the moon’s node, a the mean 
distance, and 7 the radius vector. Then the non-periodic part of < can 
be expanded in the form 
A+ Be?+Cy?+ Ee! + 2Fe?y?+Gyi+... 
where A, B, C...are functions of the solar eccentricity and of the 
ratio of the mean motions of the sun and moon ; similarly the terms in 
¢ which involve e? and y? can be denoted by He?+ Ky’, and the similar 
terms in g by Me?+Ny?. 
Then Adams’s theorems are that 
B=0, C=0, EK—FH=0, FN—GM=0. 
These results are all obtained by one process, which for the case of the 
first may be thus described : consider two moons, of which the orbit of one 
has no eccentricity or inclination, and the orbit of the other has no inclina- 
tion. It is proved that a certain function of the coordinates of the two 
moons is purely periodic ; and it is shown that this requires the vanishing 
of the quantity B. 
1 ‘Sur la stabilité du systéme solaire,’ C. R. cviii. pp. 1156-9; ‘ Mémoires sur la 
stabilité du systéme solaire,’ Annales de l’ Obs. de Paris, Mémoires, xix. 
? «Sur un point de la théorie de la Lune,’ C. R. evi. pp. 788-93. 
* Note on a remarkable property of the analytical expression for the constant 
term in the reciprocal of the moon’s radius vector,’ Monthly Notices, Roy. Astro. Soc, 
xxxviii. pp. 460-72, 
