128 REPORT—1899. 
The first paper oi the subject in the period under review was 
published by Routh! in 1875; he showed that the three equidistant 
particles are stable when the square of the sum of the masses is greater 
than twenty-seven times the sum of the products of the masses taken two 
and two together ; a result which, however, had already been stated by 
Gascheau. The stability was considered from a somewhat more general 
point of view by Liapunow? in 1889; and Gyldén® in 1884 discussed 
solutions which differ but little from the three collinear particles. 
Lagrange’s results have been generalised, and corresponding theorems 
found for the motion of more than three bodies. An attempt made in 
this direction by Veltmann‘ in 1875 is open to criticism, but Hoppe® in 
1879, and Lehmann-Filhes ° in 1891, discovered solutions in which more 
than three particles are placed at the corners of a regular polygon or poly- 
hedron, or on a straight line. Sloudsky” in 1892 claimed to have given 
some of Hoppe’s results in 1878, in a paper published in Russian. In 
Hoppe’s paper the masses of the particles are supposed to be equal, 
which detracts from the value of his results ; in Lehmann-Filhes’s paper 
the masses are not so restricted. 
Cases in which the triangle formed by the bodies is isosceles were 
discussed by Fransen § in 1895, and Gorjatschew ° in 1895-6. 
§ III. Memoirs of 1868-89 on General and Particular Solutions of the 
Differential Equations, and their Expression by Means of Infinite 
Series (excluding G'yldén’s Theory). 
From the time when it was first realised that the motion of the three 
bodies cannot be represented in a finite form by means of known functions, 
interest has centred chiefly round that division of the subject to which 
the present section will be devoted, namely, the derivation, nature, and 
properties of the infinite series by means of which the problem can be 
solved. 
The result of our observations of the heavenly bodies suggests a form 
into which we may try to put the analytical solution. It is found that 
the facts can be represented, at any rate for as far back as our records 
take us, by supposing that the planets move in ellipses round the sun. 
These ellipses are, however, not fixed, but their elements (the eccentricity, 
ec.) vary from year to year. Some of these variations, or inequalities, are 
periodic—that is to say, can be expressed by terms such as a sin (bt + ©), 
‘ «On Laplace’s three particles, with a Supplement on the stability of steady 
motion,’ Proc. Lond. Math. Soc. vi. pp. 86-97. 
2 *On the stability of the motion inaspecial case of the problem of three bodies,’ 
Trans. Math. Soc. of Krakow (8), ii. pp. 1-94. (Russian.) ; 
3 «Om ett af Lagrange behandladt fall af tre-kroppars-problemet; Ofversigt af 
K. Vet.-ak. Forhandlingar, xii. pp. 3-11; ‘Sur un cas particulier du probléme des 
trois corps, Bull. Astr. i. pp. 361-9. 
4 * Bewegung in Kegelschnitten von mehr als zwei Korpern, welche sich nach 
‘dem Newton’schen Gesetz anziehen,’ Ast. Nach. \xxxvi. pp. 17-380. 
5 «Erweiterung der bekannten Speciallésung des Dreikdrperproblems,’ Archiv 
der Math. u. Phys. lxiv. pp. 218-23. 
6 «Ueber zwei Falle des Vielkérperproblems,’ Astr. Nach. cxxvii. pp. 137-44. 
7 ‘Note sur quelques cas particuliers du probléme de plusieurs corps,’ Bulletin 
de la Soc. Imp. Natur. Moscow, 1892, pp. 437-40. 
8 ¢ Ett specialfall af tre-kroppars-problemet,’ Ofversigt af K. Vet.-ak. Forhand. 
lii, pp. 783-805. 
5 Transactions of the Inyp. Soc. of Nat. Moscow, vii. viii. 
