156 REPORT—1899. 
The same author! in 1896 showed that a theorem analogous to the invari- 
ability of the mean distances can be obtained for a general class of 
dynamical systems. 
The solution of canonical systems of equations by series was discussed 
in 1891-2 by Wand ? ; suggestions for directing the approximations in the 
problem of bodies were published in 1891 by Laska,* and in 1897 by 
Kovesligethy,t and (for solving the differential equation for the mean 
distance) in 1893 by Gyldén.’ The first part of a paper® published by 
Newcomb in 1895 contains a solution of the problem of three bodies based 
on continued approximation. Hill’ in 1893 and 1897 showed how, by 
dividing the potential function otherwise than in the old theories, an inter- 
mediate orbit may be obtained which is free from the disadvantages of 
Kepler’s ellipse ; and Krassnow § in 1898-9 obtained an intermediate 
orbit for the moon, making the suppositions of the restricted problem of 
three bodies, by integrating a Hamilton-Jacobi partial differential equation, 
in which small quantities of the third order are neglected. 
Painlevé ® in 1896 showed that the problem of three bodies can be 
integrated by means of series of polynomials, convergent for all values of 
z, except when the initial conditions are such that two of the bodies collide 
after a finite interval of time. The same author !° in 1597 showed that 
the conditions which must be satisfied in order that, after a finite interval 
of time, two of the bodies may collide, cannot be algebraical conditions. 
Brown! in 1897 discussed the properties of the general solution in 
trigonometric series of the problem of three bodies, by supposing it to 
have been derived by integrating the Hamilton-Jacobi equation. Several 
properties of the constants of the solution are deduced, including those 
previously given by Newcomb. In a second paper,!? the same method is 
applied to the Lunar Theory, and Adams’s theorems on the constant part of 
1 «Sur l’extension que l’on peut donner au théoréme de Poisson, relatif a l’invari- 
abilité des grands axes,’ C. 2. cxxiii. pp. 790-3. 
2 ‘Ueber die Integration der Differentialgleichungen, welche die Bewegungen 
eines Systems von Punkten bestimmen,’ Ast. Mach. cxxvi. pp. 129-88, cxxvii. 
pp. 353-60, cxxx. pp. 377-90. 
3 «Zur Berechnung der absoluten Stérungen,’ Sitzungsberichte der k. Bohm. Ges. 
der Wiss., Prague, 1891, pp. 147-53. 
4 «Storungen im Vielkorpersystem,’ Mathem.u, Natur. Berichte aus Ungarn, xiii. 
pp. 380-412. 
5 ‘Ueber die Ungleichheiten der grossen Axen der Planetenbahnen,’ Ast. Nach. 
Cxxxlii. pp. 185-90. 
* * Action of the Planets on the Moon,’ American Hphemeris Papers, v. Part III. 
7 «On Intermediate Orbits, Annals of Maths. viii. pp. 1-20 (1893). ‘On Inter- 
mediary Orbits in the Lunar Theory,’ Astron. Jown. xviii. pp. 81-7 (1897), 
8 «Zur Theorie der intermediiren Bahnen des Mondes,’ Ast. Nach. cxlvi. pp. 7-10 ; 
‘Weitere Mittheilung betreffend die Theorie der intermediiiren Bahnen des Mondes,’ 
ibid. cxlvi, pp. 337-40; ‘Zur Integration der Jacobi’sche Differentialgleichung fiir 
die Mondbewegung,’ ibid. exlviii. pp. 37-42. 
® «Sur les singularités des équations de la Dynamique et surle probléme des trois 
corps, C. R. cxxiii. pp. 871-3. 
1 «Sur les cas du probléme des trois corps (et des » corps) ot deux des corps se 
choquent au bout d’un temps fini,’ C. R. exxv. pp. 1078-81. 
1 ¢QOn the application of Jacobi's Dynamical Method to the General Problem of 
Three Bodies,’ Proc. Lond. Math. Soc. xxviii. pp. 130-42. There is a slight error in 
result (x.), p. 141 of the paper. 
12 «On certain properties of the Mean Motions and the Secular Accelerations of 
the principal arguments used in the Lunar Theory,’ Proc. Lond. Math. Soc. xxviii. 
pp. 143-55. 
