144 REPORT—1899; 
Gyldén in 1893 published the first volume of a work! which was 
intended to furnish, in three volumes, a complete exposition of his theory 
of absolute orbits. His death occurred in 1896 before the second volume 
was ready, but it is expected that Dr. Backlund, who has charge of 
Gyldén’s manuscript, will as far as possible carry out the original design. 
Backlund ? in 1896 described a method, founded on Gyldén’s work, 
for integrating the differential equation which determines the radius 
vector in the case of minor planets whose mean motion is nearly twice that 
of Jupiter. Brendel® in the same year discussed the relation of Gyldén’s 
series to the gaps in the distribution of the minor planets; in 1897 
Brendel‘ announced that he had found an improved process of integration, 
and in 1898 the same author published the theoretical part > of a memoir 
in which the motions of the minor planets are discussed by a modified 
form of Gyldén’s process ; the second part, which is not yet published, is 
to deal with definite applications to the solar system. 
§ VI. Progress in 1890-8 of the Theories of §§ ILI. and IV. 
A new impetus was given to Dynamical Astronomy in 1890 by the 
publication of a memoir ® by Poincaré. 
The first feature is the introduction of integral invariants. We can 
regard a system of ordinary differential equations 
May _ 
GH ee 
"hai leider n 
dt 
as defining the totion of a point whose coordinates are (x, 22)... 2,) 
in space of » dimensions. If now we consider a group of such points 
which occupy » v-dimensional region Z, at the beginning of the motion, 
they will at any subsequent time ¢ occupy a region Z. A v-ple integral 
taken over ¢ is called an integral invariant if it has the same value at all 
times ¢. Thus, in the motion of an incompressible fluid, the volume of 
the fluid which was contained initially in any given region is an integral 
invariant. 
Poincaré gives a number of integral invariants which exist in particular 
cases, and then proceeds to apply his results to the question of the stability 
of the motion in the problem of three bodies. There are, he remarks, two 
senses in which the word ‘stability’ may be taken. It may be taken to 
mean that variations in the mean distances of the bodies are always 
restrained within finite limits—Hill and Bohlin have proved that under 
Gyldén’schen Theorie,’ Archiv f. Math.og Natur, Christiania, xiv. pp. 1-10 (1890) ; 
‘Ueber die Convergenz der Annaherungen in der Gyldén’schen Storungstheorie,’ ibid. 
pp. 232-9 ; ‘Untersuchung tiber eine Gruppe von langperiodisch elementiiren Gliedern 
in der Zeitreduction,’ Bihang till k. Sv. Vet-ak. Handlingar, xvii. No. 4. 
1 Traité analytique des orbites absolues des huit planétes principales, tome 1, 
‘ Théorie générale des orbites absolues, Stockholm. : 
2 ¢Sur Vintégration de Véquation différentielle du rayon vecteur d’un certain 
groupe des petites planétes, C. R. cxxii. pp. 1103-7. 
3 «Ueber die Liicken im System der Kleiner Planeten und iiber ein Integrations- 
verfahren im Probleme der drei Korper,’ Ast. Nach. cxl. pp. 145-60. 
4 «Ueber stabile und instabile Bewegungen in unserem Planetensystem,’ 
Jahresbericht der Deutscher Math. Verein, vi. pp. 123. 
5 «Theorie der kleinen Planeten,’ erster Theil, Abhandlungen der Kin. Ges. der 
Wiss. su Gottingen, Neue Folge, i. No. 2. 
6 «Sur le probleme des trois corps et les équations de la dynamique,’ Acta Math, 
sili. pp. 1-220. 
