142 REPORT—1899. 
notion of the absolute orbit and the definitions of elementary and co- 
ordinated terms are introduced in the second part of the ‘ Undersékningar.’ 
Gyldén wrote another paper! on this in 1882, and in the same year dis- 
cussed further? one of the differential equations of his theory of inter- 
mediate orbits. 
A long series of papers dealing with the processes for integrating 
differential equations of the second order by successive approximation, and 
with the convergence of the developments, was published? by Gyldén in 
1882-96. On this see also Harzer.! 
In 1885 Gyldén® and Shdanow® applied the theory of intermediate 
orbits, which had been given in the ‘ Undersékningar,’ to the case of the 
moon’s motion. The problem is made to depend on the solution of the 
Gyldén-Lindstedt equation, and the results yield an approximation to the 
motion of the perigee. Andoyer’ also applied Gyldén’s theory to the 
moon in 1887 ; and Tisserand * in 1888 discussed the Gyldén-Lindstedt 
equation, and applied his results to Andoyer’s equations. 
Harzer ° in 1886 applied Gyldén’s ideas to the determination of the 
motion of those of the minor planets (e.g. Hecuba) whose mean motion is 
approximately twice as great as that of Jupiter. On account of this 
approach to commensurability, some of the characteristic terms become 
very important. Harzer modifies Gyldén’s original procedure in two 
respects : firstly, in using the true longitude throughout as the independent 
variable ; and, secondly, in abandoning the use of the ‘reduced time. 
1 ‘Ueber die absoluten Elemente der Planeten-Bahnen,’ Ast. Nach. ciii. pp. 
49-58. 
2 «Sur l’équation différentielle qui donne immédiatement la solution du probléme 
des trois corps jusqu’aux quantités de deuxiéme ordre inclusivement, C. R. xcv. 
. 55-8. 
- 3 *Hine Anna&herungsmethode im Probleme der drei Kérper, Acta Math. i. 
pp. 77-92; ‘ Untersuchungen tiber die Convergenz der Reihen, welche zur Darstellung 
der Coordinaten der Planeten angewendet werden,’ ibid. ix. pp. 185-294; ‘ Nouvelles 
recherches sur les séries employées dans les théories des planétes,’ ibid. xv. pp. 
65-189 ; xvii. pp. 1-168; ‘ Ueber die Convergenz einer in der St6rungstheorie vorkom- 
menden Reihe,’ Ast. Wach. cxix. pp. 321-30; ‘ Bemerkungen tiber die Convergenz der 
nach der Potenzen der stdrenden Krafte geordneten Anniiherungen im Stérungs- 
problem,’ ibid. cxxi. pp. 80-94; ‘Sur une équation différentielle du second ordre, non 
linéaire et a coefficients doublement périodiques,’ C. £. cxxii. pp. 160-5 ; ‘ Remarques 
ultérieures relativement 4 ma derniére communication 4 M. Hermite,’ ibid. exxii. 
pp. 585-8; ‘ Om bestiimningen af ojemnheter med mycket lang period i theorien for 
planeters och satelliters rorelser, Ofversigt af Kongl. Vet.-ak. For. lii. pp. 419-32 ; 
‘En transformation .af den differentialekvation, som bestiimmer ojemnheterna med 
mycket langa periorder i en planets longitud,’ ibid. lii. pp. 503-6 ; ‘ Olika methoder att 
bestiimma de horistika termerna i den differentialekvation, som férmedlar hiidled- 
ningen af ojemnheterna ien planets longitud,’ ibid. liii. pp. 421-30. 
+ ¢*QUeber eine Differentialgleichung der Stdrungstheorie,’ Ast. ach. ecxix. 
. 273-94. 
aks Sur l’orbite intermédiaire de la Lune,’ (. R. ci. pp. 223-6; ‘Die intermediire 
Bahn des Mondes,’ Acta Math, vii. pp. 125-72. 
6 Recherches sur le mouvement de la Lune autour de la Terre daprés la Théorie 
de M. Gyldén, Stockholm, 1885. 
7 ¢Contribution a la théorie des orbites intermédiaires,’ Annales de la Fac. des 
Sc. de Toulouse, i. 
* «Sur une équation différentielle du second ordre qui joue un réle important 
dans la mécanique céleste,’ did, ii, 
® *Untersuchuugen tber einen speciellen Fall des Problems der drei Kérper,’ 
Mémoires de Saint-Pétersbowrg, xxxiv. No. 12; ‘Quelques remarques sur un cas 
spécial du probléme des trois corps,’ stronomiska Iakttagelser ooh Undersikningar 
anstilda pa Stockholms Observatorium, iii. No. 4. y 
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