126 REPORT—1899. 
problem of three bodies, which seem scarcely of sufficient importance to 
be here described in detail, are those of Weiler! in 1869-70, Hill? in 
1875, Weiler # in 1879-80, Seydler + in 1884, and Duport® in 1898, 
The problem of 7 bodies can be reduced from the 6nth order to the 
(6n-12)th order, just as the problem of three bodies can be reduced from the 
18th order to the 6th. This subject has been discussed in the period 
under review by Allégret ® in 1875 (who fell in errors which were pointed 
out by Mathieu’ in 1877), by Betti* in 1877, Mathieu in 1877, Ball !° in 
1877, Dillner !! in 1877 (who attempted to use quaternions, but made mis- 
takes which were pointed out by Bruns !* in 1880), and Dillner }3 in 1882-8. 
Seydler !* in 1885 extended the analysis of Lagrange’s treatment of the 
problem of three bodies to the case of the problem of four bodies. The 
system isreduced ultimately to a system of the twelfth order and quad- 
ratures. 
The general theory underlying the work of this section has been 
developed by Lie and Mayer. A special consideration of the problem of 
three bodies will be found at p. 282 of a paper |’ published by Lie in 1875. 
In 1887 Bruns !° published a paper which will be analysed later, but 
which contains a new reduction of the problem of three bodies. 
Let 9, 72, 73 be the mutual distances of the three bodies ; and let ¢,= 
Sa,(a,+ty,)/2b\(%,+ty,), where (a, y;, 2) &c. are the coordinates of 
the bodies when the origin is taken at the centre of gravity, and the 
1 «Ueber die Elimination des Knotens in dem Problem der drei Kérper, ete.,’ Ast. 
Nach. \xxiv. pp. 81-96, Ixxv. pp. 113-28; ‘Notes sur le probleme des trois corps,’ 
Liowville, xiv. pp. 805-20, 
2 «Reduction of the Problem of Three Bodies,’ The Analyst, iii. pp. 179=85. 
8 ¢*Ueber die Differentialgleichungen der Bewegung in dem Problem der drei 
Korper,’ Ast. Nach. xcvi. pp. 161-82; ‘Das Problem der drei Korper in der neuen 
Stérungstheorie,’ ibid. xcvil. pp. 97-112, 129-44, 161-76, 193-208. 
4 «Ueber einige neue Formen der Integrale des Zwei-und-Dreikérper-Problems,’ 
Sitzungsberichte der Ak. zu Wien, \xxxix. pp. 851-72; ‘O integrovani nékterych 
rovnic vyskytrujic ich se v problemu tfi téles,’ Sitzwngsberichte d. Ges. der Wiss. in 
Prag, 1884, pp. 16-29; ‘ Dal&Si ptispévky k integrovani,’ etc., ibid. pp. 106-26. 
5 Sur le probléme des trois corps, Bull. Astr. xv. pp. 377-83. 
6 «Mémoire sur le probléme des trois corps,’ Liouville (3), i. pp. 277-316. 
7 Mathieu, ‘Sur le probléme des trois corps.’ Liowville (3), iii. pp. 216-9; Allé- 
gret, ‘ Note sur le probléme des trois corps,’ ibid. pp. 422-6; Mathieu, ‘ Réponse a la 
note de M. Allégret sur le probléme des trois corps,’ Liouville (3), iv. pp. 61-2. 
8 ¢ Sopra il moto diun sistema di un numero qualunque di punti che si attraggono 
o si respingono tra loro, Annali di matematica (2), viii. pp. 301-11. 
9 ‘Mémoire sur les équations du mouvement d’un systéme de corps,’ Liowville (3), 
ii, pp. 5-21. j 
a Note on a transformation, etc.,’ Monthly Notices, xxxvii. pp. 265-71. 
1 «Mémoire sur le probléme des 2 corps,’ Nova Acta R.SS. Upsal. vol. extra ord., 
1877, 18 pp. 
12 Jahrbuch ber die Lortschritte der Mathematik, 1877, p. 788. 
13 ¢Qm integration af differentialeqvationerna i m-kroppiirs problemet,’ Ofversigt 
af K. Vet.-ah. Forhandlingar, 1882, No. 4, pp. 13-20, No. 8, pp. 9-29; 1886, 
pp. 173-84, 217-22; 1888, pp. 367-78; ‘ Sur l’intégration des équations différentielles 
du probléme des V corps, Annali di matematica (2), xi. pp. 56-64. 
14 « Ausdehnung der Lagrange’schen Behandlung des Dreikérper-Problems auf 
das Vierkérper-Problem,’ Abhandlungen der kh. bilm. Gesellschaft der Wissenschaften, 
(7), i. No. 5, 20 pp. 
1s Begriindung einer Invarianten-Theorie der Beriihrungs-Transformationen,’ 
Math. Ann. viii. pp. 215-303. 
16 ‘Ueber die Integrale des VielkGrper-Problems,’ Berichte der hgl. Siichsischen 
Gesellschoft der Wiss. zu Leipzig, 1887, pp. 1-39, 55-82 ; Acta Math. xi. pp. 25-96. 
