. 
PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 125 
In 1868 Radau published, first in a series! of notes in the ‘ Comptes 
Rendus,’ and subsequently in a memoir? in the ‘Annales de 1’Kcole 
Normale Supérieure,’ his researches on the differential equations of the 
problem of » bodies. He finds the effect of an orthogonal substitution 
performed on the variables in the problem, and shows that Jacobi’s substi- 
tution in the problem of three bodies is a case of this. Two other cases are 
worthy of mention : firstly, a transformation which is equivalent to referring 
the second body to the first as origin, the third body to the C.G. of the 
second and third, the fourth body to the C.G. of the first three, and so on, 
at the same time modifying the masses ; and, secondly, a transformation 
which shows the existence of ‘canonical’ points, each of which has, 
with reference to the motion of (x—1) of the bodies, properties similar to 
those possessed by the C.G. for the whole system, Considering the case 
of three bodies, he deduces Bour’s equations, and also a new canonical 
system of the eighth order. 
A modification of the transformation of Jacobi and Radau was con- 
sidered in 1889 by Andrade,* and in 1896-7 Poincaré? gave another 
transformation which appears to be still better suited for effecting the 
same reduction. 
The results obtained by Allégret ° in 1874 are substantially equivalent 
to some of those in Radau’s papers. 
Radau’s researches were continued in 1869 in a number of papers," of 
which that in Liouville’s journal is the most complete ; the author dis- 
cusses the reduction of the order of a canonical system when one of the 
coordinates does not enter explicitly in the energy-function, and applies 
his results to the problem of three bodies, arriving at Scheibner’s system. 
Hesse’ in 1872 published a fresh discussion of the problem of three 
bodies, somewhat on the lines of Lagrange’s memoir ; but it was pointed 
out by Serret * in 1873 that the equations in one of Hesse’s systems were 
not independent, and consequently his results were invalid. Serret’s 
paper contains also an exposition, in an improved and symmetrical form, 
of the essential parts of Lagrange’s memoir. Other reductions of the 
"Sur un théoréme de mécanique,’ @. #. Ixvi. pp. 1262-5; ‘ Remarques sur le 
probléme des trois corps,’ idid. lxvii. pp. 171-5; ‘ Sur une transformation orthogonale , 
applicable aux équations de la dynamique,’ ibid. xvii. pp. 316-9; ‘Sur l’élimination 
directe du nceud dans le probléme des trois corps,’ ibid. lxvii. pp. 841-3. 
* ‘Sur une transformation des équations différentielles de la dynamique,’ Annales 
del’ Ecole Norm. Sup. v. pp. 311-75. 
° «Sur une réduction du probleme des » corps, qui conserve 5 ou teed 
mutuelles,’ C. A, cviii. pp. 226-8; ‘Sur les réductions du probléme des x corps, qui 
conservent certaines distances mutuelles,’ ibid. cviii. pp. 280-1. 
* ‘Sur une forme nouvelle des équations du probleme des trois corps,’ ibid. cxxiil. 
pp. 1031-5; Acta Math. xxi. pp. 83-97. 
° ‘Sur une transformation des équations de la mécanique céleste,’ C. R. cxxix. 
pp. 656-8. 
° *Betrachtungen tiber die Flichensitze,’ Ast. Nach. Ixxiii. pp. 337-44 ; ‘Weitere 
Bemerkungen tiber das Problem der drei Kérper,’ ibid. lxxiv. pp. 145-52; ‘Sur une 
propriété des syst¢mes qui ont un plan invariable,’ @. R. lxviii. pp. 145-9; ‘Sur une 
pr etomaiios des coordonnées des trois corps dans laquelle figurent les moments 
‘inertie, ibid. cxviii. pp. 1465-9; ‘Ueber gewisse Higenschaften der Differential- 
gleichungen der Dynamik, Math. Ann. ii. pp. 167-81; ‘Sur une propriété des 
systémes qui ont un plan invariable,’ Liowville, xiv. pp. 167-230. 
* «Ueber das Problem der drei Kérper,’ Crelle, Ixxiv. pp. 97-115. 
* ‘Réflexions sur le mémoire de Lagrange intitulé “Essai sur le probléme des 
trois corps,”’ C. #, lsxvi. pp. 1557-65 ; and Bull. des Sc. Math. vi. p, 48. 
distances 
