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SCIENCE 



[N. S. Vol. XXIX. No. 733 



form, ease of transformation and per- 

 spicuity in showing how the Tariables enter 

 the problem. It becomes of advantage, 

 then, to expand the function F in terms of 

 canonical elements; Charlier has given in 

 his lectures a method of expansion which 

 Noren and Wallberg carried out to terms 

 of the second degree. Stone has published 

 a simple direct derivation of the canonical 

 elements introduced into the three-body 

 problem by Delaunay and Jacobi; while 

 the formulae of transformation from 

 Cartesian to Jacobian coordinates in the 

 n-hody problem have been derived by 

 Pizzetti with the aid of linear substitu- 

 tions. 



From the standpoint of the theory of 

 integral invariants introduced by Lie and 

 Poineare the characteristic property of the 

 canonical system is the existence of the 

 relative integral invariant f^Xidpi, and 

 the covariantive correspondence between 

 the canonical system and the differential 

 expression ^Xidpi — Fdt forms the con- 

 nection between their theory and that of 

 contact transformations. A systematic 

 study of integral invariants has been pub- 

 lished by De Bonder, including his own 

 researches and those of Appell, Hadamard 

 and Koenigs. Morera has' shown in a 

 series of memoirs how this transformation 

 theory gains in generality, simplicity and 

 elegance when at its foundation we lay the 

 bilinear covariantive correspondence to 

 which allusion has just been made; Morera 

 rediscovers and generalizes the theorems of 

 Lie on the invariance of canonical systems 

 under contact transformations. The im- 

 portance of these results for the problem 

 in hand is recognized when we recall that 

 Lagrange's method of the variation of 

 arbitrary constants in the theory of per- 

 turbations leads to equations of the canon- 

 ical form; Lie's theory thus stamps the 

 history of a perturbation problem as the 

 history of a contact transformation, a rela- 



tion exhibited on the geometrical side by 

 the true orbit enveloping the successive ap- 

 proximate ones. The notion of an inter- 

 mediate orbit has been extended to canon- 

 ical systems by Charlier, who has employed 

 it in a generalization of Jacobi 's theorem. 



Lagrange showed that the eighteenth 

 order system in the three-body problem 

 can be reduced to one of the sixth order; 

 this reduction has been effected in a variety 

 of ways by other mathematicians. Poin- 

 eare employed a contact transformation to 

 reduce the problem to the twelfth order, 

 and Whittaker has used an extended point 

 transformation to carry the reduction on to 

 the eighth order. Whittaker has also ex- 

 hibited in explicit form the contact trans- 

 formations involved in Radau's direct re- 

 duction from the eighteenth to the sixth 

 order. Eouth's transformation known as 

 the ignoration of coordinates has recently 

 been generalized by Woronetz to a form 

 which includes as special cases Poineare 's 

 equations of motion, and the reductions of 

 Lagrange, Jacobi, Bour and Brioschi. 

 The discovery of the existence of a force 

 center in the three-body problem has 

 enabled Delaunay to write its equations in 

 a special form. Scholz has shown that 

 under certain assumptions regarding the 

 perturbative function the three-body prob- 

 lem can be reduced to the integration of 

 a single differential equation, and a new 

 reduction of the plane problem has been 

 given by Perchot and Ebert. The cor- 

 responding reduction of twelve units in the 

 order of the M-body problem has been 

 effected by Bennett through Poineare 's 

 transformation and a generalization of the 

 one employed by Whittaker in the three- 

 body problem. 



In the transformation and reduction of 

 the problem a principal role has been 

 played by the ten known integrals, namely, 

 the six integrals of motion of the center of 

 gravity, three integrals of angular mo- 



