Maech 6, 1896.] 



SCIENCE. 



337 



of the planets could be shown by giving 

 -superior limits to the errors necessarily in- 

 volved. 



One more remark may be made before we 

 leave Delaunay's method. In every opera- 

 tion or transformation half the integrals are 

 obtained without the intervention of the 

 time, and from these solely are obtained the 

 ranges of values for all the linear variables. 

 As no integrating divisors appear in their 

 expressions, it follows that the question of 

 stability is not affected in any way by the 

 vanishing of these . Moreover, the presence 

 of a libration in the angle of operation does 

 not necessitate any change in the procedure. 

 The integrating divisors which appear in 

 the expressions for the angular variables, 

 obtained through quadratures, may cause 

 difficulty, but this can generally be removed 

 by a modification of the parameters em- 

 ployed in the development of the coefficients 

 in series. Beyond this it does not seem ne- 

 cessary to attend particularly to the terms 

 which Professor Gylden has designated as 

 critical. 



To give a succinct idea of the scope of 

 this method, it may be said that it is appli- 

 cable whenever, in the system, the planets 

 maintain their order of succession from the 

 sun. In systems where that undergoes 

 change, as is the case with the group of 

 minor planets, supposing their action on 

 each other is sensible, it is not applicable. 



Delaunay's method has not yet received 

 all the developments and applications it is 

 susceptible of. 



The treatise of Hansen on the shortest 

 and most ready method of deriving the per- 

 turbations of the small planets was pub- 

 lished in the interval 1857-1861. But as 

 the principles on which it is founded had 

 been elaborated and communicated to the 

 public some years earlier, it is, perhaps, 

 more properly to be assigned to the first 

 half of the centurJ^ In conseqiience, I pass 

 it over with this slight mention. 



Perhaps the most conspicuous labors in 

 our subject, during the period of time we 

 consider, are those of Professor Gylden and 

 M. Poincare. We will limit our attention, 

 for the remainder of this discourse, to the 

 consideration of these investigations. 



Professor Gylden began work with the 

 methods of Hansen and was gradually led 

 to modifications of them looking towards 

 their use for indefinite lengths of time. This 

 quality has latterly become imperative with 

 him, and he has recently published the first 

 volume of what is evidently intended to be 

 a lengthy work entitled Traite Analytique 

 des Orbites Absolues des Huit Planctes Princi- 

 pales. To show the drift of Professor Gyl- 

 den's investigations, we cannot do better 

 than give an analysis of this volume. At 

 the outset the author introduces a class of 

 curves he names periphlegmatic, that is, 

 curves which surround a flame. The defi- 

 nition of this sort of curve is that it describes 

 continually the space between two concen- 

 tric spheres, and, at every point, turns its 

 concavity towards the intersection of the 

 radiu^s vector with the inner sphere. In an 

 application to the solar system, the sun is 

 supposed to occupy the common center of 

 the spheres. The investigation is at first 

 limited to the case where this curve is plane. 

 A differential equation of the second order 

 is derived which the radius vector of this 

 curve satisfies, the independent variable be- 

 ing the angle described. The perpendicular 

 distance between the spheres is called the 

 diastem. The spheres are supposed to be 

 drawn so that they touch the curve at the 

 points where the radius becomes a maxi- 

 mum or minimum. Thus, in some cases, 

 the spheres are regarded as fixed, in others 

 as movable. In the latter case, however, 

 the sum of their radii is supposed to remain 

 constant. Thence we have two groups of 

 periphlegmatic curves ; those with constant 

 and those with variable diastems. The 

 author gives examples of both these groups, 



