Maech 6, 1896.] 



SCIENCE. 



339 



then necessary an equation between the 

 time and the vaij-iable assumed as independ- 

 ent, that is, the orbit longitude, or, more 

 properly, the amount of angle described by 

 the radius vector. If we suppose the abso- 

 lute orbit to be described by the planet so 

 that equal areas are passed over by the 

 radius in equal times, it is plain that, on the 

 attainment of a given longitude, a definite 

 amount of time must have elapsed since the 

 epoch. This is what Prof. Gylden calls the 

 reduced time; and he computes the difference 

 between it and the actual time required by 

 the theory of gravity for the planet to arrive 

 at the stated direction. This mode of pro- 

 ceeding does not diifer from Hansen's, except 

 in the point that the absolute orbit is sub- 

 stituted for a iixed ellipse. 



But this gives us correctly only the orbit 

 longitude ; for the radius and the latitude, 

 which correspond in the absolute orbit to 

 this reduced time, are not quite those which 

 the planet has at the actual time. Conse- 

 quently, Prof. Gylden proposes to compute 

 two corrections, the one to be applied to the 

 product of the eccentricitj' into the cosine 

 of the true anomaly, the other to the sine 

 of the latitude. Also the reduction of the 

 orbit longitude to the plane of reference 

 must be manipulated so that it comes out 

 correctly. 



The employment of the orbit longitude as 

 independent variable throughout all the 

 integrations necessitates a mass of very in- 

 tricate transformations of terms from one 

 shape into another. Also the integrations 

 which bear on elementarj' terms must be 

 kept distinct from those which bear on non- 

 elementary terms. A degree of complexity 

 is thus imparted to the subject which makes 

 it difficult to see when one has really 

 gathered up all the warp and woof of it. 

 Prof. Gylden has nowhere removed the 

 scaffolding from the front of his building 

 and allowed us to see what architectural 

 beauty it may possess; it is necessary to 



compare a large number of equations scat- 

 tered through the volume before one can 

 opine how the author means to proceed. 



, The advantages claimed for the method 

 are that it prevents the time from appear- 

 ing outside the trigonometrical functions, 

 and that it escapes all criticism on the score 

 of convergence. The first is readily con- 

 ceded, but many simpler methods possessing 

 this advantage are already elaborated, and 

 it is not so clear that the second ought to 

 be granted. 



No completely worked out example of the 

 application of this method has yet been 

 published. The great labor involved will 

 naturally deter investigators from employ- 

 ing it. 



In 1890 was published the memoir of M. 

 H. Poincare entitled Sur le probleme des trois 

 corps et les equations de la dynamique, and 

 which obtained the prize of the King of 

 Sweden. Most of the results of this memoir 

 were worked over and presented anew with 

 greater elaboration and clearness by their 

 author in Les Metliodes Nouvelles de la Meca- 

 nique Celeste. Here we find a large number 

 of new and very interesting theorems. 



First is to be noted the class of particular 

 solutions in the problem of the motion of a 

 system of material points which are now 

 named periodic solutions. The initial rela- 

 tive positions and velocities of the several 

 points are so adjusted that, after the lapse 

 of a definite time, the latter retake them. 

 Hence is evident a method which may be 

 employed to elaborate this special case of 

 motion, viz., by the tentative process with 

 mechanical quadratures. M. Poincare has 

 divided this sort of solutions into three 

 classes, of which, however, the second and 

 third are not essentially different. He has 

 shown that, in the latter classes, the values 

 of the arbitrary constants of the problem 

 must be so adjusted that no secular inequali- 

 ties, or, as Professor Gylden calls them, ele- 

 mentary terms, may arise. The number 



