Maech 6, 1896.] 



SCIENCE. 



335 



profoundlj^ engaged in applying liis method 

 to the motion of the moOn. Tisserand's 

 exposition of this method is somewhat more 

 brief than the author's own. But when the 

 necessary modifications are introduced into 

 Delaunay's procedures, to make them ap- 

 plicable to the more general case of the 

 motion of a system of bodies, the establish- 

 ment of the formulas can be rendered still 

 more brief. 



There is one point in reference to De- 

 launay's method which, as far as I am 

 aware, has escaped notice. This method 

 consists in a series of operations or trans- 

 formations, in each of which the position of 

 the moon in space is defined by six vari- 

 ables, the number three being doubled in 

 order that the velocities, as well as the co- 

 ordinates, may be expressed without dif- 

 ferentials. The aim of the transformations 

 is to make one-half of these, which Poin- 

 car6 has called the linear variables, contin- 

 ually approach constancy, while the other 

 half, named the angular variables, contin- 

 ually approach a linear function of the 

 time. But at any stage of the process the 

 position of the moon, as well as its velocity, 

 is definiteljr fixed by the six variables pro- 

 duced by the last transformation, provided 

 that the proper degree of variability is at- 

 tributed to them, just as, before any trans- 

 formation was made, the six elements of el- 

 liptic motion, usually denominated oscula- 

 ting, defined them; the point of difference 

 to be noticed being that the more the trans- 

 formations are multiplied, the more com- 

 plex becomes the character of the expression 

 of the former quantities in terms of the 

 latter. But, however great may be the 

 number of transformations, the series 

 evolved have always one consistent trait, 

 viz., that the angular variables are involved 

 in them only through cosines or sines of 

 linear functions of these variables, the 

 linear functions being formed with integral 

 coeiBcients. Now, as in all tliis work we 



are obliged to employ infinite series, the 

 question of their convergence is an ex- 

 tremely important one. The inquiry in 

 this respect may be divided into two parts, 

 mainly independent of each other. These 

 are, convergence as respects the angular 

 variables, and convergence as respects the 

 linear variables. The first part is much 

 the more simple. Regarding each of the 

 coefficients of the series we employ as a 

 whole, that is, representing it by a definite 

 integral, it is quite easily perceived that the 

 said series are both legitimate and conver- 

 gent when, giving the angular variables 

 the utmost range of values, still no two of 

 the bodies can occupy the same point of 

 space. In the contrary case the series are 

 evidently divergent. This condition affords 

 certain limiting conditions for the values of 

 the linear variables. Could we trace these 

 limiting conditions through all the trans- 

 formations, and obtain hj comparison the 

 formulas to which these tend when the 

 number of transformations is made infinite, 

 we should be in possession of the conditions 

 of stability of motion of the system of 

 bodies. The second part of the inquirjr 

 relates to the expression of the mentioned 

 coefficients by infinite series proceeding ac- 

 cording to powers and products of certain 

 parameters which are functions of the linear 

 variables. It is well known that, in the 

 case of elliptic elements, Laplace and 

 Cauchy almost simultaneously showed that 

 the series are convergent when the eccen- 

 tricity does not exceed a fraction which is 

 about two-thirds. The determination of 

 the conditions of convergence, after certain 

 transformations have been made in the sig- 

 nification of the elements, is undoubtedly a 

 more complex problem; nevertheless, it 

 seems to be within the competency of anal- 

 ysis as it exists at present. 



The discovery of the criterion for the con- 

 vergence of series proceeding according to 

 powers and products of parameters is due 



