PREFACE, 



The present memoir may be considered as, in part, an extension and generaliza- 

 tion of two former papers by the author : the first being TMorie cles perturhations de 

 la Lime qui sonf dues a V action des Planetes, published in LiouviUe's Journal, tome 

 xvi., 1871; and the second, Sur un Theoreme de Mecanlque Celeste, published in the 

 Comptes Bendns, tome Ixxv. Notwithstanding its extent, the author is conscious, 

 in his treatment of the subject, of several gaps, which may detract from entire rigor. 

 He believes that some of these are of such a nature that the reader can readily 

 fill them, while the remainder would have led into long digressions, and thus 

 caused great delay in the publication of the paper. To the former class belong 

 (1) the analogy between the expressions for the rectangular co-ordinates x and 

 y, wliich diff'er only in that the latter is composed of products of sines, while the 

 former is composed of similar products of cosines ; and (2) the omission of all 

 considerations of the modifications growing out of the fact that in equation (1) 

 one value of h vanishes. To the latter class belong the omission of all considera- 

 tions respecting the convergence of the series encountered, respecting terms of 

 long period, and respecting the occurrence of relations among the arguments, 

 such as that known to subsist between the mean motions of three of the satellites 

 of Jupiter. These subjects will naturally come up for consideration when the 

 process of actually integrating tlie differential equations of planetary motion in 

 the most general way is undertaken. No method for the actual execution of this 

 integration is given at present, partly because the paper may be considered com- 

 plete without it, partly because the author has not succeeded in working out any 

 method satisfactory to himself. It is true that a large part of the paper is devoted 

 to reviewing the general forms met with in a certain integrating process, but the 

 actual execution of this process, even for a single approximation, may be consid- 

 ered impracticable on account of the enormous labor involved in it. It is shown, 

 by a bird's eye view, that a certain object is, in the nature of things, attainable; 

 but a practicable way of actually reaching it is yet to be pointed out. It would be 

 extremely agreeable to the author to learn that abler hands than his were success- 

 fully working to effect the actual solution of this noble problem in its most general 

 form. 



(V) . 



