GENERAL INTEGRALS OF PLANETARY MOTION. 7 



The number of coefficients to be determined is now n(n -\- 1). The total num^ 

 ber of conditions which the system must satisfy is -^ — — — ~ — ^ , but one of these 



n(n 4- 3) 

 being ah-eady satisfied by the quantities in the first line, there remain only — ^—^ 



conditions to be satisfied by n{ii -|- 1) quantities, we have therefore 



«(7i + 3) n{n — l) 

 n{n + 1) ^—^ = 2 



quantities which may be chosen at pleasure. 



The general theory of the substitution which we have been considering, and the 

 various modes in which the orthogonal system just found may be formed, have been 

 developed very fully by Radau in a paper in Annales de VEcole Normale Superieure, 

 Tome V. (1868)/ We shall, therefore, at present confine ourselves to a brief indi- 

 cation of the special form of the substitution which has been found useful in 

 Celestial Mechanics. We first remark that if we form the (ra -\- 1) equations 



by giving i in succession all values from to ?i, we shall have by the theory of 

 orthogonal substitutions the (rt -(- 1) equations 



If we suppose in the first equations 



we shall have from (5) 



Vi = ^i^ 



whence, by substituting these values of Z; and ?/j in the second equation, we shall 

 have for the expression of Xi in terms of ^q, ^j, etc. to replace equation (7) 



Xi = 4" lo + ^' ^1 + - h + etc. (9) 



The first term of this expression is common to all the values of a',, representing, 

 as it does, the co-ordinates of the centre of gravity of the system. It may, there- 

 fore, be omitted entirely, when we seek only the relative co-ordinates of the vari- 

 ous bodies, and, in any case, it will disappear from the diff'erential equations of 

 motion. 



n{ii — 1) 

 The most simple way of forming the coefficients a^ is to suppose ^ of .them 



equal to zero. Let us first suppose a^- = whenever y > i, the first line, in which 

 % =z 0, being, of course, excepted. 



The orthogonal system will then be of the form 



' Sur une Transformation des Equations Diffcrentielles de la D3'namique. 



