4 GENERAL INTEGRALS OF PLANETARY MOTION. 



The letter iS being used to express the sum of an infinite series of similar terms ; 

 Jc, i, anciy having the signification just expressed, and each system of values of the 

 integers i and^ being subjected to the condition 



^'l + *2 + 4 + • • • • + iin = 1 ,„Y 



It is evident that when x, y, and z are expressed in this form, any entire func- 

 tion of these quantities will reduce itself to the same form. 



We shall now proceed to show that the form (3) is a general one : that is to say, 

 that having an approximate solution of this form, if Ave make further approxima- 

 tions, developed in powers of the errors of this first solution, every approximation 

 can be expressed in the form (3). 



We can make no general determination of the limits Avithin which these approxi- 

 mations will be convergent, we are therefore obliged to assume their convergency. 



§ 2. Canonical Transformation of the Equations of Motion. 



If Ave put 



n, the potential of the n -\- 1 bodies, that is, the sum of the products of every 

 pair of masses divided by their mutual distance, the difterential equations of 

 motion Avill be 3(ri + 1) '™- n^imber, each of the form 



' df dXi ' 



If Ave substitute for the co-ordinates themselves their products by the square 

 roots of their masses, putting 



x^=^m\oCi; Yi^ mfy I, etc., 

 the difterential equations Avill assume the canonical form 



d^Xi__ d£l ,.^ 



'W~~dxf ^ 



We suppose the index i to assume for each of the three co-ordinates all values 

 from to 11, the value referring to the sun, and we thus have 3(ft + 1) equations 

 of the form (4) the integration of Avhich Avill give the co-ordinates in terras of the 

 time, and 6(n + 1) arbitrary constants. 



We shall noAV diminish the number of variables to be determined in the follow- 

 ing general manner: Suppose that Ave have m difi'erential equations of the first 

 order, betAveen m variables and the time t, each being of the form 



dt ^ ^ •■ 

 vSuppose also that Ave have found h integrals of these equations, each of the form 



f{x^,X2, .... x^,t) = constant. 

 Let us assume at pleasure m — Ic other independent functions of the variables, 

 each of the form 



