GENERAL INTEGRALS OP PLANETARY MOTION 5 



so that the m variables x can be expressed as a function of Ic arbitrary constants, 

 the time t, and the m — k variables 



^l!^25 • • • • C,m—k' 



Differentiating the above expression for ^i, and substituting for ^ its value X, 

 we shall have 



d^i _ ^<pi I x^ 4- x^ + 4- X -^ 



clt 6t ^ 6x^ "" 6x2 '" dx^ 



By substituting for the tc's in the right hand side of this equation their expres- 

 sions in terras of ^1, ... . ^„j_j., t, and the arbitrary constants, we shall have the 

 problem reduced to the integration df m — h equations between that number of 

 variables. 



In the special problem now under consideration, the m variables are the co- 

 ordinates X, ?/, z, and their first derivatives with respect to the time. The integrals 

 by which we shall seek to reduce the number of the variables are those of the con- 

 servation of the centre of gravity. We shall take for ^1, ^2? etc., linear functions 

 of Xi, Xn,, etc., so chosen that the reduced equations shall maintain the canonical 

 form. Let us take tlie ?i -|- 1 linear functions of the co-ordinates x: — 



^0 = a -[- 5i! = aooaJo + aoi*'! + + o.on^n 



^1 = aio^r,, + aniCi + + ai„a:;„ (5) 



where we have put for symmetry 



171- 



m^ = caoi, or ttoj = — ^, ' (6) 



c being an arbitrary coefficient, while the other coefficients are to be chosen, so 

 that the resulting differential equations shall be of the canonical form. Let us 

 represent the values of x which we obtain from these equations by 



Differentiating any one of the preceding expressions for ^, and substituting for 



d'^x 



-,-0" its value, we hq,ve 



df ' 



df ~^ m^ dx^ m-^ dx^ ^n S^n ' 



If we suppose x^, x^, etc., replaced by their expressions in ^g, ^1, etc., obtained 

 by solving the equations (5), that is, by their values in (7), we shall have 



da _ do. do. do. 



^"""^■^ +"''■ (5,^1 + ^""■'' d^,: 



Substituting these values in the preceding equation, it becomes 



