y. . (6) 



of a Finite Conservative System. obi 



and, in virtue of this and of (3) and (1), (2) becomes 

 A(o^, *, & . . . , 0, 0', %', • • •) = A Oh 0, 0, ... 0, 0, 0) 



+ l[ll0 2 + 22% 2 +33$ 2 + 440 /2 + 55%' 2 + 66$' 2 



+ 2(i20% + i30$ + u00' + i50%' + 16 y 



23%$ + 24%0'+25%%' + 26%$' 

 + 34$0' + 35$%' + 36$$' 

 + 450'%' + 460'$' 



+ 56%'$')] 



where, merely for simplicity of notation, we suppose the total 

 number of freedoms of the system, that is to say the total 

 number of the coordinates -^, 0, %, $, to be four ; and for 

 brevity put 



.(i)=». .(&)-* .(i)-^ 



27. From (6) we find, by (1), 



£ = 110 + 12% + 13$ + 140' + 15%' + 16$' "1 

 t) = 210 + 22% + 23$ + 240' + 25%' + 26$' 

 f = 310 + 32% + 33$ + 340' + 35%' + 36$' 



— f = 410 + 42% + 43$ + 440' + 45%' + 46$' 



— rf = 510 + 52% + 53$ + 540' + 55%' + 56$' 



— ?' = 610 + 62% + 63$ + 640' + 65%' + 66$' 



(7). 



> 



(8) 



These equations allow us to determine the three displacements, 

 0, %, $, and the three corresponding momentums, f , 77, £, for 

 any position on the path, in terms of the initial values 0', %', $', 

 £', ?/, f, supposed known. 



28. To introduce now our supposition (§ 24) that F' P is 

 part of a periodic path ; let Q be a position on it between P / 

 and P; and let us now, to avoid ambiguity, call it P' Q P. 

 Let P' and P now be taken to coincide in a position which 

 we shall call O; in other words, let P'Q P, or OQO, be the 

 complete periodic circuit, or orbit as we have called it (§ 2 

 above). Our path P'P is now a path infinitely near to this 

 orbit, and P' and P are two consecutive positions in it for 

 which i/r has the value zero. These two positions are infinitely 

 near to one another and to O. We shall call them Of, and 

 Oj+i, considering them as the positions on our path in which 

 1/r is zero for the iih. time and for the (i + l)th time, from an 

 earlier initial epoch than first passage through i|r = which we 



