motions in general dynamics 



183 



where, corresponding to Wj = 0, x^^, x^ are such as to satisfy 



A -f 



ry~'~ \J ^ lOX Jb-% X-l ^ Xo tZ/g ^ 



tu^ is such that the function [F-^, of x-^, x^, w^, obtained by picking 

 out the part of F^ depending on x-^, x^, y^ only and substituting 

 x^, x.^, w, for iCj, »2> Vi, tas a maximum or minimum value, namely 

 vj^ is such that 



9c7j 



= 0, 



and A-^, A2, B^ are determined, namely by hnear equations whose 

 determinant is 



d^F. 



d^F. 



8xi02 ' axi"aa;2« 



d^F. 



a-^^n 



82 [i?^j 



dxj^dx^^' dx^^^ 



so that neither of these factors may be zero, while cjg + ^2 ^^ 

 arbitrary. In other words, the origin of time-reckoning is arbitrary, 

 and either the initial value of X2 or its corresponding mean motion. 

 If Xi - <^i (0, ^2 = </.2 (0, yi = "Ai (0, ^2 = 'A2 (0 be the equations 

 of such a periodic motion, and we substitute in the differential 

 equations 



dXj. dF dyr _ dF 

 dt dy^ ' dt dXf ' 



supposing F = Fq+ fjiF^, 



values X^ - (^,. (t) + ^r, Vr = ^r (0 + Ir, 



and retain only first powers of the small increments |^, -q^, we have 

 a system of linear differential equations with periodic coefficients 

 (of period ^ttJu^) 



di 



~^l = iVx, «i) ^1 + (^1. x^) ^2 + (^1' Vi) Vi + (^1' 2/2) ^2. 



d^ 



£ = iy2, ^1) ^1 + (^2. 2^2) I2 + (^2' 2/1) ^1 + (.^2. ^2) ^2» 



= - (Xi, x^) li - («i, a'2) I2 - (^1' ^i) Vi - i^v ^2) V2^ 



= ~ (^2> ^1) SI ~" (*2' *"2) fe2 ~ (^25 ^l) ''71 "" (^25 ¥2) V2i 



d^F 



dt 

 dt 



where (w„, a;,) denotes the value of ^ ^^^ for 



x,= <j>,{t), y, = xl},{t)\ 

 these values of x^, y^ are of the forms x^9 + /xX^, n^t + w^-\- fx Y^.. 

 Since i'',, does not contain y^, y^ all of the sixteen coefficients are 



