546 Mr. C. G, Darwin on the 



ure at (# 1? j/i) (.i- 2 , y 2 ) a ^ any time. Then from (11) we 

 have 



L = im(.tV + yi a ) + iM0V + ^ 8 ) + g^OV + 2/i 2 ) 2 



+ ^j 2 M(x 2 2 + ;y 2 2 ) 2 + — - 2^3 1 r- 2 ^2 + 2/1.V2) 



+ [>i(.i'i-^ 2 ) 4- 2/1(^1-^2)] [^2(^1-^2) -r# 2 (yi— #2)] J • 



• • . (U) 

 The first transformation is 



^ 1= :X + Mf/(M + m), # 3 = X-mf/(M + ??i), 



with similar expressions for y 1? y 2 . Then X, Y may con- 

 veniently be called the centroid, though except for low 

 velocities it has none of the properties ordinarily associated 

 with the name. Then 



+ ^(M +m )(XHt V+ ^^(iX + ^ 



+ 2C 2 (M + m) 2 lt + v M? - " 



1 M»i(M ! -M.» + i» ! ) ,£. , .... , Ee 



+ 8(? - - ( ^ rlB) . - (£+*?) + - 



-p^^t-(r + ^)+(^ + ^)]} (15) 



As r 2 = f 2 + 77 2 , X and Y do not occur explicitly in (15), and 

 so we have integrals 



— - =» x —T=p (lb) 



dX l BY ly 



