If the Jacobi bracket-expression, constructed for all combinations of 

 i, j, is equivalent to a linear function of the symbols Uk/ with constant 

 coefficients, then are the symbols 



Uk /' = fk (X, y) p -f vk (x, y) q, k = 1, 2, ... r, 



the infinitesimal transformations of an 7--parjimeter group.* 



5. TJie Extended Group. An infinitesimal transformation 



U/EEUx,y) [||"] + ''(-^'y)lcryJ 

 may be extended in two ways. In the first place, the variation of the coor- 

 dinates of n points is simply the sum of the variations of the coordi- 

 nates of the separate points; hence, U /"extended in this manner becomes' 



(A). U/n= ^""1 fk (xk, yk) f-^^-''- 1 + // (xk, yk) f//-l 1. 

 k— 1 ( I dxk J ^d yk ' j 



The symbol U/may also be extended so as to include the variation of 



y' = t—' y" = ^2' • • • • ' y"" = ^- We have 

 d x d x^ d x° 



S X =z c (X, y) rf t, (5 y = '/ (x, y) (5 t. 



,5 / ^ jj dy ^ dx rf dy — dy rf dx _ d fi y — j'Ad x 

 dx dx2 dx 



= {||-y'^}'5t= |,, + y^(;;y-f,)_y'2Cy| 



dx dx J 



= '/ (X, y, yO '^ t 

 In a similar manner. 



rft 



5 y" = \^-j"^].6t = >," (X, y, y', y^O '^ t. 



dx dx J 



and so on for higher variations. 



The infinitesimal transformation U/ extended to include these higher 

 variations becomes 



Each of the members of an r-parameter group Uk/, k := 1, 2, ... r, may 

 be extended, giving the infinitesimal transformations of the coordinates of 

 n points as indicated by equation (A); or each may be extended as in 

 (B) to include the variations of x, y ,y', y^', j^^\ ... y'"'. A group of 

 transformations extended in style of (A) or (B) is called an extended group. 



'''Lie — Scheft'ers, Continuierliche Grupijen, p. 390. 



