88 



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 f ^ 5k (X, y) p -i- //k (X, y) q, k = 1, 2, ... r, 



the infinitesimal transformations of an /--parameter group.* 



5. The Extended Group. An infinitesimal transformation 



may be extended in two ways. In tlie 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). VU= s"""] ;-k (xk, yk) f-^K '^ (^'" y") I ^ ' 1- 

 k = 1 i ^ d xk J L d yk ' J 



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



y' = ^, y" -= ^iZ, . . . . , yf"' = ^-^. We have 

 "^ d X d x2 d x° 



f5 X = f (x, y; f$ t, (5 y = /; (x, y) fS t. 



5 / _ A dy dx f$ dy — dy <' dx _ d f5 y — y'drf x 



' ^ ~ d^ dx2 dx 



= .^^_y'^\,U= [;;., + y'(,y-,^.)-y'2fy|dt 



•^ dx dx J ' J 



= '/ (x, y, yO '5 f. 



In a similar manner, 



,S y" = { il' — y" ii \ rf t = ;/" (x, y, y', y^') r5 t, 

 i^ ds 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 /--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\ j" , j"' , . . . y^"'. A group of 

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



■'Lie— Scheffers, Continuierliche Gruppen, p. 390. 



