TABER. — ON SINGULAR TRANSFORMATIONS. 585 



It will be found that one or more of the roots of the characteristic 



equation of the matrix e^ tb e™ is equal to unity, irrespective of the value 

 of t. If T a is non-singular, and for every value of t each root of this 

 equation is equal to unity, T tb T & is non-singular for every value of t. 

 Let it be assumed that Ta is non-singular, and that just 5 < r of the 



roots of the characteristic equation of the matrix e^ lb e^ a are equal to 

 uuity, irrespective of the value of t. Then the values of t for which 

 T a =■ T tb T a is singular (essentially or non-essentially) are included 

 among those for which one, or more, of the remaining r-s roots of this 

 equation is equal to unity. 



§4. 



The infinitesimal transformation 2 cij Xj of group G, where the «'s are 

 quantities independent of the x's, is said to be derived lineally from the r 

 independent infinitesimal transformations X u %>, . . . X r which gene- 

 rate G. The r infinitesimal transformations 



a^ Xi + an'* 1 X„ + . . . + a r w ^ (k = 1, 2, ... r) 



are independent if the determinant 



a;<*» 



0. 



(j,k=l,2,. ..r) 



Any r independent infinitesimal transformation derived linearly from the 

 Xs also generates group G and may be substituted for the X's.* 



Group G may contain an infinitesimal transformation 2 a, Xj commu- 

 tative with each of the r infinitesimal transformations Xj which generate 

 G, and, therefore, commutative with every infinitesimal transformation 

 of G. Such a transformation Lie terms an ausgezeichnete infinitesimale 

 Transformation.^ In what follows it will be termed an extraordinary in- 

 finitesimal transformation. 



Let G contain just s independent extraordinary infinitesimal transfor- 

 mation. In this case, from what has been said, we may suppose the Xs 

 so chosen that 



(j =1,2,..'. s k = 1, 2, . . . r), 



but that 



* Lie : Transformationsgruppen, I. p. 276. 

 t Lie : Continuinerliche Gruppen, p. 465. 



