TABER. — ON SINGULAR TRANSFORMATIONS. 585 



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



equation of the matrix 6*^^*6**" is equal to unity, irrespective of the value 

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

 equation is equal to unity, 7^,j7a is non-singular for every value of t. 

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



roots of the characteristic equation of the matrix e*^''-' e^°- are equal to 

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

 7\ = T,t, Ta 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 a^ Xj of group G, where the a's are 

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

 independent infinitesimal transformations X^, X2, . . . X,. which gene- 

 rate G. The r infinitesimal transformations 



ai"--'Xi -f a^^^X^ + . . . + a/*' Z,(^ = 1, 2, . . . r) 



are independent if the determinant 



(A) 



to. 



Any r independent infinitesimal transformation derived linearly from the 

 ^'s also generates group G and may be substituted for the ^'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 injinitesimale 

 Tra7isformation.-\ In what follows it will be termed an exti-aordinary in- 

 Jinitesimal transformation. 



Let G contain just s independent extraordinary infinitesimal transfor- 

 mation. In this case, from what has been said, we may suppose the A''3 

 so chosen that 



Xj X^ = X^ Xj 



(j=l,2, . . . s fc = l,2, . . . r), 

 but that 



* Lie: Transformationsgruppen, I. p. 276. 

 t Lie : Continuinerliche Gruppen, p. 465. 



