TABER. — ON SINGULAR TRANSFORMATIONS. 587 



the infinitesimal transformation %a.j Xj is commutative with no other in- 

 finitesimal transformation of G. But, if r = 3 and this determinant van- 

 ishes, it is alway possible to find two distinct infinitesimal transformations 

 2", X; and 2 bj Xj which shall be commutative. 



Again, if r > 3, it is always possible to find two distinct infinitesimal 

 transformations of G which shall be commutative. 



The condition necessary and sufficient that two infinitesimal transfor- 

 mations 2 ctj Xj and 2 bj Xj shall be commutative is that 



(17) <f> a (b u b 2 , . . . b r ) = 0; 

 or, what is the same thing, that 



(18) </>„ (a u a 2 , . . . a r ) = 0. 



If A (( ^p 0, the necessary and sufficient condition that every transforma- 

 tion of the group G^ with a single parameter t shall be commutative 

 with T a , that is to say, that T a T lb = T tb T a for every value of t, is that 

 the infinitesimal transformations 2 Oj Xj and 2 b } X, shall be commutative. 



In certain groups G, whatever the transformation T a , provided A„ = 0, 

 it is always possible to find an infinitesmal transformation 2 bj Xj< not 

 commutative with 2 ctj Xj, which shall, nevertheless, generate a group 

 G7' 7) with a single parameter t, every transformation T tb of which shall 

 be commutative with T„. In other groups this is possible for certain 

 transformations T a for which A a = 0. 



As an example of the former we have the group 



v d „ d v d „ d 



Al==?1 ^' X * = X *d^' Xs = Xs d^> X ' = X *J^ 2 



For this group A a = if a l or a. 2 is an even multiple, not zero, of tt \/ — 1. Let 

 a v a s , a 4 be arbitrary, and a. 2 = 2w <y/^l, and let <i x b 3 — a 3 b x = 0. Then T„ T,b 

 = Ttb T a for all values of t; but 2 bj Xj is not commutative with 2 aj Xj unless 

 "■■ ! 'i — a i h = °- 1^ however, a\ — a lt a\ = 0, a' z = a 3 , a\ = 0, T a > — T a and 2 fy Ay 

 is commutative 2 a'j Xj. 



If A f , = 0, the necessary and sufficient condition that T„ T tb — T tb T a 

 for all values of t is 



(19) (e*« -I$b u b 2 , . . . b,.)=0. 



It is to be noted that this condition is always satisfied if 2 cij Xj and 

 2 bj Xj are commutative. For then 



0„ (&!, 6 2 , ... J,.) = 



. • . <f>\ (b x , b,, . . . 6 r ) = 



<£ 3 , ( (b u b,, . . . b,) = 



