TABER, — ON SINGULAR TRANSFORMATIONS. 587 



the infinitesimal transformation '^aj Xj is commutative with no other in- 

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

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

 2cf, Xj 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 a J Xj and 2 bj Xj shall be commutative is that 



(17) </>„(^i, ^o, . . . (i,) =0; 

 or, what is the same thing, that 



(18) cf>f, («i, a.2, . . . a^) = 0. 



If A„ 4= 0> tl^6 necessary and sufficient condition that every transforma- 

 tion of the group Ci**' with a single parameter t shall be commutative 

 with Ta, that is to say, that 7\ T,,^ = T^f^ T^ for every value of t, is tliat 

 the infinitesimal transformations 2 cij Xj and 2 b^ Xj shall be commutative. 



In certain groups G, whatever the transformation T^, provided i\a = O5 

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

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

 Ci'*' with a single parameter t, every transformation J'^j of which shall 

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

 transformations 7^, for which A„ = 0. 



As an example of tlie former we have the group 



V — ^ \- — ^ V _ ^ V — ^ 



^^-'■^cTTi' ^^2--i-2^^, A3_^3^^, A,_X3^^ 



For this group A,, = if a^ or (i., is an even multiple, not zero, of tt /^— 1. Let 

 «i. «3. «4 he arbitrary, and a., = "J tt \/ —1, and let n/ig — og hj^ = 0. Then Ta Ttb 

 = TtbTa ior all values of t; but 'S.hjXjXS not commutative with 2 a> A7 unless 

 "2 64 — «4 ^2 = 0. If, however, a\ — a-y, a'. 2 — 0, a'g = a^, a\ — 0, Ta' = Ta and 2 hj Xj 

 is commutative 2 a'j Xj. 



If A„ = 0, the necessary and sufficient condition that T^ T^f, = Ttt, T^ 

 for all values of t is 



(19) (A - /(J^i, 5,, . . . 5,) =0. 



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

 2 bj Xj are commutative. For then 



Cf>a (5i, b,, . . . b,) = 



.•.4>\{b,,b,, . . . b,) = {) 



^\ (b,, b„. . . b,) = 



