■>S PROCEEDINGS OP TIIF AMERICAN ACADEMY. 



Consequently, if A a | 0, and every transformation of the group G{'" is 

 commutative with /',. the :il>o\<- condition is satisfied 



Lei 



(20) T a =T a ,. 



Then every transformation of the sub-group <i. is commutative with 

 7',. Therefore, if A„ £ 0, i '/'_, -V,. is commutative with 1 </ A',. Whence 

 it follows that 



(21) T a _ a ,= T a T_ a ,= T a T~J 



is the identical transformation. 



If, however, A„ = 0, it does not necessarily follow that - a* X and 

 ^ d'j J£j are commutative ; and therefore we do not necessarily have 



/tt rp rp— 1 



-*,! — «' — ■* a •» a' • 



E. g., let r — 5, and let 



Then A„ = if either a } or <j.j is an even multiple, not zero, of it /y/— 1. 



Let a x = 0, «o = 2 i * \/ — 1 4 1 0, «3 = 0, 



a'j = 2 k' -rr \/~l, o'j = 2 £ it <v/~l ^ 0, a' 6 = a 8l 

 where h and f arc integers. Tlien A fI = 0, A„< = ; and T„ = r a ». But 2 n, X, is 

 not commutative with 2 <;', A', unless a'^ = « 4 . Moreover T„ _ „ is not the identical 

 transformation (i. e., Y'„ _„» { '/'„ V,,- - ') unless a' A = <ij. 



When T a = 7',, and A„ = 0, it does not necessarily follow that A,, = 0. 

 Thus, in the case of the group just considered, if 



ily = '2k-TT v /~ 1 : <>. „., :::'// M /-l i 0, 



a\ = 0, «' 2 ;= 0, a' a = 0, a' 4 = 0, a' c = o 6 , 

 we have T a — T u ,, and A„ = ; but A,, 4= 0. 



§7. 



The equations of the general infinitesimal transformations of the ad- 

 joined group r of group (rare in Cayley's matrical notation 



(■-'•-'i (a',, a'* . . . a! r ) = (/ + St^ga,, a* . . . a P ). 



The successive application of tins infinitesimal transformation to the 



manifold a i} Og, . . . a r gives the general transformation of F, namely. 



(23) (»,".■',' i r w )= («**8«i,o f , . . . a r ). 



