588 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Consequently, if A„ 4^ 0, and every transformation of the group Ci**' is 

 commutative with 7^„, the above condition is satisfied. 



§6. 



Let 

 (20) 7; = T„.. 



Then every transformation of the sub-group G^i'"'' is commutative with 

 7^,. Therefore, if A„ 4^ 0, 2 a! ^ X^ is commutative with 2 a^ Xj. Whence 

 it follows that 



(21) J a - a' ^= -^ a ^ - a' ^ -'a -* a' 



is the identical transformation. 



If, however, A„ — 0, it does not necessarily follow that 2 Oj Xj and 

 2 a' J Xj are commutative ; and therefore we do not necessarily have 



rp rp rp— 1 



a — a' 



E. g., let r = 5, and let 



V— '^ V- ^ Tf- <^ It'- ^ A'— ^ 



"^i-^iJ7i' "^•^-''2^; ^3-'"«J7i' ^"-""3(^x7 ^^-'^*(^:r; 



Then A„ = if eitlier Oj or a^ is an even multiple, not zero, of ir /y/ — 1. 



Let oj = 0, ao = 2 ^- T vA^ 4: 0, rtg = 0, 



a'l = 2 k' TT /y/ — 1 , a'.2 = 2 A; TT y^ — 1 4^ 0, a '5 = fls, 

 where A: and ^•' are integers. Then A„ = 0, A,,/ = ; and Ta = T^a'- But 2 «;■ X,- is 

 not commutative with 2 a'j Xj unless (('4 = a^. Moreover Ta- a' is not the identical 

 transformation (i. e., Ta-a< 4= Ta Ta'~^) unless a\ — a^. 



When T,^ = T„, and A„ = 0, it does not necessarily follow that A„, = 0. 

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



«i = 2k7r ^/^-^ ^ 0, a2 = 2 k' TT ^^ ^ 0, 

 a\ = 0, a',, = 0, a'a = 0, a\ = 0, a\ — 05, 

 we have T^ — ?„-, and A„ = ; but A^, ^ 0. 



§7. 



The equations of the general infinitesimal transformations of the ad- 

 joined group r of group G are in Cayley's matrical notation 



(22) (d'l, a\, . . . a',) = (I + 8<(^aO«n ^2, • • • «r)- 



The successive application of this infinitesimal transformation to the 

 manifold cii, a^, . . . a^ gives the general transformation of F, namely, 



(23) (a/i', «,", . . . a,^^) = (e<^«0^i, <t., . . . «,). 



