SLOCUM. — FINITE CONTINUOUS GROUPS. 99 



Let now 



a." j ^ji\^i . . . x,^, Oi . . . a^) = e"- Xf 

 {i=l,2... n), 

 and 



x'\=f,{x\ . . . x'„,b, . . . K) =e^'x\ 

 = 1, 2 . . . n), 



where /3' denotes the result of substituting the accented for the unaccented 

 variables in the X's which appear in the operator (3. Then, by what 

 precedes, we have 



^"i =fi {x'l ■ ■ • a^'«' ^1 • • . K) 



= e-^fiixi . . . x„, ii . . . b,) = e« (e^x,) 

 (l = \,2... n). 



Let the operator e"e^ be defined as follows : 



(e«e3)/ 



= (1 + (a + /i) + ^-,(a^+ 2a/? + /3'0+ ^,(aH 3a^/3 + 3a/3H^') + . . .)/ 



Then 



(e^e^) Xi — e" (e^iCt) 

 (/ = !, 2 . . . n), 

 and therefore 



x"i = (e" e^) Xi 



(i=l, 2 . . . H); 

 thus e"e^ denotes the result of the composition in the order named of the 

 transformations denoted by e" and e^.* 



By § 2, page 94, the transformation inverse to e'^ is e~«. Let 8 t 

 denote an infinitesimal constant. Since the transformation e°- + ^'v is 

 infinitely near the transformation e», the transformation e--"e* + *'Y 

 is an infinitesimal transformation. If we denote its parameters by Stbi, 

 ^tbj . . . 8 t b,., we have 



g — a ga + Sly 



= I + 8t{y-^(a, y) +^ (a, (a, t)) - ^; («, («, («, y))) + ..•}+... 



=ze«'^= 1 + Bt(3 + . . . 



in wliich (a, y) denotes the alternant ay — ya ; and neglecting infinites- 

 imals of the second and higher orders, we have 



(26) /3 = y — 2", («' y) + 3] ('^' («' y)) — 4] («» («' («^ y))) + • • • 



* Cf. Campbell: Proc. London Matii. Soc, XXVIII. 381-390. Also Polncarc: 

 Comptes Rendus, Mai l", 1899. 



