SLOCUM. — FINITE CONTINUOUS GROUPS. 99 



Let now 



X'i =fi( X l ■ • • X ni «1 • • • «,) = eax i 



(i = 1, 2 . . . n), 



and 



x" i ^f i {x\ . . . *'„, b x . . . 6 P )=ePV, 

 (i = 1, 2 . . . »), 



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 



X "i =f i (x\ . . . X' n , by . . . b r ) 



= e«/i(*i ■ • • *»> *i ' • ■ b >) = ea C^*') 



(i = 1, 2 . . . n). 



Let the operator e a e& be defined as follows : 

 (e« e 0)/ 

 = (1 + (a + /8) + i(a 2 + 2aj3 + /Q a )+ i(a»+ 3a 2 /3+ 3a/3 2 + /3 3 ) + . . .)/ 



2! v ' ' ' r/ ' 3! 



^/+(a + iS)/+^(a 2 4-2a^4-^)/+^(a 3 +3a^ + 3a/3 2 +^)/+... 



Then 



(6 a e^) a^ = e a (e^) 



(i = 1, 2 . . . n), 

 and therefore 



x"; = (e a e&) x { 



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



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

 transformations denoted by e a and e* 3 .* 



By § 2, page 94, the transformation inverse to e a is e~ a . Let S* 

 denote an infinitesimal constant. Since the transformation e a + s 'y is 

 infinitely near the transformation e a , the transformation e~ °- e a + s/ y 

 is an infinitesimal transformation. If we denote its parameters by 8tb x , 

 ?>tb 2 . . . 8 t b r , we have 



e~ a e a + & 'y 



11 1 



= 1 + 8t{ 7 --(a,y)+-(a,(a,y)) - -, (a, (a, (a, y))) + . . .} +... 



= e 5tp — i + g j £ + . . . 



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

 imals of the second and higher orders, we have 



(26) = y - ^ (a, y) + ^ («, (a, y)) - ^ (a, (a, (a, y))) + . . - 



* Cf. Campbell: Proc. London Math. Soc, XXVIII. 381-390. Also Point-are; 

 Comptes Rendus, Mai l er , 1899. 



