and their Index Symbol. 131 



to (1), namely, 



F(V).U=F(0)« + F(1)« 1 +F(2)m,+ &c. +~F{m)u m . ... (2) 



As an example, let the result of the operation of a v on U be 

 investigated. Then 



ffl V . V = u + au l + d 2 u 2 + &c. +a m u m ; 



and the interpretation of this result is readily seen to be, that 



the operation of a v upon the mixed rational function U converts 



the several variables x, y, z, &c. throughout it into ax, ay, az, &c. 



If U be supposed to contain the two distinct sets of variables 



x, y, z, &c, 



a, b, c, &c, 

 it may be exhibited in either of the two forms 

 U = M + Mi+w 2 + &c. +u m , 

 U=v -frv 1 +Vc t + &c. +v fl . 



As _*, d d d a 



V = X ^ + ^d-y +Z dz +kC -' 



S0let n««£+*i+*i*** 



da do dc 



Then, since these two symbols are commutative with each other, 

 we have 



4>(V) .¥(n) . U=¥(n) . 4>(V) • U, 

 whence the theorem 



4>(V).{^(0)7- + ^(l)?; 1 + &c.}=^(n).{4>(0)« +<I>(l)M 1 +&c.}. 

 3. Since y, z, &c. are constant relative to x, and therefore 



-=-, -r, &c. commutative, writing 

 dx dy 



d* d 2 d* 2 



A3 ,13 A3 A3 



*>'*& +*i? + &c ' + 3A ^¥ +9 *sv + &c " 



&c, 

 we have 



V(V-1) = V 2 , 

 V(V-1)(V-2) = V 3J 

 &c, 

 and generally 



V(V-l)...(V-» + l) = V„, .... (3) 

 K2 



