EEY. E, HAELEY 01^ THE METHOD OE STAIMETEIO PEODUCTS. 
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sions without rendering it necessary to examine all the results of multiplication, or to 
make a hazardous selection of those which may he deemed material. Combining as 
many terms (fi) in one as there are roots involved, the cyclical symbol reduces the labour 
71 ”• 1 . ' . 
of multiplication alone by ~^th ; and in passing from the circular to the corresponding 
symmetric function the process also affords considerable facilities. 
8. Let ^(^(0) be a function of x, and let %(y) be derived from %(0) by advancing each 
of the roots contained in its q steps in a given cycle r. Ex. gr. Suppose that 
and that we are following the first cycle, which, indeed, may be taken / 
as a type of all others, and is found in actual practice to be the most 1 
easy with which to operate. Then, according to the definition, 

%(2) — x^x^, 
— . 
y^{n) =x,x^=x(0), 
Or again, if we suppose n=5, and 
Z( ^ (^’2^5 “1" ^ 3 ^ 4 } ? 
then, following the same cycle, we have 
yXl)=a^^(XyX^-]-x,x,), 
)^( 2 ) — xXx^X2 “1“ x^Xj^ , 
yX^)=o[^X^t\X2+x,X2), 
%('5)=‘'PK^2'^5 + ^3^4) = %(0), 
and so on. 
Next, let Srx(O) or, more simply, represent the circular function 
>i(0)+%(l)+ . . . +x(n—2)+x{n—l). 
Then, since each root recurs at every wth step in the cycle, we have 
z{n) =%(0), 
x(n+l)=yXl), 
y:{n+q)=x{q); 
* When there is only one cycle involved in the operation, or when there is no comparison of cycles, it is 
not necessary to snfiix S'. 
