334 
BEV. E. HAELEY ON THE IMETHOD OE STMMETEIC PEODHCTS. 
and therefore 
Similarly, 
^'pc(i)= +?c(w) 
= {%(l)+%(2)+ . . . 1)} +%(0) 
=S'%(0). 
5:'%(2)=^'x(1)=^'%(0), 
and by generalization and induction, 
Hence 
Theokem I. — A circular function is not affected in value by the simultaneous advancing 
or receding of the roots which it contains any number of steps in the cycle to which it 
belongs. 
Whence it follows that ^'^ 2 ( 0)5 oi’ its equivalent, 
iXi(0)+Xi(l)+ • • • +Xi(^^~2)+Xi(^^~l)} -^XaC*^)’ 
may be written 
Xi(*^)-^^X2(0)+Xi(1)-^^X2(1)+ • • • +Xi(^~2).!^'x2(»^— 2)+Xi(w~1)-^'x2(^^“-1)5 
which is equal to 
X'{x.(0).^X2(0)}; 
and by simply interchanging Xi and X 25 we have also 
2'x,(0).2'x,(0)=2:'{xa(0).S'x,(0)}. 
"VMiich gives us 
Theorem II. — The product of two circular functions belonging to the same cycle is 
itself a circular function to that cycle, and is given by the application of the cyclical 
symbol to the product of either function into the initial or leading terms of the other. 
It should be remarked that %' is a symbol of cyclical operation, and subject to the 
same laws, with certain obvious limitations as if it were a symbol of quantity. Thus 
S'{x.(0)+x»(0)+ &c.}=S'x.(0)+S'x.(0)+ &C. : 
and, in general, if we develope 
{Xi(^)+X2(^)+^<^-}™ 
by the multinomial theorem, and then apply the symbol to each term, the result wdl 
be equal to 
^^(Xi(^)"f~X2(^)“l“ 
It should also be remarked that if C be a function or quantity such that it is not 
affected by the cyclical interchange of the roots, as when it is a constant quantity and 
therefore independent of the roots altogether, or as when it is a circular function and 
belongs to the same cycle as Xx(^)’ then will 
^'Cx(0)=crx(0). 
The foregoing theorems are true for any circular functions whatever, whether rational 
