EEV. E. HAELET ON THE JIETHOD OE STMIMETEIC PEODIJCTS. 
335 
or irrational, integral or fi-actional. But the following holds only for those circular 
functions which are rational and integral. 
Theoeem III. If X be a circular function of the form 
2'{%a+Zb+:^+&c.}, 
% being defined by 
. . . 4 +a'" &c., 
where a, j3, 7 , . . . | are positive integers or (some of them) zero, and m is the number of 
values of • 4 or (what is the same thing) the number of terms contained in 
• • • 4 ? then will 
2X=«»'(j2>i+l2z,+j%+&c.), 
w' being the number of values of X, and 2%^ being of the form 
(a'+a"+a"'+&c.) 2 < 44 ... 4- 
For if, as we are permitted, we fix one of the roots, and permute the remaining n—\ 
roots in all possible ways, there will arise 1.2.3... [n — I), or (say) p, corresponding 
cycles ; and if, for the moment, we represent by 2 the sum of the p expressions formed 
by applpng these cycles to any one of the values of X, then, since 2' consists of w, and 
consequently 22 ' of np, expressions of the form 
%a + )t:b + %c + &C., . 
and smce also m=np^ or a submultiple of np^ it follows that 
But 2 X=t 2X. "Whence the theorem. 
n 
9. The symbol 2' admits of an easy extension to functions of the form 
/’"(f)=4‘+f4+f"4+ . .. ’4^ 
being an wth root of unity, real or imaginary. For, if we represent this function by 
2 ^, we shall have 
or 
2' 2' (a^’”2' = 2' (^"2' ,/r”*), 
where 2 ' is the simple or ordinary cyclical function to the first cycle 
...123...(?i-l)I23...(w-l)... 
This may be proved thus : — 
and 
* The idea of extending S' to functions of this form was suggested to me by Mr. Cockle, in a letter 
under date January 8, 1859. 
MDCCCLXI. 3 A 
