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EEV. E. HAELEY ON THE METHOD OE STMMETEIC PEODTJCTS. 
13. Fixing two of the roots, and and permuting the others, the following cycles 
arise, viz. : — 
Six values of r will then be included in the formula 
to 6 
and the other values, which are in fact the corresponding values of r', will be included 
in the fofmulse, 
^ 1 , 2 , 3, 6 and ^ 4 , s ‘^ 1 ^ 4 * 
The values of X are as follows : — 
^2 ■“^2^1 (^2^4 “h ^ 3 ^ 5 }? ^5 x^x^J^ 
X 3 1 2'^5 ~1“ ^ 3 ^ 4 ) ? ^6 ~ ^6^ i (^ 2^3 ~H ^ 
And by Theorem III., art. 9, 
a result which may be verified by actual development and summation. By Theorem II. 
{^i(‘^2'^5~1~ ^3^4)^1^i(^2'^5~1“ ^ 3 ^ 4 ) } 
— > I /y* 'T* - 1— ^ <v \ S < 7*^1 Of* Of* I ft* ^ \ -I— /y» /y» _!— fv* ^ \ 
— -Wi|_ 3*^/4 J \vt/ *^3*^4/ *^'4*^5 J 
+ /Y*^l /y* /y* - 1— /y» /y» \ -I— /v^f /y» /y» -!_ on /y* \ I— Of*^f ^ ^Y* -L /y* /y» ill* 
*^3\‘^4*^2 U/gU/jy"]" w 4 ^a/ 5 tA/ 3 "]“ *^5\*^l*^4n^ ‘^2*^3y/J > 
or, multiplying out and reducing by the aid of Theorem I. *, we have 
Xi = X'l { (x\xlxl 4- xtaf^afi) + 2 (xlxlXaX^ + x^xlx^x^) 
1 /jr^ /Y*^ /v* J_ /y*3/Tri2^2^ _ /y^/y /yi 2 /M 2 /yt3/y, /y^ry^'W 
U/ ja/2*^3*^4 1 w 2 *A' 4 ^ 1 / 5 w ja/2‘^3*^5 ^r*^l*^3*^4‘^6y J 
I O ’’V /yt*^ /v^ /y» /y» /y* _ L. O /y^ /y^ /y»2 /y» /yi • 
I 46ii<ti/ j w 2 w 3 tt/ 4 M/ g I ^ |(A/ 2 ^ 3 tV 4 vvg 9 
* In effecting reductions by the cyclical process, the greatest exponent is made the leading exponent. 
