EEV. E. HAELET ON THE IMETHOD OE SYMMETEIC PEOHTJCTS. 
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and 
and therefore to 
which, when m=l, coincides (as it ought to do) with the above result. 
11. The expansion of X'x’^x^X'x’l^x^ niay, by Theorems I. and II., be effected thus, — 
I 
^ t/C 2 U /2 ^ tX/ 2 »<v 3 ^ ^2 V*^l *^3 iX /2 U /3 U/g ^ i ^ *^2 / 
/y^m /y,D7l /y^n I /y»27yt /yyT^ I /y*^ /yt^ mITI /mSTW/mW^ _l /}^2?7Z^W2 /mW2 \ 
■ I \ 1 2 3 ' I iX- 2 3 5 I *X /2 *X' 2 ^' 3 «X/^ | tX/ 2 tX^ 2 4 | <X/ 2 iX/ ^ tX- 5 I 
/y»2W2/ /ytffl .ytJfl I rvt^ nrJ!^ I ryiTfl /%-jn /y^^/y»^\ /v»^^ /v»^ /v*^ • 
— w 2 I <^2 *^3 "^”*^2 *^4 ' I iX/g tX/g "l” tX/j J ^^tX' 2»X/2‘X/3tX/4 y 
or since 
/yPTl /y%tT^ /vi^ /y»^ /y*^ /y*^ /y»^ a-j^Wi /y*^ /y*^ fy*^ /y»^ ^ /y^Tfl MfTl /y\TXl 
^ 0^2 ^•^2 *^3 * 1 ” tX ^2 ‘'^4 “l *'^2*^5 ”l tV 3 tX/^ ”p“ 1 X /3 iX '4 tX /5 J ^tX /2 ^^2 *^3 9 
• l ^f^2m/’ ^m \ ^m^m\ 
• • 4 ^ iX' 2 2 1 3 ' “f«X/ 2 w 2 w 3 j 2 2 3 4 """ w 2 I *X/ 2 5 f tX 3 tX 4 / ♦ 
Consequently 
/'"((y) •f’"(^^)-f’"(‘^^) .f’”{<!>)*) = {Xx^”‘f — Xx^"‘%T'^X2 
I /y^TTi /yjn /yiJTl ytffi /yxftX \2 1 fC A^Wl ,y»W / mWI / y>?M /y^2Wt/ Mffl /y^Yl /y*f^ ^yjfl\ • 
”l* v 2 w 2 *X' 3 1 M^«X 2 w 2 y ' | ' X/ ^^t.v 2 w 2 3 4 ^ * 4 / 2 V •X' 2 ^X/ 3 | tX 3 w 4 y j 
which, making m=l, gives the resolvent product for quintics <r^{x) 
= b'Zxlx^^ — ( 2 ^ 1 . 172 )®+ bSx^x^x^x^ — 6 %'a^i(x 2 Xi-{-XsX^) ; 
or, expressing the symmetric portion of this value in terms of the coefficients of the 
quintic, and wilting X for the unsymmetric portion 'S'x%X 2 X^-\-X 3 Xi}, we have 
-rj(a:) = ^ ( — 1 5aV+ ^a^hd + — 'baTfc + ¥) — hX. 
I remark also that 
rri = '^x^x^x^x^ = Xxlx^Xs + Xx^x^^x^ — X = -^ ( — ?>ae ■^hd)—X. 
12. The number of unequal values of the several functions r, r', rri, X and ’ttJ^x) may 
be determined by the following considerations. Since the coefficients of the several 
terms in the expression for r are equal, we may, in forming its values, regard one of the 
roots as fixed, while the others are permuted inter se. Thus we shall have 1 . 2 . 3 . 4 
cycles, giving rise to 24 corresponding exjiressions for r; but since, for the first cycle, 
r=%'XiX2=%'x,X^, 
and similar relations obtain for the other cycles, therefore these 24 expressions may be 
grouped in pairs, the members of each pair being equal. Hence r has only 12 values. 
In the same way it may be shown that r' has only 12 values. And since the several 
values of r' may be referred to the same cycles which arise in the formation of the values 
of r, and since also r and ri are complementary to each other (for their sum ^XXiX., a 
one-valued function), it follows that rri, and therefore also X and '^^{x), are six-valued 
functions. An independent proof that the resolvent product ‘!!'^(x) is six-valued may be 
found in my original memoir, Section II., art. 15. 
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