TPIE THEOET OF EQUATIONS OF THE FIFTH OEDEE. 
275 
Then we have ^ 
where 
?)=12345-24135, 
and 
1 2 3 4 5 = 1 2 -}- 2 3 -j- 3 4 + 4 5 + 5 1 = “i3 "t" /3 7 H- y ^ T“ -f- s a , 
24135=24 + 41--}-13+35d-52=/3§+^'^+®57+ys+s|3, 
and where ^=(12345)~(24135), 
and (12345) = 123 + 234H-345+451 + 512 = a/37+/3y^+7^£+§£a+2a/3, 
(24135)=241+413+135+352+524=/3^«+Sa7+a7£+ys/3+£^. 
29. In fact, r 
cl>=:^|(12345)-(24135)j, 
(1^45) =123 + 234 T-'m + 451 +^2, 
(24135) =m + 413 + 135 +'^2 +'^, 
where 123, &c. denote resjiectively 
&C., 
and (12345) — (24135) thus presents itself as a cubic function dmded by But in this 
cubic function the coefficients of x^y vanish. For the coefficient of any power of a’ 
will be 
123 + 234+345+451 + 512-241-413-135-352-524, 
where, first, for x\ 123, &c. denote respectively unity; the coefficient of x^ therefore 
vanishes. Next, for 123, &c. denote respectively —(1+2 + 3), &c. ( = a+|3 + y), 
and the coefficient of x!^y also vanishes. But for xy^^ 123, &c. denote respectively 
12+23 + 31(=aj3+j3y+ya), &c. respectively; the positive terms are 
(12+23 f31)+(23+34+42) + (34 + 45+53)+(45+51+14)+(51 + 12 + 25), 
which are 
= 2(12+23+34+45 + 51)+(24+41 + 13+35+52) 
= 2.12345+24153; 
and the negative terms, taken positively, are 
(24+41+12)+(41 + 13+34) + (13+35+51)+(35 + 52+23) + (52 + 24+45), 
which are 
=:(12+23+34+45 + 51)+2(24+41+13+35+52) 
= 12345+2.24135; 
so that the difference, or coefficient of xy"^, is 
=12345-24135, 
which is —<p. 
And for +, 123, &c. denote respectively — 123(= — a/3y), &c., so that the coefficient 
of+ is =x. 
2 p 2 
