143 
Next, the functions 
|('^)i(B) i +(A) j (C) i +(B)j(G) j }+{(D) j (E)i+(D) j (F) j +(E)j(I')i}=P j , 
{(A),(B) i +(A),(C) j +(B) i (C) i }{(D) ( (E),+(D) j (F) i +(E) j (F) i } =Q„ 
are rational functions of the coefficients of the quintic F=rO ; for 
the former is the sum, and the latter is the product, of two func- 
tions of x x x 2 x s x 4 , of which either is changed into the other by any 
odd number of transpositions of those variables, while neither is 
changed by any even number of such transpositions. 
Wherefore 
(A) i (B) < +(A) j (C),+(B) j (C) i 
is the root of a given quadratic, whose coefficients are rational 
functions of those of the given quintic. 
Finally, 
(A) i (B),(C) i +(D) i (E) i (F) i rzzR, and 
{(A) ( (B),(C j )}{(D) i (E) i (E) i }=S < 
are both rational functions of the coefficients of the quintic F=rO, 
for the reasons above given, and (A) i (B) i (C) i is again the root of a 
given quadratic. 
Hence we can form the cubic whose roots are (A) f , (B) i5 and(C) 8 -, 
whatever be i ; for its coefficients are given irrational functions of 
the coefficients of F=0 with ambiguous radicals. 
Forming, then, and solving these cubics, for i= 1, i=2, i— 3, 
i— 4, we obtain (A) i =Y i , Y* being a given irrational function of 
the coefficients with ambiguous radicals. That is, we have 
H 1 +H 2 -fH 3 +H 4 =V 1 , 
H?+H^H|+Hf=V 2 , 
Hi+H|+Ht+HJ=Y 4 , 
By the aid of these equations we can form the quartic whose 
roots are H 1? H 2 , H 3 , and H 4 , by a well-known method; and, as 
we can also solve this quartic, we can write its four roots (H^, 
(H 2 ), (H 3 ), (H 4 ), all given irrational functions of the co-efficients of 
FmO, in the above expression for x 0 , in the places of H 4 , H 2 , H 3 , 
and H 4 ; and thus we have the long-sought algebraic root of the 
given quintic F=0. 
