Question of solving the Equation of the Fifth Degree. 541 
the two remaining roots x, x, of the equation (1.). We shall 
therefore take the first of the three factors of (22.), namely, 
the equation 
O = hgthys  cseccccccee coveceese (23.) 
which reduces the two equations (20.) and (21.) to the two 
following, obtained by elimination of /,, 
0 = hy (xP —axP) +h, (xe?—aX537) 5 veeeee (24) 
Oe. fiy2 (a, 4074) ehg® (hg ai) om sonore (2s) 
These two last equations give, by elimination of /,, 
O = h,?(x1+24){ (41+ x4)(%1— 24)? + (®q + %3)(%g—H3)"} 5 (26-) 
in which we cannot suppose the factor x,-+., to vanish, be- 
cause the relations 
oo #£°+D2,+E, 0) =2'+Da,+k, (27.) 
give D = —(a+a%a,+e7x2+a, oye 28.) 
B= (x, +24) (v,7+247) x, 245 : 
and we have supposed that E does not vanish; and since, for 
a similar reason, we cannot suppose that x,+.3 vanishes, we 
see that we must conclude 
h,=0, 4,=0, hk; = 0, hy = 0, (29.) 
unless we can suppose that the third factor of (26.) vanishes, 
that is, unless 
(a, +24) (@1— 24) + (Xo +23) (2-43) = 0. — (30.) 
Let us then examine into the meaning of this last condition, 
and the circumstances under which it can be satisfied. 
If we put, for abridgement, 
Lot Lz = —a, X v3 = Bs Secvceces (31.) 
the condition (30.) will become 
0 = «f—a2) 1,—2,2°+r3—8+40 8; (32.) 
and we shall have, in virtue of the relations (11.) (12.) (13.), 
two other equations between x,, 7,, a, 6, namely, 
0 = P44, (x,—a)+x7—2,0a+0°—B8, (33.) 
and 
0 = rp—xr ata, (a*—6)—a?+2aB; eee (34.) 
between which three equations, (32.) (33.) (34.), we shall now 
proceed to eliminate v, and 2,. For this purpose we may 
begin by multiplying (33.) by 2,, and adding the product to 
(32.); a process which gives, by (34), 
0 = rP—4742, 44+2342AB, secoseeee (35.) 
a relation more simple than (32.). In the next place we may 
