140 RESOLUTION OF SOLVABLE EQUATIONS 
Then the values of wu, w3, u8, uw’, obtained from the expression for 
diy in §3, are 
4-8) ED} 
tf) SEHD} 
if 
so )-JlE-V} 
4 
wo a 
4-7 {-1+/G)t avG ae 
Hence, 7, = u, + u, + u + uw, = — 1.52887 — 2.25035 + 
2.48413 — 3. 65639 — — 4 95148. 
WHEN ANY RELATION Is ASSUMED BETWEEN THE SIx 
UNKNOWN QUANTITIES. 
S17. In the case in which 2 U4 Was taken equal to uw, w3 a relation 
was in fact assumed betwixt the six unknown quantites a, c, e, h, 9, ¢; 
for, as we saw, to put Uy U4 = U2 UZ is tantamount to putting a = 0. 
Hence. as was noticed in §7, we had only five unknown quantities 
to be found from six equations. Now, when any relation whatever ~ 
is assumed betwixt the six unknown quantities, the root of the quintic 
can be found in terms of the given coefficients po, ps , ete., without 
any definite numerical values being assigned to the coefficients, 
because six rational quantities can always be found from seven 
equations, 
Tur GENERAL CASE. 
§18. We have hitherto been dealing with solvable quintics, assumed 
to be subject to some condition additional to what is involved in their 
solvability. We have now to consider how the general case is to be 
dealt with. That is to say, we here make no ‘supposition regarding 
the equation of the fifth degree /(x) = 0 except that it wants the 
second term and is solvable algebraically. In this case it is impossible 
to find the roots in terms of the coefficients p,, p,, etc., while these 
coefficients retain their general symbolic forms. But the equations in 
§3 enable us to find the roots when the coefficients receive any definite 
numerical values that render the equation solvable. For, we have 
the six equations (15) to determine the six unknown quantities 
a, ¢, €, h, 0, g ; and we can eliminate five of the unknown quantities, 
