OF THE FIFTH DEGREE. 131 
include z, g, &; because z = 1 + e?; and g and & are known by (6) 
and (10). To find the six unknown quantities we have six equations, 
which are here gathered together. 
ps = — 204 + 5g? + 15072, 
ps = — 4B + 40acz, 
Be = i, 
Bl =< (0) (15) 
hz (92 + g? z+ 209) = + cz —g (g? — a 2) 
h (0? + g? 2+ 20gz)= 2ke — a (9? — a? 2). | 
The first two of these equations are the equations (12) and (13). 
As to the third and fourth, it was proved in the ‘ Principles ” 
that the form of wis m+n /2+ J (he +h 2), m andn 
being rational. This is saying in other words that B’= 1 and B’=0. 
The last two of the equations (15) are the equations (9). 
§5. The criterion of solvability of the equation / («) = 0 may now 
be stated in a general way to be that the coefficients p2, p3, ete., 
must be so related that rational quantities, a, ¢, 6, 9, e, h, exist 
satisfying the equations (15). We also see what requires to be done 
in order to find the roots of the equation /' (x) = 0 in terms of the 
given coefficients. By (3), 7 is known when 2%, #2, U3, U4, are 
known. But, B’ and B” being respectively unity and zero, 
m= B4+BVYzt+/3 mae B—-BVYz+V 4H, 
Beene iT fe — S's ee Ba Ve = VS fh. 
Therefore, to find 7; we need to find B, B’, z and h; which is equiva- 
lent to saying that we need to find the six unknown quantities 
a, c, 0, 9, e¢, h. Before pointing out how this may be done in the 
most general case, I will refer to some special forms of soluble quintics. 
OASE IN WHICH wy Ug = U2 Us. 
§6. A notable class of solvable quintics is that in which uw, u, = uw, U;- 
It includes, as was proved in the “ Principles,” all the Gaussian 
equations of the fifth degree for the reduction of a” — 1 = 0, 7 prime. 
It includes also other equations, of which examples will presently be 
given. Now, when w uw, = w2%3, the root of the quintic can be 
found in terms of the coefficients pz, ps , etc., even while these coeffi- 
cients retain their general symbolic forms ; in other words, the root 
can be found in terms of pg, p3, etc., without definite numerical 
values being assigned to p2, p;, etc. This I proceed to show. 
