OF THE FIFTH DEGREE. 139 
Hence the quintic becomes 
Bete tao) 
F(x) = 2 + jae + Iba? 2 ba 
af. | = = (02 —p? 2) + dace | = 0. (28) 
The criterion of solvability of a trinomial quintic of the kind under 
consideration is therefore that the coefficients p, and p, be related in 
the manner indicated in the form (28); while at the same time the 
last four of the equations (15), modified by putting g = k = 0, 
subsist between the rational quantities a, c, e, h, 0, ¢. From these 
data, the three following equations may be deduced, v being put for 
} 
ar? 
8ev® — 42v? + 2 (3 — 4e)v — 2 = 0, ) 
Bg Ber -250s, 
" ee t (29) 
Av (ze + 4ev — 8v’) -(- 32 + fe {2 + 4v(e—1)+ 8va}. 
The first of these equations is obtained from a comparison of the two 
equations (9), the second is obtained by putting p, and p, respectively 
equal to the values they have in (28) ; and the third is obtained by 
putting p, equal to the coefficient of the first power of « in (28). 
§15. lf any rational values of e and v can be found satisfying the 
first of equations (29), let such values be taken. Then, from the 
second and third of (29), a? and ac can be found. Therefore a and ¢ 
are known. ‘Therefore, by (21), and g are known. Therefore, by 
(9), 2 is known. In this way all the elements for the solution of the 
quintic are obtained. 
$16. For example, the three equations (29) are satisfied by the values.. 
1 nahn 5 
ene ge 8 
4.\> heed on x6 258 
a—d, BOs ras aa 
When these values are substituted in (28), the quintic becomes 
625a 
4 
a + + 3750 — 0. 
