227 
33] 
U IS A FUNCTION OF THE FOURTH ORDER. 
Suppose q — a is one of the values of q; the equation becomes 
+ 14q + l) 3 _ (a 2 + 14a + l) 3 (ß s + 14/3 4 + l) 3 
q(q-iy ~ a (a — l) 4 ’ /3 4 (/3 4 -l) 4 ’ ü a ~ ^ • 
Now if 
then 
1-/3 
1+/3 
(q~ + 14 q +1) — 
16 (ß 8 + 14/3 4 + 1) 
<7-1 = - 
8/Ö (1 + ß 2 ) 
(1+/3) 8 ’ a * (1+/3) 4 
which values satisfy the above equation: hence also, identically, 
or the values of q take the form 
1 
/1-/3V 
( l+ßy 
fl-ßiy 
/1 + ßiv 
/3 4 ’ 
U +ßJ ’ 
\l-ß) ’ 
\l+ßi) ’ 
U -ßi) 
(Comp. Abel. Œuv. tom. I. p. 310 [Ed. 2, p. 459].) 
The equation 6 3 — M6 + M = 0 
has its three roots real if 27 — 4ilf is negative, and only a single real root if 27 — 471/ is 
positive. Writing the equation under the form 
(0 + 3) 3 — 9 (0 + 3) 2 + (27 — M) (0 + 3) — (27 — 4if) = 0, 
we see that in the former case 6 has two values greater than - 3, and a single value 
less than — 3. Writing the equation under the form 
(0 — l) 3 + 3 {6 — l) 2 + (3 — M) (6 — 1) + 1 = 0, (3 — M is negative) 
the positive roots are both greater than 1. Hence, in this case, q has four positive 
values and two imaginary ones. In the second case 6 has a single real value, which is 
greater than — 3 and less than 1. Hence q has two negative values and four imaginary 
ones. In the former case, P — ^lJ 2 is positive, and the function U has either four 
imaginary factors or four real ones. In the second case, I s — 27./ 2 is negative, or the 
function U has two real and two imaginary factors. 
29—2