24 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
cbse evga a9 (Uh-oh 
$= l—/p) +2 (1—W/p) ; 
se Wy +2 ey: 
where z is a definite third root of unity, distinet from unity. In the 
three first of these equations, in order that ¢,, $,, and d,, may be 
definite, we must take a definite value of ./p, and then also a definite 
co|to 
eto 
value of (1+ py. Asa new surd, (JI — yp)? occurs in the three last 
equations, we must fix upon some definite value of this surd, retain- 
ing the definite value already assigned to yp; and then $,, ¢,, and 
$s, will be definitely determined. Had we assumed 
f (p)=(pt+ P= 1) + (p—“p?—1) (p+ vp? iy, 
we should have got six cognate functions ; but three of them merely 
a repetition of the other three; for the three which result from 
taking vp? —1 with the negative sign are the same as those which 
result from taking it with the positive sign. 
Def. 7. Suppose that we form the cognate functions of f (p), as 
described in the previous definition, with this difference, that we 
now proceed as though certain surds, Y,, Y,, &c., (in such a series 
all the subordinates of any surd mentioned are necessarily included), 
were rational. In other words, attach no indefinite numerical multi- 
pliers, (as z,, 2, &e.,) to any of the surds, Y,, Y,, &c.; but con- 
sider each of these surds as having a single definite value. The 
cognate functions of f(p), so obtained, may be termed the cognate 
functions of f ( p), taken without reference to the surd character of the 
surds Y,, Y,, &c. or instance, let 
1 
4 = 4 
Ff (py=24+p ) +1+vp) +(+p) +p; 
then the cognate functions of f (p), taken without reference to the 
1 Rope | 
2 
5 57 
surd character of the surds, “p, DP, (2+p ) , are, 
1 1 
a A 2 
=2+p) + (+p) + +p) +; 
al 
q|- 
o,=(Qtp) +2 (4p) +2 U+/p) +p; 
