OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 539 
Pp 2P, ,3P,,4P,|, 
£27 3P) oP B 
and also = 
that is 
(en eg ego ey 
awe, ¢,@,u8¢\=0">(a—f8)?) 1°, ° 0 |}. 
Aie., (6, 6. 0a y+6, 1 
v5 ,yt+d 
0, 76 
‘ 4.3 ; : : 
Hence the following =) 6 results for biquadratic—the first three being 
NeEwtTon’s quadratic elements— 
48(b’—ac)=a@> {(a—fB)| 1 , iY = {(a—B)}, 
a | 
y+ 
y+8, 1 |} =e2{(a—f)ly+3—y9)}, 
oe) ,yt+d 
144(?—bd) =aS {(a—f) 
yS ,y+8|} =w2 {(a—B)’7'8} , 
48(d’— ce) =a’2 {(a— 8B)’ 
2", v9 | 
and, 
1 , 0 |} =a*3 {(a—B)(y+8)}, 
—48(be—ad) =a°E {(a—B)? 
ys 2 ) 
16(bd — ae) =a {(a—B)’ 
, 
BE es ty te ae 
7) 
Oy 
The equation of the fifth degree gives similarly, the co = )10 results 
ee Ge eae 
=o Oe Cd, Cd 
Big. OoO,¢, @&é¢ 
=a {(a—B)’ 1 : 0 ar 
yt d+e , 1 | 
yotyet+de, yt d+y | 
yoe , yot yet de 
0 ’ ye | 
and that of the sixth degree the co = is results 
