378 On Cubic Equations. [May, 
Articxe VII. 
On Cubic Equations. By Mr. Horner. 
{To Dr. Thomson.) 
DEAR SIR, Bath, Jan. 17, 1817. 
Havine frequently, when reading or investigating subjects con- 
nected with cubic equations, experienced the advantage of having 
by me a compressed syllabus of the simple relations existing between 
the roots and the numeral parts, and conceiving that the same con- 
venience will be acceptable to others, I beg to submit-the following 
specimen through the medium of the Annals of Philosophy. To 
avoid the pedantry of unnecessary reference to authorities, I have 
indicated elementary sources of each transformation, in a connected 
series. ; 
Let the proposed equation be 
MW — De — CH Ovcccecccccccreccees RO eee eaee (TY 
Compare it with 
(w@ —R) x (@w@ +1) x (2+ 0) = 
a? — (R—r —e)2*— (Rr+ Re —re)x—Rre =O.. (2) 
—=_— 
This comparison gives us, 
First, Ro— 7. — 9 =O .nnogecseee § ay espace od 4p Siete 
Or, R=r +e 7= R— ep eH R—-T wooeccereescee (4) 
—=— 
Secondly, b= Rr + Re—7re srevevevsscees suid Paster (5) 
Which, by substituting from equations 4, becomes 
b= R? —re,orRr+ o,orRe + 7°........ ote iaena(8) 
The sum of these, compared with eq. 5, gives 
b= 4 (Ro + 2 + O°) cnn se cneeccens Hh Se ERA te tisca ee Mat 
Comparing the square of (5) with (3), 
b= J (Ri? + Re? + 77 6%) «0.1 cee ees sisi Sus 5 tle a Mee 
Comparing this with the square of (7), 
Ri+ r+ 
Df eee ee eee glee sete sptatehten 
Substitute (4) in (6), and we have 
b= R?— Rr +7°,0rR? — Ret oor? +7re+o°.... (10) 
which is equivalent to 
R3 + 73 R3 + <3 ri— 3 
b= Rer? Ree? or a Pl eeeeseecee (11) 
Substitute R, — 7, — e, for x in (1), and 
Us Ry or $F, OF Gt = oreo es, sieferte é- hiaietaioy CA) 
—__— 
Thirdly, c=Rre CROC ores ee ees eHeeeoereorevrene? (13) 
