1817,] On Cubic Equations. 379 
Or, on comparing with (4), 
o= Rr — R72, or Re — Ro or72 9 + 71 erscccssecee (14) 
By equation (1) or (6) or (12), 
c= RS — DR, or — # + Or, or — ee + Le wecveneeoe (15) 
Comparing the sum of these with (3), 
cles 2.(Ro — 8 — 0) decedccsees ae sera e's gp ectea's o/s alge (EOP 
The sum of equations (14) gives 
e—2(h ra hee Be Re +o + 7 oe)... 00 (la) 
Comparing (15) with (7) 
c=1R (R°—r?—¢*), or tr (R?—7* + 9°), or 2.0 (R°+7°—o0").. (18) 
For reasons, which your printer will by this time conjecture, I 
refrain from multiplying these beautiful and interesting analogies to 
the utmost. Still excluding binomial and more complicated func- 
tions of J and ¢, I shall close this table with a few of those in which 
the reciprocals of the roots are concerned, 
Dividing (5) by (13), 
b HSIN, J 
HS TT Pr crete ee eene cers ecscererececes (19) 
i 
Dividing the square of (8) by that of (13), 
a 1 1 1 
eat at ee wmererseeeeresr esr es esos ere Fevneneves ee (20) 
Dividing eq. (3) by eq. (13), 
1 1 1 
aaa = O Oe ee ee ee ee © ee ee | (21) 
Dividing twice (7) by the square of (13), 
2b 1 1 1 
Peed Or RUS oar eu on (22) 
Dividing thrice (16) by the cube of (13), 
3 1 1 1 
ee tas ok Bs? GG eeereevrerereeareretes eoeeeen (23) 
Dividing (9)* by (13)*, 
2 1 1 1 
PS Le re ee re cece peace Fen cas asennad 
Dividing eq. (14) by the square of (13), 
BART) (bak) pti riggh vugtorgyy yitde 
RRs e? rice R? C batt R eraeereeeres (25) 
Dividing (17) by (13), 
R r R e r 4. 383 j ; 
te ee ee Hier Waa pies aves se . (26) 
The reader who will take the trouble to refer to my papers of 
October and November last will perceive the facilities gained in 
that investigation by employing the 12th and Gth of the present 
series of equations. Asa further exemplification, I shall here only 
notice the elementary case, in which one root, as R, being known, 
the general values of the other two roots are required. These are 
enveloped in the quadratic equation 
(x2 +7) x (u +o) = w+ (r +e) a+ re sO vveeeees (I.) 
6 
