444 Dr. hutton on Cubic 'Equations 
any power, but unaccompanied with any appearance of 
o 
the idea of thus reducing the one cafe of the cubic 
equation to the other. 
129. It is hardly neceflary to remark, that any ge- 
neral feries of each of the above four forms, is fumined 
by means of the fum or difference of the roots of thefe 
two equations x 3 — 3'v^. b* ± c z . x = 2-b, and that by fob- 
llituting particular numbers for b and c, we may thus 
fum as many feries of thofe forms as we pleafe. 
1*30. Ex. 1. We may now illuftrate thefe formulas by 
fome examples. And firft in the equation x 3 — 1 5 x — 4.. 
Here 2 b — 4, and 3v / ^ a + c r =15, confequently b— 2, 
and c l - 5 3 — b 1 = i 25 — 4 - 1 21 = 1 1% and 
x = vfo + b — \/ c-~b — J / 1 3 - J / 9 = *27 12508 the root 
of the equation x 3 — ^vb z — x — ib or 
x 3 + 3v / 1 1 7 . x = 4. And as b is lefs than c, this equa- 
tion belongs to the two feries in the latter cafe for find- 
ing the leaft root. Hence, the terms of the two feries 
agreeing with the pofitive and, negative terms of the fer- 
ries in Art. 1 06, they will Hand thus t 
