I

TRACT 28.

AND INFINITE SERIES.

in

roots call R and r respectively. Then, by adding and sub*

tracting the first and second, as also the third and fourth,

there result these four equations :

, , 2.5.8c" 2.5.8 . 11 . 14 . 17.20c»

R+x=Vix:l-3 Tyr g~ iS i-

R-X = 4J<'5 X

_2c*

3.66 s

3 . 6.2.12 . 15 . 18.21 . 2« 82.5.8 . 11 . 14c 6

&C.

4 b

W =V^ x. **x+r = v? *

3.6.9 . 12.15.184«2.5.8.114"

+ &C.

+ &C.

1 _

3 “ 3 . 6.9 . 12.15c"

2.54* 2.5 . 8.11 . 14.174® .

57 + 3.6.9.12.1 6 i« <21 ,-6 + KC.

3.6.9c* 1 3.6.9.12.15.18.21c®

127. And hence, by equal addition or subtraction, ive•find these two different expressions, both for the greatestand least roots of a cubic equation, in which c 1 or b 1 4- as isnegative, namely,

2.5.8c" 2.5.8.11 . 14 . 17 . 20c'

3.6.9.12.15.18.21.244* ‘ XC > 0r

R= —x+4 f/bx :1-

3.6.9. 124"

a = x+4 !/bx :~^ +

44 1 2.5.8.114"

r==X “^ X: 3 + 3 -

S ~ ~ X + X ‘ 3 . 6.9c*

2.5 . 8 . 11 . 14c s

+ &c;

2.5.8.11.14.17.20.234'

6 . 9 . 12 . 15c" '3.6.9.12.15.18.21.24.27c'2.54* . 2 . 5 . 8 . 11 . 14 . 174*

&c, or

+ &c;

3 . 6 . 9 . 12 . 15 . 18 . 21c'

where r is the greatest, and r the least root of the equation.r 3 - 3 ax = 2 b, or x z — 3.J/(c* + b 1 ) x = 2b, and x the onlyreal root of the equation x 3 + S^c* — i 1 ) x — 2b-, inwhich, as well as in the above series, c 1 denotes a positivequantity.

128. And hence it can no longer be said that Cardan’srule is of no use in the solution of cubic equations, thathave three real roots ; since they have here been reduced tothe other case, in which the equation has only one real root,which case is always resolvable by that rule. And the firsthint of such reduction I received from Francis Maseres,Esq. Cursitor Baron of the Exchequer, he having done methe favour to communicate to me the second of the abovefour forms for the greatest root, in a letter of the 17th ofJuly 1779; the investigation of which formula, togetherwith those of the other three, nearly as above, I had the