102

ON CUBIC EQUATIONS

TRACT 28

found tobe^/ J = l/1q = —l/\b = — 2 l/\b, and the othertwo, or the greatest and least roots, are — f r x (1 ± \/S) =

85. Hence, by a comparison of these two different formsof the Same roots, we find

a / 3+1 2 2 . 5.8 . 2 . 5 . 8 . 11 . 14

23/2 ~ 3^6 3 . 6 . 9 . 12 ' 3.6 . 9 . 12 . 15 . 18

and

a / 3-1

2.5 2 . 5 . 8.11 0

-j- ~—;—;———— — &C

&.C — A,

B.

2 J /2 3 3 . 6 . 9 1 3 . 6 . 9 . 12 . 15

86. And by addrng and subtracting these two, we find

2.5 2 . 5.8

^=i + ! + -!

3/2 3 ^ 3.6 3 . 6.9 3 . 6 . 9.12

-p -1 -See, and

L _ i _ 1 | 2 | 2 - 5

3/2 ~ 1 3 3 . 6 +

= c.

2 - 5 - 8 _ + + _ _ &c

3.6 1 3 . 6.9 3 . 6 . 9.12 ' 1 ’

87. Also, because I_I x is = —which is =

x i s = J_

23/2 23/2 2 J/ 4 ’

\ x (—) z ; therefore the mean proportional between the two

series a and b, is to the series c, as the side of a square isto its diagonal.

88. Further, to and from the two series A and b, addingand subtracting the two series in Art. 74, namelv, j?/2 or

3/2 2 2 . 5.8 ^1 2.5 2 . 5 . 8.11 , r

2 _ 1 3.6 3 . 6 . 9.12 ° 3 *" 3 ,~6 . 9 3 . 6 . 9 . 12.15 CiC ’

we obtain the four following series:

^3 + 3 / 4+1 _ , 2 . 5.8 2 . 5 . 8 . 11 . 14 . 17,20

43/2 — _ 3 . 6 . 9 . 12 3 . 6.9 . 12 .'15 . 18 . 21 . 24

a /3 + 5 / 4 - 1_1 , 2 , 5,8 . 11

&C,

2 . 5 . 8 . 11 . 14 . 17 . 20 . 23 „

—-+:&C,■m v.n 9

• 13/2 3 1 3.6 . 9 . 12 . 15 1 3 . 6 . 9 . 12 . 15 . 18 . 21 . 24 . 27

a / 3 - 3 / 4+1

43/2

a/3-5/4-1

2

3.6 1 3 . 6.9 . 12 . 15 . 18

2.5 2 . 5 . 8 . 11 . 14.17

&.C.

43/2 3 . 6.9 3 . 6.9 . 12 . 15 . 18.21

89. It also appear*, that the series

, 1 , 2 , 2.5 2 . 5.8 2 . 5 . 8.11

~3 3 Ti "*■ 3~. 6 . 9 3~7<T79. 12 ~ 3 . 6 . 9 . iaT 15

is the reciprocal of the series

1 2 . 2.5 2 . 5.8

&C,

2 . 5 . 8.11