462

ROOTS AND RECIPROCALS

tract 24,

the scries of roots and differences, with the calculation fromthem.

Now the general form of the series for ^/(« 3 + ?t), or thecube root of « 3 + «, is a + £ - ^ + ~ - ^^there-fore, expounding n by '1, 2, 3, &c, the series of the cuberoots of r/ 3 , a? -f 1, a 3 -f 2, a 3 + 3, See, with their 1st, 2d, 3d, Seedifferences, will be as below:

Nos.

a 3

a 3 + 1a 3 +‘2a 3 + 3a 3 -f 4

Cube Roots.

9o5“*"Sla rt,2 4 , 40

«+ ~+ ~

.3 9 133

« + -— — + —

4 16 , 320

a + —

1 st Diffs.

3a 2 9a aT 81a 81 3 . 35

— - T+ —

2__ 5 -j_ 95

_L_ 7 I 185

. T .

2d Diffs.

0 10

3d Diffs.

9ai~ 27a 8

10

2 20

27a 8

10 &c

2 30

,

_-—7-

Now here all the series converge faster than the like seriesfor the square roots; because here the denominators, havinghigher powers, are larger than those in the former; conse-quently fewer terms will suffice in this case, than were re-quisite in the former, for an equal degree of accuracy, in allthe differences and roots. The calculation for a few termshere follows.

First, To find the Diffs.

-1 = -0033333333

3a*

ni = .... mu

ya s

.±1 .. c

81<z 8 _

1st dif. -0033322228

— = *0000022222

9n s

~ — =. 31

27u 8 ;

2d dif. '0C00022185

10

3d dif. . . 37

Then for the Roots.

3d Dif.■0 8 37

2d Diffs.

1st Diffs.

•0822185

•0033322228

22148

33300043

92111

33277895

22074

33255784

22037

33233710

22000

33211673

21963

33189673

21926

331677iO

21889

33145784

21852

33123895

21815

33102043

21778

33080228

33058450

Cube Roots.

10-0000000009

0033322228

0066622271

0099900166

0133155950

0166389660

0199601333

0232791006

0265958716

0299104500

0332228395

0365330438

0398410666

0431469116