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Tchibyscheff's Ordinate Table.

NUMBER

OF OR-
DINATES,

DISTANCE OF ORDINATES FROM MIDDLE

OF BASE, X , IN
FRACTIONS OF HALF THE BASE LENGTH,

2 3 4 5 6 7 9 10

.5773
* , .7071
.1876, .7947
*, .3745, .8325
.2666, .4225, .8662

*, .3239, .5297, .8839
H, .1679, .5288, .6010, .9116
.0838, .3127, .5000, .6873, .9162

The employment of this rule to find the volume of displacement and the other elements usually tabulated on the displace ment sheet is shown on the attached Tables. The number of stations used is ten, as in the case of Simpson's rule, but for clearness the after body five are indicated by Roman numerals, and the fore body ones in Arabic. The displacement length is 600 feet, therefore by taking the fractions given the preceding table for ten ordinates and multiplying them by 300, we shall obtain the distance of the displacement sections apart. These distances from the half-length and the sections are here given as used for the Table, but it will be observed that the water lines are spaced to suit Simpson's first rule for the vertical sections as no advantage would be gained by the use of Tchibyscheff in this direction, owing to the fewer number of water lines generally necessary. The various operations in the Table will be clearly understood from the headlines of the respective columns.

As already pointed out, the great value of this rule is in the calculations to obtain cross curves of stability, specimen tables of which are also given. The fewness of the sections necessary, and the fact that the integrator saves the calculator the tedium of adding up, tells greatly in favor of the adoption of this rule for these calculations both as a time saver and an eliminator of the chances of error.

T. S. S. "LUCANIA"
BODY SECTIONS FOR DISPLACEMENT ETC. BY TCHIBYSCHEFF'S RULE

(FOR CALCULATION SEE TABLE)
ORDINATES FROM AMIDSHIPS:-
BEFORE ABAFT
1.--&__.-= 25. 14

WATER-LINES 3.833' APART
2..-.&.-..-= 93. 90
3.-.&.L.Ill.-= 150.00
4.-_&_VI.-= 206. 10
5_-.&-V__= 274.80

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FIG. 8.

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A = Distance of Ordinates.
* 10 = number of stations. † 3= Simpsons' multiplier.

Tchibyscheff's Rule.

Mo

$265.62

VERTICAL SECTIONS.
5
6

7
Func- Differ-

Levers.
1
2

$
tions. ences.

ments. 32.50 32.40 32.35 32.50 64.80

16.18 328.48 32.50 32.40 32.30 32.50 64.80

16.15

328.35 .13 1.0838 .109 31.45 31.50 31.45 31.45 63.00

15.73 314.31 30.25 30.35 30.40 30.25 60.70

15.20

300.71 | 13.60 .313 4.259 28.55 29.10 29.25 28.55 58.20

14.63 24.65 25.10 25.40 24.65 50.20

12.70 236.04 29.48 1.500 14.740 21.00 22.45 23.70 21.00 44.90

11.85 179.93
16.10 16.90 17.45
16 10
33.80

8.73 145.69 34.24 1.687 23.523
5.75 6.90 8.25
5.75
13.80

4.13

43.91
4.10 4.50 4.80
4.10
9.00

2.40

30.41 13.50 1.916 12.367 226.85 231.60 235.35 = 21

DISTANCE

54.998

OF 226.85 463.20 117.68 2,173.31 WATER LINES = 3.833 2 1

0 453.70 463.20 0 7,023.71 of Water Lines : X 2.

CENTRE OF BUOYANCY. 7,023.71 X 3.833

below 27,222.00 27,792.00 28,242.00

= 12.39 2,173.31

W.L. 7 per Inch : 420.

54.998 X 600 64.814 66.171 67.243

= 7.59' abaft *

2,173.31 X2 2,173.31 = 666,445.25

{

2,173.31 = 19,041.29

42

X(3x0+10x0,-03) = Moments.

21x480X2= Area of Water Lines.

2938

[blocks in formation]

Keel 5.40 W.L. } 160.15 W.L. 1 183.19 $

W.L. 1 183.19 1
W.L. 2 204.30 2
W.L. 3 214.80 Ž

W.L. 3 214.80 1
W.L. 4 221.65
W.L. 5 226.85 1

1.35 0
160.15 1 80.07
45.80 45.80

125.87
207.31 125.88 3.833 x

= 2.328

207.31
91.60 1 91.60
408.60 2 817.20
107.40 3 322.20

1356.87
814.90 1,356.87 3.853 x

6.383

814.90
107.40 3 322.20
443.30 4 1,773.20
113.43 5 567.15

4019.46
1,479.03 4,019.46 3.833 x

10.420 113.43 5 567.15

1479.07
463.20 6 2,779.20
117.687 823.76

8189.57
2,173.34 8,189.57 3.833 x

= 14.450 2173.38

W.L. 5 226.85
W.L. 6 231.60 2
W.L. 7 235.35

DISPLACEMENT IN

2 X 600 X 2 X 3.833 Keel to W.L. 1:

x 207.31 = 3 x 10

2 X 600 X 3.833 W.L. 1 to W.L. 2 :

x 2,335.55 = 10 X 12

2 X 600 X 2 X 3.833 Keel to W.L. 3:

Х 814.90 = 3 x 10

3 X 600 X 2 X 3 833 W.L. 1 to W.L. 4:

x 1,662.14 = 3 x 10

2 X 600 X 2 X 3.833 Keel to W.L. 5 :

x 1,479.03= 3 x 10

600 X 2 X 3.833 W.L. 4 to W.L. 6:

x 1,360.65 = 3 X 10

2 X 600 X 2 X3.833 Keel to W.L. 7:

x 2,173.34 = 3 x 10

X (5X0, +8X0,--03) = Area by rule. 12

24

X(3X0, + 10X0,-03 = Moments.

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