The Complete Work of Charles Darwin Online

RECORD: Darwin, C. R. & George Darwin. n.d. The stream flows in a cylindrical bed. CUL-DAR52.C1-C4. (John van Wyhe ed., 2002-. The Complete Work of Charles Darwin Online, http://darwin-online.org.uk/)

REVISION HISTORY: Transcribed by Christine Chua and edited by John van Wyhe 10.2022. RN2

NOTE: See record in the Darwin Online manuscript catalogue, enter its Identifier here. Reproduced with permission of the Syndics of Cambridge University Library and William Huxley Darwin.

[C1]

The stream flows in a cylindrical bed & the width is 5 times the depth. The vel: is deduced from Eytelwein's law i.e vel = 10/11 √ (fall in inches in 2 miles) x (hydraulic depth of stream)

width of stream	when the velocity is 25 inches per sec (or 1.42 miles an hour), the fall is	when the vel: is 50 inches per sec (or 2.84 miles an hour) the fall is	when the vel: is 75 inches per sec (or 4.26 miles an hour) the fall is	when the vel: is 100 inches per sec (or 5.68 miles an hour) the fall is
10 feet	2 ft 0 3/4 in per mile or 1 in 2640	8 ft 3 in. per mile or 1 in 640	18 ft 7 in per mile or 1 in 280	33 ft 0 1/2 in per mile or 1 in 160
20 feet	1 ft 0 1/4 in. per mile or 1 in 5280	4 ft 1 1/2 in per mile or 1 in 1280	9 ft 3 1/2 in per mile or 1 in 560	16 ft 6 in per mile or 1 in 320
50 feet	5 inches per mile or 1 in 12672	1 ft 8 in. per mile or 1 in 3168	3 ft 8 1/2 per mile or 1 in 1427	6 ft 7 1/4 in per mile or 1 in 802
100 feet	2 1/2 inches per mile or 1 in 25344	10 inches per mile or 1 in 6336	1 ft 10 1/4 inches per mile or 1 in 2854	3 ft 3 3/4 in per mile or 1 in 1604
300 feet	4/5 of an inch per mile or 1 in 76,032	3 1/3 inches per mile or 1 in 19,000	7 2/5 in per mile or 1 in 8,562	1 ft 1 1/4 in per mile or 1 in 4,812

NB the falls in feet & inches are accurate; in the other form only approx but the whole has been looked thro' with care. GHD

[C1v]

width 20 ft [calculations not transcribed, page crossed]

[C2]

Let the stream flow in a cylindrical channel & let BAC = 90°

[Diagram]

A
45° 45°
a inches    a inches
B    E    C
D
Then
BE = a sin 45° = 5a/7
BC = 10a/7
ED = a - 5a/7 = 2a/7
∴ width of stream = 5 times its depth

Hydraulic depth = area BECD / arc BCD = πa2/4 - a2/2 / πa/2 = a/2 (1 - 2/π)
let b be the fall in inches per mile
Then by Eytelwein's law vel:(v) of steam in inches per sec = 10/11 √ 2b.a/2(1-2/π = 10/11 √ ab(1 - 2/π
log 2 = .30303 [-] log π = .49715 [=] ī.80388 = log .6366    1.0000 [-] .6366 [=] .3634
∴ 1 - 2/π = .3634
∴ b = (11/10)2.v2/a(1 - 2/π) = 1.21 x 3 v2 / a x .3634 nearly    log 1.21 = 08279 [-] log .3634 = ī.56038 [=] .52241 [=] log 3.33
   = 3.6v2 / a    3 1/3 v2 / a very nearly

100 [÷] 63 [=] .158

[C3]

Let the width of stream be 10 feet

[calculations and draft for table in C1]

[calculations and table not transcribed]

[C3v]

Theory

Beds of Gravels flowing over frozen snow.

[C4]

The depth of the stream is supposed to be 1/5th of its width and the stream to flow in ~~a cylind~~ the bottom of a cylindrical bed

Then by Eytelwein's law

The depth of the stream is supposed to be 1/5th of its width and the stream to flow in the bottom of a cylindrical bed
The by Eytelwein's law


width of stream 5 ft	vel: in inches per sec	5	10	15	20	25	30
	fall in inches per mile	1/3	1 5/11	3	6	9	13
width of stream 10 ft	vel: in inch per sec	5	10	15	20	25	30
	fall in inches per mile	1/6	16/22	1 1/2	3	4 1/2	6 1/2
width of stream 20 ft	vel: in inch per sec	5	10	15	20	25	30
	fall in inches per mile	1/12	1/3	3/4	1 1/2	2 1/2	3 1/4
width of stream 30 ft	vel: in inch per sec	5	10	15	20	25	30
	fall in inches per mile	1/18	1/4	1/2	1	1 2/3	2 1/6
width of stream 50 ft	vel: in inch per sec	5	10	15	20	25	30
	fall in inches per mile	1/30	1/8	3/10	3/5	9/10	1 3/10
width of stream 300 ft	vel: in inch per sec	5	10	15	20	25	30
	fall in inches per mile	1/180	1/40	1/20	1/10	1/7	1/5

10 in /12 a sec 1/60
17.60 10/36 in yd — 1/3600.
3600
1 / 17.

Citation: John van Wyhe, ed. 2002-. The Complete Work of Charles Darwin Online. (http://darwin-online.org.uk/)