The South Staffordshire Coalfield/Appendix

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The South Staffordshire Coalfield
by Joseph Beete Jukes

Appendix

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2800419The South Staffordshire Coalfield — AppendixJoseph Beete Jukes

APPENDIX.

In speaking of the inclination of a bed, or of a fault, or any other plane, it is common in South Staffordshire to describe it by saying how many inches it deepens, or dips, in a yard. In many other districts this is done by saying how many feet or yards it dips or inclines in Geologists usually describe the dip by stating the number of degrees of the angle included between the plane of the bed, &c. and the plane of the horizon. It is often very useful in field surveying to know at once, roughly, how these things correspond, for which purpose I wrote out the following table for my own use, and add it here, as it may be useful to others:—

Nearest degree of dip, or each 1 in 100, answering to each inch in a yard.

Inches in a yard.	In 100.	Nearest degree.
1	2.78	1½°
2	5.56	3°
3	8.34	5°
4	10.12	6°
5	13.90	8°
6	16.68	10°
7	19.46	11°
8	92.24	12°
9	25.02	14°
10	27.80	16°
11	30.50	17
12	33.36	19°
13	36.14	20°
14	39.92	21°
15	41.70	23°
16	44.48	24°
17	47.26	25°
18	50.04	26°
19	52.82	28°
20	55.60	29°
21	58.38	30°
22	61.16	32°
23	63.94	33°
24	66.71	34°
25	69.50	35°
26	72.28	36°
27	75.06	37°
28	77.84	38°
29	80.62	39°
30	83.40	40°
31	86.18	40½°
32	88.96	41°
33	91.74	42°
34	94.52	43°
35	97.30	44°
36	100.00	45°

When the inclination is greater than 45° it is commonly sufficient to say that a bed, &c, dips two yards in a yard, three yards in a yard, &c., &c.

Now	1 in 1 = 45°
	2 in 1 = 63°	nearly.
	3 in 1 = 71°	nearly.
	4 in 1 = 76°	nearly.
	&c. &c.

Another way of describing the inclination of the beds is also not unfrequently used, namely by saying that they dip at the rate of one yard or foot in so many yards or feet. The following table gives the latter number for each of the angles mentioned.

Angle of dip.	Incline of
01°	1 in 57
02°	1 in 29
03°	1 in 19
04°	1 in 14
05°	1 in 11
06°	1 in 10
07°	1 in 8
08°	1 in 7
09°	1 in 6
11°	1 in 5
14°	1 in 4
18°	1 in 3
26°	1 in 2
45°	1 in 1

Another table that is often found useful in geological surveying is one that for every degree of dip of a bed, Ac. will give its depth from the surface (supposed to be a horizontal plane) at a distance of 100 feet or yards, measured in the exact direction of the dip. In the following table this is given for every degree up to 20°, and for every five degrees after that; and also the thickness of any set of beds thus inclined, measured, not perpendicularly to the surface but perpendicularly to the dip, in other words, the thickness they would have if they were horizontal.

Horizontal distance =100.

Angle of dip.	Depth.	Thickness.
1°	1.7	1.7
2°	3.5	3.5
3°	5.3	5.3
4°	7.0	7.0
5°	8.8	8.7
6°	10.6	10.5
7°	12.3	12.2
8°	14.1	13.9
9°	16.0	15.6
10°	17.7	17.4
11°	19.5	19.1
12°	21.4	20.8
13°	23.2	22.5
14°	25.2	24.2
15°	26.9	25.9
16°	28.7	27.6
17°	30.7	29.2
18°	31.8	30.9
19°	34.5	32.6
20°	36.6	34.2
25°	46.9	42.3
30°	58.0	50.0
35°	70.5	57.4
40°	84.2	65.6
45°	100.0	70.7
50°	119.0	76.6
55°	143.0	81.9
60°	174.0	86.6
65°	214.0	90.6
70°	275.0	94.0
75°	368.0	97.0
80°	575.0	98.0
85°	1143.0	99.0

As this table is one giving the solution of a right-angled triangle for each angle specified, it may be readily used to find any dimension which can be stated in the form of a right-angled triangle, as for calculating the space between the outcrop of two beds, of which the angle of dip is known and the thickness between them; the distance which any bed, of which the depth and inclination are known, will require before its outcrop at the surface can occur; and so on.

By means of this table, also, the probable "throw" of faults can be ascertained, where the broken ends of a bed on opposite sides of a fault can be found, and a certain mean angle of dip assigned to the whole mass.

Fig. 31.

If, for instance, Fig. 31, there be a set of beds, including one particular bed A A' dipping at 30° in one direction, traversed by the fault B C running in the direction of the dip, and the ends of the bed A A' on opposite sides of the fault be any distance apart, say 200 yards, then, inasmuch as the bed A would by the table be twice 58=116 yards deep at D on one side of the fault while it is at the surface of the ground (supposed to be a horizontal plane) on the other side of the fault, it is obvious that the fault B C has a "downthrow" of 116 yards towards D.

If the fault traverse the beds obliquely to the strike, as in the following figure—

Fig. 32.

—we must, instead of measuring along the fault B C, of course, measure A D along the dip, and then proceed as before.

Conversely when the amount of the throw of any fault, and the angle of inclination of the beds, are known, if the place of the bed on one side of the fault be ascertained, that of its outcrop on the other side of the fault may be calculated, and so on.

Oblique Sections.

Although it has no especial reference to the district of South Staffordshire, I would yet take this opportunity of printing and publishing the additional table below.

In the year 1850 we were engaged in running sections across a very contorted district of North Wales, so contorted that it was impossible to contrive any long continuous section that should not in some part of its course cross both beds and cleavage planes very obliquely. It became important, therefore, to know what correction to apply to the observed angle of dip of those beds and planes, so that they should be drawn in the sections correctly with the dips they would actually appear to have in a vertical cliff if one were formed along the line of section. Although the little trigonometry I ever possessed had long grown rusty from disuse, I yet contrived to puzzle out a formula which should express this correction, and from that calculated the table.

Subsequently, however, I lost the clue which had led me to the results, and became doubtful as to their correctness; I therefore applied to my friend Mr. Hopkins, then President of the Geological Society of London, and he, with his usual kindness, favoured me with the following solution of the problem, which I was glad to find gave the same result as that at which I had arrived by a more roundabout and empirical course.

Fig. 33.

Let O A be a horizontal line on the surface of a bed, it will be the direction of the strike: O C the direction of the section as given by the compass, O C being also horizontal.

Draw A C in the same horizontal plane as O A and O C, and at right angles to O A, A C will be the direction of the dip as given by the compass.

Draw C B, vertical, to meet the surface of the bed in B, and join A B and O B.

The angle C A B will be the real dip, and C O B the apparent dip, of the bed, as seen in the face of the supposed cliff or section.

Let O C A = x, the angle which the section makes with the direction of the dip,

C A B = y, the real dip,

C O B = z, the apparent dip,

then tan. y = BC/AC

but B C = O B sin. z

and A C = O C cos. x

= OB cos. z cos. x

∴ tan. y = tan. z/cos. x sec. x

or to radius r

(1) r tan. y—tan. z sec. x

and ∴ log. tan. y = log. tan. z + log. sec. x-10;

(2) or tan. z = tan. y/sec. x

∴ log. tan. z = 10 + log. tan. y— log. sec. x.

(1) Giving the true dip if the apparent dip were observed in a cliff.

(2) Giving the apparent dip that ought to be drawn in the section when the true dip is known.

From this formula the following table has been calculated:—

Oblique Section Table.
Angle between the direction of the dip and that of the section.	Angle of the Dip.
	10°	15°	20°	25°	30°	35°	40°	45°	50°	55°	60°	65°	70°	75°	80°	85°	89°
10°	9°51′	14°47′	19°43′	24°40′	29°37′	34°36′	39°34′	44°34′	49°34′	54°35′	59°37′	64°60′	69°43′	74°47′	79°51′	84°56′	88°59′
15°	09°40′	14°31′	19°23′	24°15′	29°09′	34°04′	39°02′	44°01′	49°01′	54°04′	59°08′	64°14′	69°21′	74°30′	79°39′	84°50′	88°58′
20°	09°24′	14°08′	18°53′	23°39′	28°29′	33°21′	38°15′	43°13′	46°14′	53°19′	58°26′	63°36′	68°40′	74°05′	79°22′	84°41′	88°36′
25°	09°05′	13°39′	18°15′	22°55′	27°37′	32°24′	37°15′	42°11′	47°12′	52°18′	57°30′	62°46′	68°07′	73°32′	78°59′	84°29′	88°54′
30°	08°41′	13°04′	17°30′	22°00′	26°34′	31°14′	36°00′	40°54′	45°54′	51°03′	56°19′	61°42′	67°12′	72°48′	78°29′	84°14′	88°51′
35°	08°13′	12°23′	16°36′	20°54′	25°18′	29°50′	34°30′	39°19′	44°19′	49°29′	54°49′	60°21′	66°03′	71°53′	77°51′	83°54′	88°47′
40°	07°41′	11°36′	15°35′	19°39′	23°51′	28°12′	32°44′	37°27′	42°23′	47°35′	53°00′	58°40′	64°35′	70°43′	77°02′	83°29′	88°42′
45°	07°06′	10°04′	14°25′	18°15′	22°12′	26°20′	30°41′	35°16′	40°07′	45°17′	50°46′	56°36′	62°46′	69°14′	76°00′	82°57′	88°35′
50°	06°28′	09°46′	13°10′	16°41′	24°21′	24°14′	23°20′	32°44′	37°27′	42°33′	48°04′	54°02′	60°29′	67°22′	74°40′	82°15′	88°27′
55°	05°46′	08°44′	11°48′	14°58′	18°19′	21°53′	25°42′	29°50′	34°21′	39°20′	44°49′	50°53′	57°36′	64°58′	72°55′	81°20′	88°15′
60°	05°02′	07°38′	10°19′	13°07′	16°06′	19°18′	22°45′	26°33′	30°47′	35°32′	40°54′	46°59′	53°57′	61°49′	70°34′	80°05′	88°00′
65°	04°15′	06°28′	08°45′	11°09′	13°43′	16°29′	19°31′	22°55′	26°44′	31°07′	36°12′	42°11′	49°16′	57°37′	67°21′	78°19′	87°38′
70°	03°27′	05°14′	07°06′	09°03′	11°10′	13°26′	16°00′	18°53′	22°11′	26°02′	30°39′	36°15′	43°13′	51°55′	62°43′	75°39′	87°05′
75°	02°37′	03°58′	05°23′	06°53′	08°36′	10°16′	12°15′	14°30′	17°09′	20°17′	24°08′	29°02′	35°25′	44°01′	55°44′	71°20′	86°09′
80°	01°45′	02°40′	03°37′	04°37′	05°44′	06°56′	08°17′	09°51′	11°41′	13°55′	16°44′	20°25′	25°30′	32°57′	44°33′	63°15′	84°15′
85°	00°53′	01°20′	01°49′	02°20′	02°53′	03°30′	04°11′	04°59′	05°56′	07°06′	08°35′	10°35′	13°28′	18°01′	26°18′	44°54′	78°41′
89°	00°10′	00°16′	00°22′	00°28′	00°35′	00°42′	00°50′	01°00′	01°11′	01°26′	01°44′	02°09′	02°45′	03°44′	05°59′	11°17′	44°59′

The use of this table is obvious without much explanation. A single example will suffice. Suppose a section be drawn running north-east and south-west, and it crosses certain beds dipping north at 35° (or certain cleavage planes, or a vein, or a fault, or any other plane having that dip), what angle ought we give to those beds in the section in order to give a true representation of the apparent dip they would have in a cliff running parallel to the section?

In this case the angle between the direction of the dip and that of the section, or between north and north-east = 45°; which we look for in the vertical column on the left of the table, the angle of the dip = 35°, which we find in the horizontal column at the top of the table. At the intersection of these two lines in the body of the table we should find 26° 15', the angle required.

In practice the minutes of the angle are never required, but as it involved no extra trouble to insert them they are given, as the table might possibly be of use in other ways where more minute accuracy is requisite.

It is plain that the table can be equally used to find the true dip where the apparent dip only can be observed in a real cliff, provided the angle between the line of section and the strike (and therefore the direction of the true dip) of the beds can be ascertained. This, however is a case which rarely occurs in practice. When it does, of course the nearest angle to the observed apparent dip will be sought in the body of the table, on the line opposite to the angle between the direction of the cliff and that of the strike ± 90°, and the angle of the real dip answering to it will be found at the top of the table.