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ABBREVIATIONS.

+ Increased by.

Diminished by. X Multiplied by. • Divided by.

Equal to.

Since, or seeing that. .. Hence, or therefore.

: Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quantities.

:: Indicates an equality of ratios, and connects equal ratios in a proportion. Thus, a :b::c: d indicates that a b=0 id; or it may be read, a is to b as c is to d.

( ) Brackets indicate that the operations embraced by them shall first be performed, and the result treated as a single factor in the remaining processes required by a formula. Thus, (a X 0) = (a + b) requires that the product of a and b shall be divided by their sum.

A2. A small secondary figure annexed thus to an expression is called its exponent. It requires the principal to which it is attached to be used as many times in continued multiplication as there are units in the exponent. Thus, A2 = A X A; A? = A X A X A, which is called the cube, or third power, of A.

This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken.

If preceded by a small secondary figure, called the index, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken; and so on.

v If the index be fractional, as in the marginal figure, it requires that the square root of the thirıl power of the quantity covered shall be taken. B. M. Benchmark : any fixed reference point for the level,

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as outcropping ledge, water-table of building, or other permanent object. Usually a blunt conical seat for the rod, hewn on a buttressed tree-base, having a small nail sometimes driven flush in the top of it, and a blaze opposite, on which the elevation is marked with kiel.

T. P. Turning-point : usually marked O in the field-book.

P. I. Point of intersection; as of tangents, which are to be connected by a curve.

A. D. Apex distance : i.e., the distance from the P. I. to the point where a curve merges in the tangent.

P. C. Point of curve : the stake-mark at the beginning of

à curve.

P. T. Point of tangent: the stake-mark at the end of a

curve.

P. C. C. Point of compound curvature : the stake-mark where a curve merges in another of different curvature, turning in the same direction.

P. R. C. Point of reverse curvature: the stake-mark where a curve merges in another turning in the opposite direction.

B. S. Backsight, in transit work; or the reading of the rod to ascertain the instrument height in levelling.

F. S. Foresight, in transit work; or the reading of the rod to ascertain elevations in levelling.

H. I. Height of instrument : elevation of the level above the datum or zero plane,

H. W. High water.
L. W. Low water,

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XII. Setting slope stakes

30

XIII. Vertical curves

36

XIV. The transit

40

To adjust the level tubes

40

To adjust the vertical hair so that it shall re-

volve in a plane, and mark backsight and fore-

sight points in the same straight line

40

To adjust the needle

41

XV. Miscellaneous .

42

The vernier

42

To read an angle

43

To re-magnetize a needle

44

To replace cross-hairs

To fix a true meridian

44

PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE.

XVI. Propositions relating to the circle

49

XVII. Circular curves on railroads

50

XVIII. To find the radius, the apex distance, the

length, the degree, &c., of a curve

52

Given the intersection angle I and radius R, to

find the tangent T

52

Given the intersection angle I and tangent T, to

find the radius R

53

Given the intersection angle I and chord AB=C,

connecting the tangent points, to find the

radius R

54

Given the intersection angle I and the degree of

curvature or deflection angle D, with 100-feet

chords, to determine the length of the long

chord C, the versed sine V, the external secant

S, or the tangent T

54

Given C, V, S, or T, of any curve, and D, the

degree of curvature, to find the intersection

angle I.

55

Given the intersection angle I and deflection

angle D, to find the length of the curve.

Given any radius R and chord C, to find the de-

flection angle D

57

Given any radius R and chord C, to find the de-

flection distance d

57

Given any radius R and chord C, to find the

tangential angle T

57

Given any radius R and chord C, to find the

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XIX. Ordinates

58

Given any radius R and chord C, to find the

middle ordinate M

58

Given the radius R, chord C, and middle ordi-

nate M, to find any other ordinate

59

Ordinates of a 1o curve, chord 100-feet

60

TRACING CURVES AND TURNING OBSTACLES IN THE FIELD.

XX. To trace a curve on the ground with the chain

only

63

XXI. To trace a curve on the ground with transit and

100-feet chain

66

XXII. Turning obstacles to vision in tangent

71

XXIII. Turning obstacles to measurement in tangent 73

SUGGESTIONS AS TO FIELD-WORK AND LOCATION-PROJECTS.

XXIV. Suggestions concerning field-work

79

XXV. The curve-protractor and the projecting of loca-

tions

84

Table showing the distance, D, in feet, at which

a straight line must pass from the nearest

point of any curve struck with radius r, in

order that a terminal branch having a radius

R=2 r, and consuming a given angle, x, may

merge in said straight line

88

Table showing the distance, d, in feet, at which

curves of contrary flexure must be placed

asunder, in order that the connecting tangent,

T, may be 300 feet long

89

PROBLEMS IN FIELD LOCATION.

XXVI. How to proceed when the P. C. is inaccessible . 93

XXVII. How to proceed when the P. C. C. is inaccessible, 95

XXVIII. To shift a P. C. so that the curve shall termi-

nate in a given tangent

96

XXIX. To substitute for a curve already located one of

different radius, beginning at the same point,

containing the same angle, and ending in a

fixed terminal tangent

97

XXX. Having located a curve A B C, to find the point

B at which to compound into another curve

of given radius, which shall end in tangent

EF, parallel to the terminal tangent of the

original curve, and a given distance from it 98

XXXI. To shift a P. C. C. so that the terminal branch of

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