time inverted on C' T', and a third time on TC. If the distance of the middle offsets 92 92 be less or greater than 1 chain, that distance, and consequently the following ones, must be made equal to 1 chain, as in the preceding case.

= =

30+1.875 30.2344 ch. 30.23 ch. nearly and by trig.

ZBTC 165° 38', whence the position of

the tangent T C' T' becomes known; it will be unnecessary in this case to find TT C.

=

By table 4 the first offset on BT, viz. P1 91 1.65 in., from which the other offsets may be found by multiplying by 4, 9, 16, &c. which may be laid out as in the last case, p. 93 being the 30th offset from B, and meeting the corresponding offset p ́3 93 at 93, the middle of the curve B C'. The method of making the distances equal on the second part C' C of curve, if required, is the same as in Case I.

NOTE. If the curve be a long one, it may be divided, in this manner, into 3, 4, or 5 parts, according to its length.

CASE III. When the Length of the Curve exceeds the Limit adopted in Case I., but is considered too short to be divided into Two Parts, as in Case II.

Let the curve be one of 80 chains radius, and between 32 and 38 chains in length, then to avoid the trouble of adding another tangent, the offsets beyond the 8th must be calculated from the following formulæ :

:

If the radius of the curve be 160 chains, and its length about 50 or 60 chains, the offsets must be calculated, as above, beyond the 12 22 32 2 r' 2 r' 2 r

15th. The formula to the true lengths of the offsets, within the limits assigned, but beyond these limits the errors of the offsets begin gradually to augment, till they become too considerable to be overlooked. See Demonstration to Case I.

&c. being a very near approximation

PROBLEM IV.

To lay out the Curve, where Water, Swamps, Quarries, or other Obstructions prevent the use of the Chain, by Two Theodolites.

The position of the tangents A B, D C to the curve at its extremities B and C, its radius O B or O C, and the B OC supple

T

=

ment of the angle made by the tangents when prolonged, being determined; join B C, and take several equidistant points 91 92 93 &c. in the curve, from which points draw lines to B and C.

б

Take the 29, C B = an arc whose sine is wherein d

2 r d

= 91 92 = &c. and r = O B, or, because is, in practice, usually

required to be = 1 chain, take 49, CB. arc whose sine is

The several angles may then be arranged as below.

1

2 r

BOC-291 CB,

Therefore if theodolites be fixed at B and C, and the angles 91 C B, 91 B C be taken at the same time, the intersection of B 1 and C 91 will give the point 91. In the same manner by taking the angles 92 C B, 92 B C, the intersection of B 92 and C 92 will give the point 92, &c.

Ex. Let the radius O B = 80 chains, and the angle between the tangents A B, D C, when prolonged to meet, be 160°; then its supplement BOC= 80°, and BOC 40°. Also 49, CB

=

arc to siner (

= BOC

=

1

160

=

C B =2491 CB= 2 × 0°

40°.

0° 21′ 29′′ = 39° 38′ 31′′; ≤92

21′ 29′′=0° 42′ 58′′ and 92 B C =

229, CB = 40° - 0° 42′ 58′′ 39° 17′ 2", &c. The

=

angles would be best arranged for use as follows, the opposite ones to be taken at the same time.

-91

This list of angles must be continued till n q1 C B can no longer be taken from BO C, and one angle in each column being taken at the same time, as 49, C B at C and q1 B C at B will give the point q1: and so on for the other points 92, 93, &c. See the following

notes.

NOTE 1. Where the obstructions are such as to prevent the stumps being put down at the consecutive points q,, 92, 93, &c., it would be best to take every fourth angle, thus obtaining the points 44' 8' 12, &c., which will be 4 chains apart on the curve, the intermediate points being left to be put in when the work of the line has progressed so far as to present a better opportunity. Or, if it should be thought preferable to avoid taking a multiplicity of angles, every fourth angle may be taken in any case, as this method is equally available whether obstructions exist or not.

NOTE 2.- Mr. Rankine's method of setting out the curve. As this method is a modification of the one just given, it will be proper to explain it here. It depends on

the property already given, i. e. sin. 4q, C B

of tables = 1. A theodolite being fixed at B, and the Lq, BT being taken, the distance d, or 1 chain, is set off from B to q, in the direction of the axis of the instrument; the 4g, B T is next taken 2 L B T, and the distance 9, 92 chain applied between q, and the direction of the axis of the instrument, and so on to the end of the curve. It will be seen that this method, though elegant in theory, is quite as objectionable in practice as the common method, given in Problem II., since the least errors, at the commencement of the operation, will gradually multiply as the work proceeds, whether the errors be in taking the angles q, B T, q2 B T, &c., or in laying out the distances B 41, 41 92, &c., or in both. Whereas in the method by two theodolites, just given, an error in taking one angle does not affect any

other part of the work, but simply the point to which the erroneous angle refers, the position of which point may be easily corrected. Besides, Mr. R.'s method will be found impracticable where hills, woods, buildings, &c. are within the concavity of the curve, his method having been ostentatiously put forth as a universal method.

Demonst, to Prob. IV.-The angles B, C, Bq2 C, Bq, C, &c., being in the same segment, are well known to be equal and constant, and also equal to half the supplement of 4 B O C, and consequently the sums of the angles of the triangles Bq1 C, Bq, C, &c. adjacent to B C are equal to 4 B O C, whence 4 q, B C = { 4 BOC — Lq, CB, 492 B C = 1 4 BO C L2 C B (because equal angles stand on equal arcs) 4 BOC-24g, CB, &c. Also join O q, and put B O = r, and B q1: d, then it is well known that 4 BO q1 Lq, C B, and, because the sides B O, q, O rad. x d qr

=

=

of the triangle B O q, are equal, sin. 1⁄2 4 B O q1

=

sin. 4q, CB=

1, and 8 = 1 chain, as required in practice, sin. 49, C B

=

; or, by

1

=

2r

PROBLEM V.

To lay out a Railway Curve by means of Ordinates or Offsets from its Chord or Chords, no material Obstruction being supposed to exist on the concave Side of the Curve to prevent the use of the Chain. Let B C C be a portion, or the whole, of a curve of a railway; A T, T' D ́ and C D tangents to the curve at B, C' and C; O the

centre, O B the radius, B C' and C' C chords of the curve; which chords must not exceed 40 chains, if the radius be 80 or 120 chains, but they may be 60 chains, if the radius exceed 120 (this limitation is necessary to prevent the offsets P1 91, P2 92, &c. being too long, as was observed with respect to the offsets from the tangents, in Prob. JU.): and let the radius 92 O bisect the curve and chord in q3 and p3

respectively. Then from the right-angled triangle B p. O, in which BO and A p3 = B C' are given, the ▲ B O p3 = LTBC may be found, which determines the position of the chord B C'; also Op3 may be found from the same triangle being = √O B2 − B p23 VO B2B C'2. Put O Br, Ops, and the chord B p3 =n chains, then

=

93 P3 being supposed to be the offset at the middle point of the curve B C', not the third offset as shown in the figure, it being impossible to draw all the offsets without confusing the figure or making it unnecessarily large. After reaching the middle point 3 of the curve the same offsets are repeated in an inverted order till the curve shall have been set out to C'. The same operation may be repeated as often as necessary, till the whole curve be completed, observing to make the BC' C 2 complement of TB C', which has been already found. See the following Notes.

NOTE 1. When B C' is the whole curve, and its chord Bp, C includes a fractional part of a chain, the distance of the offsets on each side of the middle of the curve will be less than one chain; therefore that distance, and consequently the following ones, must be made equal, as was shown with respect to the distances in Prob. III.

NOTE 2.-If the last chord, which suppose to be C' C, be less than the preceding chord or chords, the OC' C must be found, and added to the LOB C' or O C' B, which will give the 4 B C' C, showing the direction of the chord C' C.

NOTE 3. This method of laying out the curve is seldom used, on account of the calculations it involves. It may, however, be used with advantage where a winding river or cliff is close to the convex side of the curve, or protrudes in some places a little through the curve; thus preventing the use of any other method, except that in Prob. IV., which requires two theodolites.

Definition of the Compound Curve. (See figures to Prob. VI.)

The compound curve B C C' C", joining the tangents A B, D C", is composed of three circular arcs B C, C C, C' C", having common normals O C, O'C' at their points of junction C, C'; and therefore

* Demonstration.

=

=

1 chain. =

(n 1); whence

- Draw OQ parallel to B C', q,Q to O Q, and join 9,0; and let n = number of chains in B C' or p ̧С', and C'p1 Then O Q = n P3 P1 1; and (Euc. I. 47.) 91 Q √22 √r2 — (n3 — 1 )o —p1Q: but p, Q = P30 = s, .'. q¡ P1 91P1 = Similarly 92 P2 is found

= √/r2 − (n2 − 2)9

=

s, &c. Q. E. D.

√/p2 - (n − 1 )3 — s.