for computation, and diminished by 180°;* the angles for computation as deduced above are those at which the lines a, b, c, are inclined to the first line a, or to the parallels thereof. After the deduction of the angles for computation, it is necessary to calculate the following terms:

the projections of a, b, c,

..... on the axis x; similarly y', y", y"...... being the projections of the same lines on the axis y, are called perpendicular co-ordinates; the signs of these coordinates depend upon the magnitude of the angles from which they are respectively derived, and these signs will be readily found by a reference to the following Table:

Table exhibiting the Signs of the direct and

Perpendicular Co-ordinates.

Magnitude of Signs of the Co-ordinates.

the angles for
computation.

After affixing proper signs to the direct and perpendicular co-ordinates, collect the former into one sum and the latter into another; the former of these sums augmented by the first distance (a), is the numerical value of x, while the latter sum is the numerical value of y.

After the values of x and y are determined, ✪ and p may be deduced by the following formulæ.

Now is of the same sign with y, and may be either positive or negative. In the former case, the terminus is to the right and in the latter to the left of o1; knowing the value of

* The established symbol for a semicircle or 180° is π.

.

e, as likewise its position with respect to 1, it is easy to trace the direct line of the route. For this purpose, put up the theodolite at the origin and take a reading to € 1. To this reading apply, according to its sigr, the 0; the resulting reading or the telescope set thereto, will point to the terminus of the survey. Again, if it be required to trace the route from the terminus, it may be done thus: According as is positive or negative add it to, or subtract it from, From the sum or difference so obtained, (augmented when less than the subtractor by 2) deduct the last angle for computation, the remainder will be the 0' at the terminus between the origin and the last station of the route. The e' may be of any value from 0° to 360°; it is likewise always positive. Adjust the theodolite over the terminus, and take a reading to the last Station; to this reading add the 0': the resulting reading will be the required direction of the origin from the terminus.

It is evident that p, determined as directed above, is in terms of the perambulator, calling R the value of the same distance as derived from a trigonometrical operation, it follows that R is the error of the Route Survey.

ρ

Without making any assumption as to the cause of this error, it is evident that this discrepancy must be expunged, before the details furnished by a Route Survey can incorporate with those of a trigonometrical operation.

The simplest and perhaps the only method of performing this, is by the following rule of proportion.

As the direct perambulator distance of the route (p)
The trigonometrical value thereof (R)

:: Each measured perambulator distance
: Its corresponding trigonometrical length.

Correcting by this proportion all the perambulator distances, as well as all the co-ordinates deduced therefrom, the resulting elements will obviously be in terms of the unit of the Trigonometrical Survey.

The following is an example of the field notes and computation deduced therefrom.

Mag. north, L00

02

314 11 0
132 24 0

...

10 30 10 30314 10 40 178 13 5 178 13 40
23 30 23 45 132 23 45

9 0134 9 10 178 14 15
23 15312 23 25

9 15 9 30 312 9 35 178 27 15 178 27 23 36 45 36 45 130 36 50

R130 41 45 301 40 15

35 30 36 15310 36 10 170 59 15 170 58 49
35 30 35 15121 35 25

58 30 58 30 162 58 40 215 54 10 215 54 0

8 10 215 53 50

2 0343 2 0

4.074 1.122