AD sin CAD 1866.15 sin 59 20"

or CD = ---------- = ------------------- sin SCD sin 85 9" 53"

or log CD = log 1866.15 + L sin 59 20" - L sin 85 9" 53"

= 3.2709456 + 9.9345738 - 9.9984516

= 3.2070678. " . CD = 1610.90 ft

The distance between any two positions of the float can be obtained by calculation in a similar way to that in which the length C D was obtained, but this is a lengthy process, and is not necessary in practical work. It is desirable, of course, that the positions of all the stations be fixed with the greatest accuracy and plotted on the map, then the position of the float can be located with sufficient correctness, if the lines of sight obtained from the angles read with the theodolites are plotted, and their point of intersection marked on the plan. The distance between any two positions of the float can be scaled from the plan.

The reason why close measurement is unnecessary in connection with the positions of the float is that it represents a single point, whereas the sewage escaping with considerable velocity from the outfall sewer spreads itself over a wide expanse of sea in front of the outlet, and thus has a tangible area. The velocity of any current is greatest in the centre, and reduces as the distance from the centre increases, until the edges of the current are lost in comparative still water; so that observations taken of the course of one particle, such as the float represents, only approximately indicate the travel of the sewage through the sea. Another point to bear in mind is that the dilution of the sewage in the sea is so great that it is generally only by reason of the unbroken faecal, or other matter, that it can be traced for any considerable distance beyond the outfall. It is unlikely that such matters would reach the outlet, except in a very finely divided state, when they would be rapidly acted upon by the sea water, which is a strong oxidising agent.

CHAPTER XV.

HYDROGRAPHICAL SURVEYING.

Hydrographical surveying is that branch of surveying which deals with the complete preparation of charts, the survey of coast lines, currents, soundings, etc., and it is applied in connection with the sewerage of sea coast towns when it is necessary to determine the course of the currents, or a float, by observations taken from a boat to fixed points on sh.o.r.e, the boat closely following the float. It has already been pointed out that it is preferable to take the observations from the sh.o.r.e rather than the boat, but circ.u.mstances may arise which render it necessary to adopt the latter course.

In the simplest case the position of the boat may be found by taking the compa.s.s bearings of two known objects on sh.o.r.e. For example, A and B in Fig. 37 may represent the positions of two prominent objects whose position is marked upon an ordnance map of the neighbourhood, or they may be flagstaffs specially set up and noted on the map; and let C represent the boat from which the bearings of A and B are taken by a prismatic compa.s.s, which is marked from 0 to 360. Let the magnetic variation be N. 15 W., and the observed bearings A 290, B 320, then the position stands as in Fig. 38, or, correcting for magnetic variation, as in Fig. 39, from which it will be seen that the true bearing of C from A will be 275-180=95 East of North, or 5 below the horizontal, and the true bearing of C from B will be 305-180=120 East of North, or 35 below the horizontal.

These directions being plotted will give the position of C by their intersection. Fig. 40 shows the prismatic compa.s.s in plan and section. It consists practically of an ordinary compa.s.s box with a prism and sight-hole at one side, and a corresponding sight-vane on the opposite side. When being used it is held horizontally in the left hand with the prism turned up in the position shown, and the sight-vane raised. When looking through the sight-hole the face of the compa.s.s-card can be seen by reflection from the back of the prism, and at the same time the direction of any required point may be sighted with the wire in the opposite sight vane, so that the bearing of the line between the boat and the required point may be read. If necessary, the compa.s.s-card may be steadied by pressing the stop at the base of the sight vane. In recording the bearings allowance must in all cases be made for the magnetic pole. The magnetic variation for the year 1910 was about l5 1/2 West of North, and it is moving nearer to true North at the rate of about seven minutes per annum.

[Ill.u.s.tration: FIG. 37.--POSITION OF BOAT FOUND BY COMPa.s.s BEARINGS.]

[Ill.u.s.tration: FIG. 38.--REDUCTION OF BEARINGS TO MAGNETIC NORTH.]

[Ill.u.s.tration: FIG. 39.--REDUCTION OF BEARINGS TO TRUE NORTH.]

There are three of Euclid"s propositions that bear very closely upon the problems involved in locating the position of a floating object with regard to the coast, by observations taken from the object. They are Euclid I. (32), "The three interior angles of every triangle are together equal to two right angles"; Euclid III. (20),

"The angle at the centre of a circle is double that of the angle at the circ.u.mference upon the same base--that is, upon the same part of the circ.u.mference,"

or in other words, on a given chord the angle subtended by it at the centre of the circle is double the angle subtended by it at the circ.u.mference; and Euclid III. (21),

"The angles in the same segment of a circle are equal to one another."

[Ill.u.s.tration: Fig. 40.--Section and Plan of Prismatic Compa.s.s.]

Having regard to this last proposition (Euclid III., 21), it will be observed that in the case of Fig. 37 it would not have been possible to locate the point C by reading the angle A C B alone, as such point might be amywhere on the circ.u.mference of a circle of which A B was the chord. The usual and more accurate method of determining the position of a floating object from the object, itself, or from a boat alongside, is by taking angles with a s.e.xtant, or box-s.e.xtant, between three fixed points on sh.o.r.e in two operations. Let A B C, Fig. 41, be the three fixed points on sh.o.r.e, the positions of which are measured and recorded upon an ordnance map, or checked if they are already there. Let D be the floating object, the position of which is required to be located, and let the observed angles from the object be A D B 30 and B D C 45. Then on the map join A B and B C, from A and B set off angles = 90 - 30 = 60, and they will intersect at point E, which will be the centre of a circle, which must be drawn, with radius E A. The circle will pa.s.s through A B, and the point D will be somewhere on its circ.u.mference. Then from B and C set off angles = 90-45 = 45, which will intersect at point F, which will be the centre of a circle of radius F B, which will pa.s.s through points B C, and point D will be somewhere on the circ.u.mference of this circle also; therefore the intersection of the two circles at D fixes that point on the map. It will be observed that the three interior angles in the triangle A B E are together equal to two right angles (Euclid I. 32), therefore the angle A E B = 180 - 2 x (90 - 30) = 600, so that the angle A E B is double the angle A D B (Euclid III., 20), and that as the angles subtending a given chord from any point of the circ.u.mference are equal (Euclid III, 21), the point that is common to the two circ.u.mferences is the required point. When point D is inked in, the construction lines are rubbed out ready for plotting the observations from the next position. When the floating point is out of range of A, a new fixed point will be required on sh.o.r.e beyond C, so that B, C, and the new point will be used together. Another approximate method which may sometimes be employed is to take a point on a piece of tracing paper and draw from it three lines of unlimited length, which shall form the two observed angles. If, now, this piece of paper is moved about on top of the ordnance map until each of the three lines pa.s.ses through the corresponding fixed points on sh.o.r.e, then the point from which the lines radiate will represent the position of the boat.

[Ill.u.s.tration: Fig. 41. Geometrical Diagram for Locating Observation Point Afloat.]

The general appearance of a box-s.e.xtant is as shown in Fig. 42, and an enlarged diagrammatic plan of it is shown in Fig. 43. It is about 3 in in diameter, and is made with or without the telescope; it is used for measuring approximately the angle between any two lines by observing poles at their extremities from the point of intersection. In Fig. 43, A is the sight- hole, B is a fixed mirror having one-half silvered and the other half plain; C is a mirror attached to the same pivot as the vernier arm D. The side of the case is open to admit rays of light from the observed objects. In making an observation of the angle formed by lines to two poles, one pole would be seen through the clear part of mirror B, and at the same time rays of light from the other pole would fall on to mirror C, which should be moved until the pole is reflected on the silvered part of mirror B, exactly in line, vertically, with the pole seen by direct vision, then the angle between the two poles would be indicated on the vernier. Take the case of a single pole, then the angle indicated should be zero, but whether it would actually be so depends upon circ.u.mstances which may be explained as follows: Suppose the pole to be fixed at E, which is extremely close, it will be found that the arrow on the vernier arm falls short of the zero of the scale owing to what may be called the width of the base line of the instrument. If the pole is placed farther off, as at F, the rays of light from the pole will take the course of the stroke-and-dot line, and the vernier arm will require to be shifted nearer the zero of the scale. After a distance of two chains between the pole and s.e.xtant is reached, the rays of light from the pole to B and C are so nearly parallel that the error is under one minute, and the instrument can be used under such conditions without difficulty occurring by reason of error. To adjust the box-s.e.xtant the smoked gla.s.s slide should be drawn over the eyepiece, and then, if the sun is sighted, it should appear as a perfect sphere when the vernier is at zero, in whatever position the s.e.xtant may be held. When reading the angle formed by the lines from two stations, the nearer station should be sighted through the plain gla.s.s, which may necessitate holding the instrument upside down. When the angle to be read between two stations exceeds 90, an intermediate station should be fixed, and the angle taken in two parts, as in viewing large angles the mirror C is turned round to such an extent that its own reflection, and that of the image upon it, is viewed almost edgeways in the mirror B.

[Ill.u.s.tration: Fig. 42.--Box-s.e.xtant.]

It should be noted that the box-s.e.xtant only reads angles in the plane of the instrument, so that if one object sighted is lower than the other, the angle read will be the direct angle between them, and not the horizontal angle, as given by a theodolite.

The same principles may be adopted for locating the position of an object in the water when the observations have to be taken at some distance from it. To ill.u.s.trate this, use may be made of an examination question in hydrographical surveying given at the Royal Naval College, Incidentally, it shows one method of recording the observations. The question was as follows:--

[Ill.u.s.tration: Fig. 43.--Diagram Showing Principle of Box- s.e.xtant]

"From Coastguard, Mound bore N. 77 W. (true) 0.45 of a mile, and Mill bore, N. 88 E, 0.56 of a mile, the following stations were taken to fix a shoal on which the sea breaks too heavily to risk the boat near:--

Mound 60 C.G. 47 Mill.

[Greek: phi]

Centre of shoal Mound 55 C.G. 57 30" Mill.

[Greek: phi]

Centre of shoal.

Project the positions on a scale of 5 in = a mile, giving the centre of the shoal." It should be noted that the sign [Greek: phi] signifies stations in one line or "in transit," and C G indicates coastguard station. The order of lettering in Fig. 44 shows the order of working.

[Ill.u.s.tration: Fig. 44.--Method of Locating Point in Water When Observations Have to Be Taken Beyond It.]

The base lines A B and A C are set out from the lengths and directions given; then, when the boat at D is "in transit" with the centre of the shoal and the coastguard station, the angle formed at D by lines from that point to B and A is 60, and the angle formed by lines to A and C is 47. If angles of 90 - 60 are set up at A and B, their intersection at E will, as has already been explained, give the centre of a circle which will pa.s.s through points A, B, and D. Similarly, by setting up angles of 90-47 at A and C, a circle is found which will pa.s.s through A C and D. The intersection of these circles gives the position of the boat D, and it is known that the shoal is situated somewhere in the straight line from D to A. The boat was then moved to G, so as to be "in transit" with the centre of the shoal and the mound, and the angle B G A was found to be 55, and the angle A G C 57 30". By a similar construction to that just described, the intersection of the circles will give the position of G, and as the shoal is situated somewhere in the line G B and also in the line A D, the intersection of these two lines at K will give its exact position.