The simple operation of levelling from a known point to another whose level is required may, if the distance is short, be obtained from one pair of sights taken with the instrument at one spot. Thus if the known level of a point A be 40'00 feet, and the unknown point B be at a distance of 350 feet from it, the instrument may be set up at any convenient spot from which A and B are visible, the level-staff being held up successively at A and B, and readings on them being obtained 5:25 and 675 feet respectively, the required level of the point B is hence 38.50 feet. It is however better in this as in every case to use equidistant sights, that is, that the instrument should be set up at a point equidistant from A and B, or nearly so, but not necessarily in the line AB and at 175 feet from each; as greater accuracy is thus obtained, the distorting effects of curvature and refraction in the two sights being thus equalised and neutralised in the resulting difference.

The circumstances and limits under which a difference of level may be thus taken by one pair of sights are, that the greater distance sighted shall not exceed the powers of the telescope for clearly reading the graduations under the actual atmospheric condition, that the fall or rise in the sum of the two distances sighted shall not be so great that the line of sight may pass below the zero of the one staff or above the top of the other; and lastly, under any circumstances in ordinary work it is rarely advisable to take any sight to a distance more than 650 feet, and hence the two points A and B should not be more than 1300 feet apart. This limit is so arranged that the effect of curvature and refraction may be neglected in levelling operations.

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First, to consider the effect of curvature. earth being a spheroid, which for these purposes may be considered to be a sphere, any truly horizontal line of considerable length is a circular arc described from a centre which is the centre of the earth; the line of sight directed on to a level-staff through a levelled telescope is (if the effect of refraction be neglected) a straight line tangential to this arc at the eyepiece which hits off a graduation on the level-staff above the actual point of equal height; the difference of height is the correction to be deducted from any reading on account of the curvature of the earth, and is a third proportional to the diameter of the earth and the distance between the instrument and the staff, or if taken in sufficiently close numbers,

Correction in feet (distance in feet)2+42 000 000;

=

hence to need a correction amounting to -o'o on this account the distance for the sight would be about 650 feet. In this case the cause of error is invariable, the correction can at any time be calculated and subtracted from any reading, and if nearly equidistant sights are adhered to, it may be entirely neglected.

Next, to consider the effect of atmospheric refraction. The amount of error due to such refraction varies with the condition of the air, and though it would be misleading to apply a mean value which might possibly never be justly due to actual conditions more than once or twice in a lifetime, yet the maximum value, which is about half of the error due to curvature but of opposite sign, can be taken into account and guarded against. The extreme correction on this score hence would be +0.005 in a sight of 650 feet, and this is just an amount appreciable in ordinary reading.

Lastly, reflex refractive action. When the visual ray of the observer passes close to the surface of the earth in its direction to the object, it is in long sights liable to a certain amount of distortion, the amount of correction for which cannot be estimated or calculated according to any law or formula, but may either be positive or negative in sign. It has been noticed by the author more especially in levelling in South America and India, and occasionally in very hot and bright weather in England and Wales; and it is evidently distinct from the simple refraction before mentioned as due to mere quality of the air. From its variation with the nature of the surface, from dry parched sand up to very damp vegetation of heavily irrigated standing crops and rank lagoon growth, as well as with the temperature and hygrometrical condition of the air, it appears that the cause is a dynamic change, a hygrometrical movement (in distinction to a hygrometrical condition) which causes a reflex refractive action on the visual ray that comes within range of this movement. An increased elevation of the line of sight by a foot, and sometimes only half-a-foot, may be sufficient to free the ray from this distortion. Further experiment on an extended scale will probably result eventually in the determination of some correct law and formula by which this class of error may be rigidly defined; in the interim its existence is here indicated by way of warning, and its mean value as estimated by the author lies between +00025 and -0.0025, even at comparatively short distances. (No mention of this phenomenon has been found recorded.) In England it is generally almost inappreciable in amount during cool weather; and probably also in any country when working over dry naked soil.

These two latter elements of uncertainty, added to the first, of alteration, combine to form the limit to levelling sights suited to all ordinary possible conditions, that has already been given. There are, also, other sources of error which when they combine to produce a cumulative result may seriously affect a single pair of sights, but as they generally 'do not, they will be treated subsequently with reference to extended series of sights, in a section devoted to that subject.

Levelling to obtain solely the relative height of a distant point is effected by taking a continuous series of pairs of sights from the given up to the distant point, each pair of sights being taken in the manner and under the limits already explained. The reduced level of the distant point then the reduced level of the starting point+sum of all the back sights-sum of all the fore sights.

Levelling to obtain a correct section of a continuous surface is generally similar to the last operation, but varies from it in that the intermediate points and the changing points through which the levelling proceeds must be correctly chosen, and also have their levels determined so that all variation of slope may be truly shown on the section. In the attached figure (21) A is the given starting point, C, D, F are changing points, and B and E are intermediate points, while G is the last point of the section. The sights to A, B, and C are taken from one position of the instrument, and the level of A being given, those of B and C are thus obtained; the level-staff being kept at C the instrument is moved to its second position, whence sights are taken to C and D; the staff at C having been

turned round to face the instrument, this pair of readings gives the level of D; from the third position

FIGURE 21. METHOD OF LEVELLING A SECTION.

of the instrument sights are taken on D, E, and F; and from the fourth position sights on F and G, from which the levels of all these points are deduced by addition and subtraction. The back sights are the readings on the starting and changing points A, C, D, F, taken in the direction of the starting point; the fore sights are the readings on the changing points, and the final point C, D, F, G taken in the direction of the final point or onward direction; and it is evident that an error in any one of these readings will affect the reduced level of any subsequent point obtained through them; for this reason the level-staff should be held truly vertical with plumb line, or swung gently backwards and forwards by the holder to get a minimum reading at changing points; while any error in observing on an intermediate point B or E will merely affect the reduced level of that point and no other, hence less care is necessary in intermediate readings. The resulting section will not only depend on the actual ground surface, but also on the positions chosen for the level-staves; for if the staves had in this case been held at A, a, b, c, d, G instead of at ABCDEFG, the dotted line shown in the figure would in that case be the resulting section; and as this would not truly represent the most marked changes of