base AG, call the difference d. Generally
Log. AC = Log. AB + Log. sin B - Log.
cosec. C, but since Log. AC has to become
Log. AC - x, x being the small logarithmic correction due to a gradual accumulation of error, then Log. AC + x = Log.
AB + (Log. sin B -

w

2
) + (Log.
cosec. C. -

x

2
). Take out the Log-
arithmic difference due to one second
for these angles in every triangle of
the series as indicated by a dot in the
diagram and call the sum of these Log.
differences s; then x will be an arc of

s

correction in seconds to be applied to each
of the above mentioned angles. When the
accumulated error causes the side AB to
exceed its proper value, this correction is
subtractive from those angles to which the
sine is applied and additive to those taking
the cosecant, but the contrary is to take
place when side AB is found to be short of
its proper value. The sides may either be
recomputed with these new angles or cor-
rected in proportion to the angular correc-
tions applied, and the discrepancy in the side
AB will then totally disappear. These correc-
tions are only to be applied when the
errors are extremely minute and within the
probability of errors in observation. (See
instructions for topographical surveying by
Lt.-Col. Waugh, Bengal Engineers).

Polygons succeeding in order to be computed on the same principles.

34. Successive polygons CDEHIK, &c.,
    should be operated upon in the same man-
    ner, observing in every case that two sides
    have always become unalterably fixed, the
    one affording a base for continuation whilst
    the other serves as a check... In this man-
    ner any number of triangles may be ex-
    tended and the work verified during pro-
    gress until the measured base of verification
    is reached. When the triangles form into
    Quadrilaterals as GLMF they present no
    check unless the bearings of both diagonals
    have been observed. Elimination of error
    in this latter case is analogous to the method
    just described, computing the triangles in
    the following order FLG, GML, GFM.

Test of Triangulation by base of Verification.

35. One of the severest tests that a Tri-
    gonometrical survey can be submitted to is
    a comparison of the computed length of
    the second measured base, brought up in
    the manner described with its actual mea-
    sure. The maximum error allowable is
    one foot per mile, but in practice it has
    been found to be considerably within this
    limit. Should, however, the errors be large
    but not exceeding the above limit, their
    elimination on the principles before men-
    tioned should be accomplished; this pro-
    cess entails the reworking of the computa-
    tions.

Meridianal and Perpendicular Co-ordinates.

36. The next process is to compute the
    meridianal co-ordinates of every station
    from some one station generally chosen in
    a central part of the survey. If, however,
    one of the stations is also that of a previously
    executed trigonometrical survey, it would
    be proper to adopt it for this purpose, since
    it will furnish a point of departure and
    direction of the meridian in terms of the
    former survey and an ascertained height
    above sea level. The bearing of any one
    side in the triangulation having been deter-
    mined upon those of all the rest are
    obtained by the application of the corrected
    mean angles of the computations. Suppose
    B to be the point of departure, first com-
    pute the meridian distance of A with the
    given bearing and length of side AB, then
    that of C from the two stations A and B,
    and in like manner those of D, E, F, &c.,
    consecutively. It is to be remarked that
    the two resulting distances for any station
    should uniformly agree to a decimal of a
    link, since there exists no inconsistency
    between the angles and sides of the trian-
    gles. The differences between the meridian
    and perpendicular distances between any
    two stations afford data for computing the
    bearing and length of side, and by com-
    parison with the observed bearing presents
    another check upon the accuracy of the
    work.

Plotting.

37. The scale to be adopted for the map
    is 80 chains to an inch. Previous to plot-
    ting the paper should be divided into
    squares representing meridian and perpen-
    dicular lines six miles apart. These lines
    besides facilitating the protraction of the
    work serve as measures of latitude and
    departure for the connection of sheet to sheet
    and eventually become convenient basis
    for setting off the true meridianal and longi-
    tudinal lines. Also, with the aid of a table
    of Natural Tangents bearings may easily
    and accurately be protracted from them,
    and they preserve a uniform scale for re-
    ference notwithstanding the expansion or
    contraction of the paper caused by changes
    of temperature. It is recommended to plot
    the stations from their computed Meridianal
    Co-ordinates and then to verify each by the
    protraction of its bearing and distance from
    other stations. The bearings and lengths
    of the sides must be neatly written upon all
    the lines specifying whether derived from
    observation, computation, or measurement
    conformable with the rules furnished in
    Appendix. The Meridianal Co-ordinates
    are to be written against every station.

Differences compared between plane and spherical measurements as affecting the Survey.

38. As the spherical excess has been
    omitted to be taken into account in the
    computations of the triangles,