^{page 849 note *} To write the plus-or-minus sign before average errors, as is done throughout the paper, is unorthodox. An average error is usually defined as the arithmetical mean of the separate errors taken regardless of their signs, and is expressed in books as . It is, however, equally logical to express it as , and hence it appears to have the same right to the double sign as, say, the probable error. Moreover, to give an average error the positive sign only, and then to apply the usual theory of errors to it (as is generally done) seems to the writer inconsistent, since there is, in such an application, a tacit admission that the error is minus as often as plus

^{page 856 note *} Compare with Holman's, S. J.Discussion of the Precision of Measurements, New York, 1901, second edition, p. 69.Google Scholar

^{page 867 note *} All such lines as a, d, g, etc., which are not bounding lines of the area covered by the scheme, are spoken of here as internal sides.

^{page 869 note *} More exactly, a chance of about 9 to 7, owing to the average error being equal to 1·18 of the probable error.

^{page 869 note †} Compare with Johnson's, J. B.Theory and tradice of Surveying (Wiley, New York).Google Scholar

^{page 870 note *} An assumption often made. In this case it is evidently of no use unless the tape has recently been standardised, for if not, constant error of comparatively large magnitude may be introduced which would render nugatory any such calculation as that above. The writer has seen calculations for probable error made on two measurements ; the result (though believed to be valuable) was of course quite worthless. A base should be measured quite five times before the error can be analysed—and the mathematician will probably say that this number is insufficient.

The method of evaluating the average error in angle, given earlier in the paper, and based on the summation of the angles of a triangle, makes an attempt to assess the real error. Hence the need of obtaining the real average error in the base, as distinct from the apparent error, since the linear and angular errors are compounded in many of the expressions derived. From the strictly mathematical point of view there are objections to compounding errors determined in so different a way; however, the conclusions reached will be sufficiently near the truth to be of service in practice.

^{page 870 note †} The usual tendency is, perhaps to exaggerate the value of distributing error. It can be shown that the average angular error, v, is only reduced to or 0·8υ, by adjusting the angles of a triangle by means of one equation of condition. (See Crandall's, Chas. L.Text-book on Geodesy and Least Squares, 1907.)Google Scholar

^{page 874 note *} See Middleton, and Chadwick's, Treatise on Surveying, 1904, Part I., p. 193, for instance.Google Scholar

^{page 874 note †} For example, see “Notes on Railway Surveying,” by Albrecht, C. J., Min. Proc. I.C.E., cliv. p. 262Google Scholar, where a compass instrument is advised for running trial lines in country overgrown by forest, or in broken ground.

^{page 876 note *} Vide Weitbrecht's, W.Ausgleichungsrechnung nach der Methode der Kleinsten Quadrate, p. 63.Google Scholar