870 MR HENRY BRIGGS ON 



and 1014*58 feet, of which the mean is 1014*57 feet. By subtracting each result from 

 the mean, the residual errors are respectively found to be : "05, '02, '06, '07, "03, '01 

 ft., their signs being disregarded. By taking the arithmetical mean of these — which 

 is found to be '04 ft. — the apparent average error of a single measurement is obtained. 

 Assuming that the same line of argument can be applied to apparent as to real errors, * 

 the theory of errors states that the real average error of the mean can be determined 

 from the apparent error of a single measurement by dividing the latter by sfn — 1 , 

 where n is the number of observations. Hence in this case the real average error of 

 the mean equals '04— *Jb, or '018 ft. The average fractional error is therefore '018 

 h- 1014, or about 1 in 56,300. 



Besides the fact that reasonable care will give a degree of precision in the base 

 measurement very superior to any stated in the third column of the above table, a 

 further slight increase of accuracy will result from the distribution of error in the 

 angles, t If the values of v in the table are adhered to as nearly as possible, the pro- 

 bability that the error in the check-base will work out under 1 in 12,000 is therefore in 

 actual circumstances considerably more than one-half. 



The Accuracy of Triang illation as a Method of Transmitting Distance. 



Having determined how the various lines of a triangulation scheme are affected by 

 error, it remains to discuss in what way these several errors combine in disturbing the 

 positions of the stations. 



Fig. 4 shows a chain of approximately equilateral triangles springing from a measured 

 base, c. Let station 1 be taken as origin of co-ordinates, and line c as arbitrary meridian. 

 An error in any point is stated with regard to station 1 as a fixed point and to the 

 direction of line c as a fixed direction. 



In dealing with the average total error affecting, say, the point 7 it is necessary to 

 regard separately the influence of angular and of base error. The effect of the latter 

 is easy to assess, since a fractional error in the base will cause the whole scheme to 



* 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. 



t 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 » *y' y or °' 8v > Dv adjusting the angles of a triangle by means of one equation of 

 i ondition. (See Chas. L. Cranjjaj.l's Text-book on Geodesy and Least Squares, 1907.) 





