876 MR HENRY BRIGGS ON 



quantities could be studied by means of (17) and (18), roughly the direction of the 

 closing side being taken, say, as arbitrary north for co-ordinates. (A preliminary 

 plotting with protractor and scale, or perhaps a measurement from an old plan, will 

 enable the direction of the missing line to be obtained sufficiently closely for this 

 purpose.) Then the result as given by (17) would evaluate the average error in the 

 closing side, while that given by (18) divided by the length of the line would express 

 the average error in bearing in radians. Considering that this mode of analysis could 

 be of great service, particularly in mine-surveying, it is a pity that is so tedious to 

 carry out and that it involves the calculation of co-ordinates, which, after the analysis 

 is completed, would not usually be of service in plotting, since they are computed with 

 reference to no definite meridian. 



(f) A traverse running more or less in a straight line will on the average be affected 

 by a greater error than any other — a closed traverse, for instance — having the same 

 number and lengths of lines (16). 



(g) Though equation (24) is not new,* the writer believes the inference drawn from 

 it, to the effect that theoretically the best shape of a triangle for triangulation purposes 

 is an isosceles one with an apical angle of 67° 30', has not previously been published. 



(h) With reference to the question as to what may 'be considered a, permissible shape 

 for a triangle, it is shown [(25) and curves, fig. 5] that the answer depends on the 

 relative accuracy of the base and angles. The limits usually given — 30° and 120° — for 

 angles of a main triangle are satisfactory when the base and angles are measured with a 

 similar degree of precision, but should the accuracy of angular measure be considerably 

 superior to that of the base, these limits may be set further apart with advantage (p. 865). 

 A consideration of (40) will show that to measure the angles more precisely than the 

 base is not so illogical a proceeding as might at first be supposed. 



(j) Too great a stress can be laid on considerations of shape of triangles ; even when 

 the base and angles are measured with the same order of precision, isosceles triangles in 

 which the apical angles lie between 50° and 90° are all almost equally well-conditioned 

 (p. 866 and curves, fig. 5). 



(k) Values for the average error in the calculated length of a check-base are obtained 

 [(27) and (30)], and also^for the average difference between the calculated and measured 

 length [(28) and (29)]. From the former, suitable average values are deduced for the 

 errors in angular and linear measurements so as to result in a given average error in a 

 check-base. These values»are set forth in tabular form (p. 869). 



(I) Having regard to the relative accuracy in base and angles usually attained in 

 practice, it is shown to be generally more desirable to increase the accuracy of the 

 angular rather than of the linear measurements when a greater precision in distance- 

 transmission is desired of a triangulation survey (40). 



(m) It is dangerous to consider that the fractional error in the check-base, as 

 determined from the measured and calculated values, expresses even approximately the 



* Vide W. Weithrkcht's Ausyleichungsrechnung nach der Methode der Kleinsten Quadrate, p. 63. 



