872 MR HENRY BKIGGS ON 



If Z N be the distance of station N from the origin, the average fractional error 

 in that distance, due to angular error in the scheme, is : — 



y , , / I Sum of the squares of the coefficients of vc for ) 



%i = ± Za/ I Nations (N - ] ) and (N - 2) plus i (N - 2) j ' ' < 39 ) 



We have seen already that tlie average fractional error in Z N due to an error, c u in 



base measurement is ±- , and we are now in a position to compound these errors to 



obtain : — 



Average fractional error in Z N due to \ I j /7j' n \ 2 A'A 2 ) 



linear and angular errors combined j ~ ~ \f ( \Z^/ \c~/ ) ' 



In this result it will be noticed that, while the effect of the angular error increases 

 as the number of triangles increase, the influence of the fractional error in base 

 measurement is constant ; hence where greater precision of distance-transference is 

 desired in an extensive scheme, it will generally pay better to devote attention to 

 improving the angular rather than the linear measurements. 



Another fact of the first importance established by the last result is the necessity 

 of using as few triangles as possible in the scheme to cover the area under survey. As 

 accuracy in distance-transmission is desired of every survey, this conclusion is hardly 

 second in importance to the one that the triangles should be well-conditioned. 



TJie Value of the Check-base as a means to Assessing the error in Distance 



Transmission. 



The value of the check-base in allowing of the error in the sides of the triangles 

 being roughly assessed has already been shown ; we have yet to examine whether the 

 wider claim — that the accuracy of the triangulation as a ivhole is determined by the 

 discordance (expressed as a ratio) between the measured and calculated length of the 

 line — is justifiable. 



If the average fractional error in the check-base were exactly equal to the fractional 

 error in Z N (the distance of station N from station 1 ), it would follow that the right- 

 hand sides of equations (29) and (40) would be equal, and it is evident that this is 

 not generally the case. Since, then, the above-stated claim cannot be established in 

 full, it remains to determine between what limits it is admissible. 



The table below is drawn up to compare the average fractional error in Z^ with 

 the average fractional difference between the calculated and measured values of the 



O 



check-base, z, for the scheme of fig. 4. It is assumed that the check-base forms a side 

 of the triangle whose apex is station N. 



