AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 873 



Station N. 



Average Fractional 



Average Fractional Difference 





Error in Z N . 



between Measured and Calcu- 



x 2 - r 2 . 







lated Check-base. 







(X). 



in 





Apex of 3rd triangle 



s /{ -2-33„3 + (i.) ! } 



v/I^GOX*)'} 



0-33u 2 - feY 



Do. 5 th do. 



-J{ i «**<$f) 



y|s,3, +( i.)%( ? y} 



0-52z; 2 - feY 



Do. 7th do. 



v /{ «» t (j)'J 



ji^c^m 



l-Slut-fey 



Do. 9th do. 



x/! nw+ &) ! } 



ji^y+m 



5-79^-^Y 



Do. 11th do. 



v/]^w + (a)*.} 



y{ 7 w + (^ +(? ; } 



14-75„2_^\ 2 



The condition to be satisfied in order that the fractional error in the position of, say, 

 the apex of the ninth triangle shall be equal to the proportional difference between 



the measured and calculated check-base is that 579v 2 shall be equal to ( - ) . That 



z. 



is to say, if v were 6 sees, (see table, p. 869), - would have to be about I in 14,250 — a 



degree of accuracy inferior to that usually attained in practice in the measurement of 

 a check-base for a scheme of this size. We may therefore conclude that if the triangles 

 were arranged as in fig. 4, it would not be safe to consider the actual degree of agree- 

 ment between the two values for the length, of the check-base as a measure of the 

 accuracy in position of the apex of the ninth triangle. Again, fig. 4 shows an excep- 

 tionally favourable arrangement of triangles for the rapid transference of distance ; 

 almost every other arrangement of the same triangles would result in a diminished Z N , 

 and therefore in an increased fractional error in Z N . So far, then, this investigation 

 has shown that the average fractional error in the check-base is always less than, 

 and in many cases will be much less than, the average fractional error in distance- 

 transmission when the number of triangles is nine. On the other hand, it can be 

 shown in like manner that if the scheme is composed of only two or three triangles 

 the accuracy in distance-transmission is superior, on the average, to that in the check- 

 base. There must therefore be a value of n, the number of triangles, at which the 

 average fractional error in each of these quantities is about equal, and, to be on the 

 safe side, the writer places this at n= 5. 



Thus we are led to conclude that the fractional error in the verification base, as 

 determined by comparing its measured and calculated length, may be taken, in general, 

 as a reliable check on the accuracy of distance-transmission only if the number of 

 triangles is less than five ; but that when the triangles exceed five in number the check 

 ceases to have value, owing to the rate of increase of the error in Z N being more rapid 

 than that in the check-base. 



TRANS. ROY. SOC. EDIN., VOL. XL VII. PART IV. (NO. 28). 128 



