AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 871 



expand or contract in that proportion ; hence, if Z 7 be the distance of the point 7 from 



the origin, the average fractional error in Z 7 , due to base error only, will equal ± — . 



Dealing next with the average displacement of the stations owing purely to angular 

 errors, (30) may be written in the form — 



since the sides of the triangles are roughly equal ; and if the influence of the base error 

 be eliminated from (31) we have — 



/2ra 



3!=±Wyy (32) 



which measures the average error in the length of a side of the nth triangle due purely 

 to error in the angles of the scheme. 



Now the apex of the first triangle, namely the station 3, is disturbed by errors 

 transmitted along a and b and of equal amount ; hence from (32) and (1) we have — 



Average displacement in station 3 due to angular errors only = ± vc*/ - . . (33) 



This displacement will be transmitted to station 4 via line d ; it will, as it were, 

 gather on its way the error in d itself. Station 4 will also be disturbed by error 

 transmitted along e, which in this case will merely be e l5 since station 2 is unaffected 

 by angular error. Applying theorem (1) to sum these displacements, we obtain : — 



y4 4 4 

 - + - + 

 o o o 



=±*\/y • • (34) 



Similarly the error influencing station 5 is compounded of those influencing stations 

 3 and 4 together with the errors in the lines /and g, and may be expressed — 



/no 



Average displacement of station 5 due to angular error onl ij = ± vc*/ ^- . . (35) 



In like manner the average displacements in the remaining stations due purely to 

 angular error may be calculated ; they are as follow : — 



/56 

 Average effect of angular error on station 6 = ± wW — . . . . (36) 



Do. do. do. 7=±w^/ 1 ^ 4 .... (37) 



. ». , - ,i / ( Sum of squares of coefficients of vc for stations ) 



Average effect of angular \ I ) n , f 



error on station N I id) (N - 1) and (N - 2) plus 1 (N - 2) f ' ^ 



