﻿Wire or Tape including the Effect of Stiffness. 105 



In the preceding investigation we have assumed that the 

 tape is used on level ground, and also that the tension used 

 is so great that the curve in which the tape hangs is so flat 

 that even at the ends we may consider sim/r and tan^r=i|r 

 where -\]r is the angle of slope. The greatest value of this 



angle is 9 T where I is the whole length of the tape, and we 



have seen that with a -J- in. tape 10 chains long under the 



rather low tension of 14 lb. this angle is only about 9° when 



the whole 10 chains is in sag. In actual practice, however, 



tapes are used on considerable slopes up to 45°, or even 60°, 



with the horizontal. It might be assumed, and it Mall be 



proved later, that the ordinary sag-correction — defined as 



the difference between the curve and its chord — can be 



obtained in these cases by writing w cos yfr for id in the 



wH 6 

 formula -yjm 2 , and the same remark applies to the correction 



for stiffness because, even in the extreme case of constrained 

 ends, the angle through which the tape is bent is not the whole 

 angle yjr but the difference between the slope of the chord and 

 -^, and this angle is less the steeper the slope, being given 



sufficiently closely by ^ g^T ^ g Q f ar ag t ^ e correc tion 



for stiffness is concerned, therefore, the case of chord hori- 

 zontal is the most unfavourable that occurs in survey practice, 

 and the solution found can be used for slopes even steeper 

 than that in Professor Maclaurin's example. But we will 

 now proceed to consider the general case dealt with by 

 Professor Maclaurin. 



(Fig. 2.) Professor Maclaurin starts with the following 



fic. 2. 



fundamental equations of equilibrium of a small element of 

 the tape : 



, — U-£ — wsmylr-0 .... (1.1) 

 as ds l 



T i± + <m_ w =0 _ \ m _ (12) 



as as 



^ L + U =0 (13) 



as 



