﻿Wire or Tape including tlie Effect of Stiffness. 113 



No. 8 o£ the Report for 1892 of the United States Coast and 

 Geodetic Survey. And lastly, in 1912 in No. 1, New Series, 

 of the Professional Papers of the Ordnance Survey there 

 appears a Discussion on the Theory of Measurement by 

 Metal Tapes and Wires in Catenary, by Professor 0. 

 Henrici, F.R.S., and Captain E. 0. Henrici, ft.E. The last 

 mentioned paper investigates thoroughly the effect of the 

 elastic extension when the tape has been standardized in 

 catenary under tension, and shows that there will then only 

 be a very small correction when the tape is used on a slope. 

 The sag formulae in this paper apply also more particularly 

 to the case where the tape is standardized in catenary and 

 used in base-line measurement under the same conditions, 

 the slopes being obtained by measuring the difference of 

 level of the supports. It is desirable, however, to have 

 formulae which contain only quantities observed in the field, 

 and in traverse work the surveyor generally uses a tape 

 which has been standardized under a certain tension on the 

 flat, and he usually applies the same tension in the field either 

 at the upper or lower end when working on slopes. He 

 observes on the vertical arc of his traversing theodolite the 

 actual angle of slope or inclination of the chord of his tape, 

 and the lengths of the bays in sag depend to a great extent 

 on the ground he is working over. 



The following investigation of the catenary at any angle is 

 a development of that given in the above mentioned paper of 

 Mr. Knibbs : 



Let 5^2, XiX%, and y^y 2 refer to two points on the catenary, 

 the subscript 2 referring to the higher, so that s 2 — s 1 = l the 

 whole length of the tape, and let & = the length of the chord of 

 I and f the angle it makes with the vertical. We have then 



y 2 = c cosh -? ; y 1 = c cosh — ; 



®2_ .1*1 



. 1 to 2 .i «f-i 



s 2 = c sinn — ; s 3 = c sinh — ; 



c c 



s 2 — s 1 = l; d' 2 — x x = k sin f ; y 2 —y l = k cos £. 

 Therefore Jc cos f = c I cosh — — cosh — I, 



I = c I sinh — — sinh— V 



p - p cos 2 ?= 2c 2 (cosh •'' 2 = 5 _l) _ 2,.-' (cosh - J5* - 1 ) 



,, . , to ¥ sin*? ife'sin'f 



Phil. Maq. S. 6. Vol. 2'X No. 169. Jan. 1915, 1 



