[ 166 J 



XYIII. The Influence of Stiffness on the form of a Suspended 

 Wire or Tape. By Richabd C. Maclaurin, M.A,, LL.M., 

 Felloic of St. Jolni's College^ Camhridc/e, Professor of 

 Mathematics, Wellington, JS.Z.^ 



SOME ot the greatest improvements in modern surveying 

 are due to the substitution of a steel tape or wire 

 for the old surveyor's chain. The newer instrument can, 

 with proper precautions, be made an exceedingly accurate 

 measurer of distances. So minute have been the corrections 

 applied in some recent surveys that it has been questioned 

 whether we may, witb propriety, any longer regard the form 

 of the curve in which the " chain " hangs as a catenary. It 

 is true that the surveyor's tape is so thin as to be very 

 flexible, but for some purposes there are advantages in using 

 a circular wire, which is, of course, more rigid than the tape 

 for the same weight. It may be thought that, at any rate 

 for the circular wire, the hypothesis of perfect flexibility (on 

 which the investigation of the form of the ordinary catenary 

 rests) may introduce an error comparable with those for 

 which corrections are applied in the best modern surveys. 

 The object of this paper is to settle the matter by investigating 

 the correction that must be applied when the rigidity of the 

 wire or tape is taken into account. 



Writing down the equations of equilibrium of a small 

 element of the chain in the usual wav, we have : — 



ds ds 



(1) 



'if .f-«.eo.^ = (2) 



Moreover, L = EI/p = EI-~- , which with (3) gives 



EI^+U = (4) 



In these equations T is the tension, U the shear, L the 

 bending moment, s the length of the curve measured along 

 the arc from some fixed point, afr the angle that the tangenir 

 makes with the horizontal, ijp the curvature, ic the weight 

 of the chain per unit length, E Young's modulus, and I 



* Communicated by the Author. 



