﻿Wire or Tape including the Effect of Stiffness. 9 7 



Professor Maclaurin then proceeds to form from first 

 principles the differential equation for the curve in which 

 a flexible wire hangs under its own weight and an applied 

 tension. This equation, which is of the fourth order, he 

 proceeds to solve by approximation, and eventually arrives 

 at a formula for the effect of stiffness on the ' Sag Correction ' 

 as deduced from the ordinary catenary formula. In an 

 example which would be considered extreme in survey 

 practice, i. e. a steel tape 10 chains long, J inch wide, -jfa inch 

 thick, wholly suspended under an end tension of 14 lb., this 

 formula gives a stiffness correction of only 0*000000023 inch 

 — equivalent to 0*184 inch in a million miles — from which 

 Professor Maclaurin concludes that the neglect of the 

 stiffness of the chain need cause little anxiety to the 

 surveyor. 



Before the writer had seen Professor Maclaurin's paper, 

 his attention had been drawn to the effect of stiffness while 

 he was engaged in measuring with a steel tape a base line 

 in connexion with the Trigonometrical Survey of the 

 Federated Malay States, and he had worked out a formula 

 for this correction on the assumption that the sag was small,, 

 as it generally is in practice. The tape was in fact regarded 

 as an elastic beam subjected to an end tension in addition to 

 its own weight, and supported " clamped " or " free V at 

 the ends. On comparing his results with Professor Mac- 

 laurin's, they were found not to agree, and for many years 

 the writer was unable to account for the discrepancy, failing 

 to discover any flaw in Professor Maclaurin's investigation 

 or his own. He at length handed over the question to 

 an assistant, Mr. D. T. Sawkins, B.A. (Cantab.), who was 

 employed for some time in the Malay States Survey Depart- 

 ment as a surveyor, and who succeeded in discovering the 

 cause of the discrepancy, which appears to be due to Professor 

 Maclaurin having neglected to use the complementary 

 function in the approximate solution of his differential 

 equation. This complementary function comprises the 

 larger part of the correction, and though the whole cor- 

 rection is so small as to be seldom or never appreciable in 

 practice, it is much larger than would appear from Professor 

 Maclaurin's paper. The object of the present paper is to 

 investigate the correct formula for the stiffness correction, 

 to show where the error occurred in Professor Maclaurin's 

 solution ; afterwards to give some results in the general 

 catenary formula and its variation when the elastic extension 

 of the tape is taken into account; and, finally, to give the 

 sag correction when a heavier tape is included in the series 



Phil. Mag. S. 6. Vol. 29. No. 169. Jan, 1915. 11 



