﻿Wire or Tape including the Effect of Stiffness. 107 



The investigation and formula? contain no stipulation as to 

 the end conditions. 



Mr. Sawkins pointed out that multiplying each member of 

 equation (15) by cost/t, each is a perfect differential, which 

 Professor Maclaurin does not seem to have noticed. 



Integrating them we have 



w cos 2 y _ 



EI 



(?cosi/r-f^sim/r J + A, . (17) 



where A is a constant of integration. Putting -^ = in this 

 and equation (12), we see that A is the tension at the point 

 tJt = if there is such a point, i. e., the horizontal component 

 of the tension = T . 



Integrating again, we find 



ws=-EI-^- +T tan^r + B. . . . (18) 



COSi/r T v 7 



If we take the origin so that 5=0 when i/r = 0, then B = 0; 

 so finally 



72 f 



EI -t^t = — ivs cos -\|r + T sin \jr, . . . (19) 



which is the equation to be solved in the general case. 



This equation might have been derived straight away 

 from statical considerations. It is simply the statement of 

 Professor Maclaurin's (14) that the differential coefficient 

 of the bending-moment is equal to the shearing force which 

 is evidently equal to T sin-^r — t^cosi/r, where T is the 

 horizontal component of the tension or the tension at the 

 point yfr = and s is measured from the same point. In 

 actual practice, when the tape is used on a slope there may 

 not actually be such a point, but in applying the formula? 

 which follow, the tape is supposed to be continued to such 

 an imaginary point from which the distances s are supposed 

 to be measured. The actual length I of the tape is in such 

 cases either s 2 + 5 i or s 2 —Si according as the vertex of the 

 curve occurs in the tape or not. When Si is equal and 

 opposite to 5 2 , we have the symmetrical case or case with 

 chord horizontal. 



If in equation (19) we put cosi/r=l and sin^ = i/r, we 

 have 



or 



dfy _ To . _ wi 

 ds* EI*~ EI 



