Influence of Stiffness on a Suspended Wire or Tape. 167 



the moment of inertia of the cross-section of the chain 

 about a line through its centre of gravity perpendicular to 

 the plane in which the chain hangs. The equations give 



IC cos -Ur ic cos -Ur + Ji/ i - , T? T V 



^_ ^ ds __ ^ ds" _ ic COS y + Jiii Y 



ds ds 



where the . denotes differentiation with respect to 5. 



Hence .. 



dT . , »' cos "v/^-vir -j^-^ylrylr — ylryl' 

 ^— = —w sill ylr— ../ ^ 4- EI ' ^ . ^ 



But from (1) and (4) 



— = ir sin yfr + Uyjr -- %c sin i|r — El^Ir-^ 



/. COS-Ur-T-^ -I- 2 sin'v^= — -dfdr + -^^ . . (^j 



which is the differential equation of the curve in which the 

 chain hangs. When the chain is perfectly flexible the right- 

 hand side of (5) is zero, and we get 



cosi/r^ ^2sin-v/r = 0. 

 "Whence 



7;=:-2tanl|r.^i^; l/p = ^=^^^, 



which, of course, represents the common catenary. 



In the surveyor's tape or wire the flexibility is so nearly 

 perfect that we may proceed to solve (5) by approximation, 

 substituting in the right-hand side of that equation the value 



COS^ "^ 



y^= ~ obtained by neglecting the rigidity. 



cos- ^lr 

 ^ G 



2 sin ■\lr cos'^ -^ 



^= -. 



'\f^ 73 cos** a/t [cos- -v/t — 3 sin^ -v/r] , 



•■;• 24 sin -vir cos'^ -v/r r 2 f • 2 f 1 



i/r = —^-j — — ^ [cos'' Y — siir y\r\ . 



c 



.\ \ir 'v|r+ ^ ^ yy _ ^jj^^cOS'^ \|rr5cos-iir — 21. 



