﻿100 Mr. A. E. Young on the Form of a Suspended 

 To find the sag correction we have with sufficient accuracy 

 5 — a'=- j(-^— \ dx, and in the present case for the sag 

 correction over the whole span we have 



Putting for simplicity \ /Wf = o,, we have 



dy _ w sinh ax ivx 

 Zx ~ aT cosh^ +T »' 



and 



i 



iv 2 fz / 2 2a 1 sinh ax sinh 2 ax \ 



°J° \ a cosh ~ a 2 cosh 2 W 



w 2 p^ 3 2x cosh ax 2 sinh ax sinh 2ax x __'— 1| 



T ° 2 [_^ a 2 cosh J a 3 cosh ^ 4a 3 cosh 2 a ± 2a 2 cosh 2 J J 



„ r- 73 7 5 tanh -^ -^ 



2 



The first term of this expression T 2 is independent of 





 the stiffness, and is the ordinary sag correction on the 



supposition that the curve in which the tape hangs is the 



catenary or, which is the same thing to the order given, the 



parabola or circle. The remaining terms give the correction 



to this due to the stiffness. When dimensions are given in 



inches the quantity a is of the order 5 in ordinary surveyors' 



tapes ; in the example taken by Professor Maclaurin, a J in. 



tape under 14 lb. tension, a = 3'l. Taking I in the same 



unit, it will be seen that both sinh— and cosh — are sensibly 



at £ ■ Z j 



equal to ^e 2 and enormously large, so that tanh-o=l. 

 The third term can therefore be neglected in comparison 



