﻿110 Mr. A. E, Young on the Form of a Suspended 



whence 6 = A sinh as + B cosli as - — -=- ; 



and since = when i/r = and 5 = 0, we have B=0. 



Thus 6 = Asmhas- 2h l s , 



ar 



and yjr = <b + = Un~ 1 bs + A mih as— -— s-< 



1 T a 2 



Also 



t /,'= I cos yfrds= I (cos cj> cos 6— sin <f> sin #)</s 



= 1 coscfads— 1 [— + 0sin<J>Ws nearly. 



The first integral is the ordinary catenary formula, and 

 the second is the correction to this due to the rigidity. 

 If we put tan" 1 6s and sin (f> = bs as a first approximation 



2b ?) s . 

 and neglect — ^— in comparison with bs, we find A from the 



equation yjr = bs + A sinh as, and then the sag correction due 



to the rigidity is given by 



i 



C 2 / 9 b 4 s 2 \ 



2 I ( Abs sinh as + ^A 2 sinh 2 as— ^^-jds. 



The first two terms will be found to give the expressions 

 already quoted, and the last gives 



6a- ~ 6T 5 ' 



which is the value found by Professor Maclaurin. 



Through the kindness of Professor H. H. Turner, F.R.S., 

 the writer is indebted to Professor A. R. Forsyth, F.R.S., 

 for suggesting the following method of carrying the ap- 

 proximation further. 



2b*s 

 Write 6 = A sinh as — ^- +- v 



a" 



and substitute this in equation (20) expanded to the first 

 term in b 2 s 2 , or 



we have 



^=a 2 (l + i&V) (0- |°) +2bh{l-2b*s* 



d 2 v 1 . a 2 



- , ., — a 2 v= -r a 2 b 2 As 2 sinh as— -x A 3 sinh 3 as — 5b r °s*. 



ds* 2 6 ' 



