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



The integrals are rather tiresome to evaluate, but many 

 terms cancel out, and the following fairly simple result is 

 obtained for the Sag Correction: 



i ^ ? [^ + 12Wr{»(l-«)-^}+12«V{n(l-«)-;jJ|. (42) 



V 



If we make n= ~, we have the case of heavier tape at one 

 end of the series, and the formula reduces to 



9 ^ 2 [ if 2 + 2 ww'tfr 2 - 2f») + wV(4r - 3r 2 j ] . . (43) 



It we make r=l, we get simply — ,„ 2 , as we should. 



If in equation (42) we write w'rl=Wthe whole extra weight 

 of the heavier tape and then make the r*s within the brackets 

 vanish, we obtain the formula for the effect of a concentrated 

 weight W attached to the tape in any position, thus 



-J^[wn 2 +12wWn(l-n) + 12W 2 n(l-n)], . (44) 

 — "A J- o~ 



which of course is greatest for n = i, or weight at centre of 

 tape. Putting w=0, we see the effect of a concentrated 

 weight at the centre is 3 times as great as the effect of the 

 same weight uniformly distributed. 



Returning to equation (43), we see that the error intro- 

 duced by using a slightly heavier tape at the end of the 

 series is as a fraction of the whole length in sag approximately 



l~vw r~ 



- T ., . The maximum value of this error occurs when the 

 44o" 



whole chain of heavier tape is in use, so that when 1 — 2 chains 



r= J, 7 = 3 chains r = ^ } and so on. Thus l 2 r 2 = constant =1, 



and as in the case for which the investigation was made 



io = 5*l oz., iv' = 2'2 oz., T = 20 lb., we have the above 



fraction = . 5 '^ 2 '^ 2 = *}'* > SSL J X in 40 ^ 000 - The 

 4x2(Hxltr 409, bOO 



a rev age effect occurs when only half the heavier tape is in 

 use and is £ the above ratio, or about 1 in 160,000. As an 

 accurac}' of only about 1 in 30,000 was aimed at in the 

 traverse work on which these tapes were used, this correction 

 was therefore negligible. 



Formula (44) might be of use when for some purpose an 

 attachment is made to the tape such as a thermometer to 

 obtain the temperature at a certain point : and in this con- 

 nexion the writer worked out in a similar way the effect of a 



