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



We can now correct this expression for the assumption 

 that h = l cos f nearly, whereas h=k cos f exactly. Putting 



\ = l — l- ^ye have strictly 



2 T 2 2 /' (/-X) 2 cos 2 f\ fA w(Z-X)co8?Y «^ 2 1 



c= .a 1 ? — ; iv 1 w 2 — ; -si?/' 



and expanding this to the first power of X we have 



., T, 2 . , r / »•? C os n 2 «- 2 / 2 1 /, , 2x co? 2 r\ 



Taking for A, the value derived from the second term of the 

 expansion in equation (30), and carrying the factor in X to 

 the numerator, we see that this correction can be included 

 in the term multiplied by cos 2 f in the third term of the 

 expansion (30), which now becomes 



■p, "-^sin 2 *: 



"«v{(.-s!gt)r-s5 



»- 4 / 4 sin 2 £ /_3_ _oos 2 |\ 



T 4 f M ^cosgV ^ 2 1 3 \640 ~ 384 7~ " ' ' 

 12 IV 1 2T, / ~4T 9 2 i 



(31) 



which is rigorously correct so far as it goes. Of course if 

 T 2 the lower tension were used we should have to change the 



minus sign of ' T in the denominators into plus. 



ivH 2 

 If we carry both the factors in -™- in the denominator of 



the second term to the third we obtain the following : 



k = l 



w 2 P sin 2 f 



24T /(l-^^) 



u-H* sin 2 £ _ cos 2 n j 



T ,4(i_l l ^A?pl920 128/ •••/ 



(32) 



from which it will be seen that the sag correction is given 

 with sufficient exactitude for all practical purposes by 



'£!£™l?(l ± ^^i) where T is the full end tension, and 

 the upper or lower sign is used according as the tension is 



