On the Form of the Trailing Aerial. 733 



Hence 



77)1°© 77j (t an a + sm 0) (cot a— sin 0) 



OL f COS 



a \tan a ■+■ si] 



tan a — cot a ( cos cos <9 



tan a + cot a \tan a -f- sin cot a — sin 0) 

 _ <i , tan a + sin# 



= — COS 2« —-lOg : : -r, 



t/tf to COta — SllJ' 



and therefore 



ds,, , • m/ , • m n /tana+ sin(9\- cos2a 



-v^rtan a + sinfc/Ucota — sm0) = Ci — ; -. — ^ ) 



dd K v y \cota— sin 0/ 



Hence the complete integral is 



_ p (' ^ m 



' S °~ J (tan«+ sin ^^"(cot,*-- S m0) 2si » 2 «" W 



The tension T and the radius of curvature p at any point 

 of the wire are connected by the relation 



T = ~Kv 2 p (tan a -f sin 0) (cot « — sin 0) 



= QIW 



/ tansc-f sinfl \- cos2a 



\cot a,~ sin (9/ ^ ' 



3. The integral in (2) seems by no means too complicated 

 to use directly for the construction of a curve by tangents 

 according to the method employed by Captain Hollingworth. 

 Since a is near 45°, tana, cot a, 2 cos 2 a, and 2 sin 2 a are all 

 near 1. Instead of using the accurate first integral, how- 

 ever, he substitutes an approximation (X.) by the partial 

 neglect of powers of p above the first, and in doing so 

 commits an error. 



To the first order in p we have 



^=-2cot2a, a = l(7r+p), 

 tana = 2 sin 2 a = l + ^p, cot a = 2 cos 2 cc = l—^p. 

 Hence 



(tana-f sin0) 2cos2a 



= (l + lp+sin#)--^, 



= (1 -f sin + \p) exp. [ — \p log (1 + sin 0)] 



= (l+sin0)[l + ip(l+ sin0)- 1 ---V;>lo«r(l+sin0)], 



(cot a.— sin 0y ain ~ a 



= (1- sin (9)[l-|p(l- sin 0) "*+ Jp'log (1- sin 0)], 



(tan a + sin 0)- Wa (cot oc - sin 0)- 2si * a * 



= sec 2 fl +p sec 2 sin + lp log- ! + Sm ^ 1 . 

 L 1— sm 6>J 



P7«7. il%. S. 6. Vol. 38. No. 228. Dec. 1919. 3 E 



