[ 732 j 



LXXVI. On the Form of the Trailing Aerial. 

 By H. C. Plummek *. 



1. nPHE problem o£ the form assumed by an aerial trailing 

 JL from an aeroplane, which has been discussed by 

 Captain Hollingworth (p. 452), is in essence scarcely so new 

 as he seems to imagine. For in principle it resembles the 

 problem of Bernoulli f , which is concerned with the shape 

 of a narrow sail held between two parallel yards and filled 

 with wind. The solution of this latter problem, when the 

 wind is supposed to find an immediate issue and the weight 

 of the sail is neglected in comparison with the pressure of 

 the wind, is known to take the form of the common catenary 

 with its axis horizontal. Captain Hollingworth attempts to 

 take the weight of the wire into account, to a first approxi- 

 mation, but an unfortunate slip makes it appear that the 

 apparent agreement of his results with observation must be 

 largely illusory. It seems probable that for practical 

 purposes the common catenary will of itself give a sufficient 

 approximation. 



by Captain 



IV. 



2. The equation 



to be solved, as given 



ollingworth, is 





d 2 s 



n ds n p+ sin 



: '2 = P.OS 8 . =*— - : ~ . 



d0 2 d0 'cos 2 0-psm0' 



where p=w/J£.v 2 , ic is the weight of the wire per unit length, 

 v is the constant horizontal velocity of the aeroplane, and K 

 is a constant depending chiefly on the altitude and the 

 diameter of the wire. This may be written 



p cos p cos -f 2 sin cos 



cos 2 —p sin cos 2 —p sin 



or 



d /, ds\ 



reV^dd) 



4 log [I (cos 2 6-p sin 0)1 = ^ 



d0 °L/# V L J ] l—psmV- 



pcosO 

 dO™* LdO x ^ r ~~ vy J ~~ 1 -^sintf- sin 2 



Let 



1 — p sin — sin 2 0= (tan « + sin 0) (cot a — sin 0) , 



so that 



p= tana— cot a— — 2 cot2u. . . . (1) 



Thus, p being small, a is an angle slightly in excess of 45°. 



* Communicated by the Author. 

 t Routh's Statics, i. p. 334. 



