for the form assumed by a cable used to tow a heavy body below an aeroplane have 

 been developed only for the case where the cable is concave downwards. In addition 

 it has been assumed hitherto that the frictional resistance of the fluid is negli- 

 gible compared with the resultant forces of lift and drag acting on the cable. 

 Mathematical expressions and numerical solutions have therefore been derived here 

 suitable to the present problem in which the cable is concave upwards and where the 

 angle of the cable with the horizontal is so small over a considerable length of 

 cable that frictional resistance cannot be neglected. Hence in addition to the 

 physical assumptions used in reference (1), an assumption regarding the magnitude 

 of the frictional resistance must be introduced. 



General Analysis 



Consider a flexible cable of weight W per unit length. Let R* be the drag 

 per unit length of the cable when at right angles to a stream of velocity V; F* the 

 drag per unit length when parallel to the stream. When the cable is inclined at an 

 angle <J> to the stream, the force per unit length on the cable due to the stream will 

 be assumed to consist of a component R sin 2 4> at right angles to the length of the 

 cable, and a component F along it. These assumptions are approximations to the 

 actual experimental results. Their experimental basis is reviewed in Appendix I. 

 No other physical assumptions will be made in the analysis of the problem. The 

 general shape of the cable is shown in Fig. 2, the origin being taken at the point 



FIG-. 2 



of zero slope. Associated with each point P (or Q) of the cable there is the ordi- 

 nate Y, the arc length S measured from 0, and the angle of inclination <j> of the 

 cable with the horizontal at the point P (or Q). These variables are related by the 

 equation 



dY/dS = sin ♦ (1) 



*See Appendix II 



