r u c 2 r u b 



y = log sec u - c J u tan u du + ■*? \ u a tan u du — ... 



(8c) 



and similarly from (12b) 



y = log cosh v + c 1 v tanh v dv + -rj \ v 2 tanh v dv + (12c) 



The coefficients of c 3 and higher powers were not computed because, in general, c 

 will be small. 



For values of c = .02, .04, .06, .08, .10, .12 and .14, tj, t 2 and y were 

 computed from (6b), (10b), (8c) and (12c) respectively. Taking values of t.. and 

 t a corresponding to the same y, T../FL was computed from (14). Fig. 3 is a plot of 

 T./FL against tan <J>/2. To facilitate the use of Fig. 3 to compute the tensions as 

 functions of <J>, Table 3 was constructed, giving values of tan $/2, sin $ and cos <fc 

 against <|>. To illustrate the method, the tension T.. and its vertical and hori- 

 zontal components are computed against <j> for a hypothetical cable 1000 feet long, 

 weighing one pound per foot moving at a speed such that F = f = 1 lb. per ft., 

 p = 20, and hence c = 2f/p = .10. The results are shown in table 4. It is seen 

 that the vertical component T.. sin $ is nearly constant for values of *}> between 60° 

 and 90°. 



Conclusion 



V/e can now formulate the following procedure for answering the questions 

 raised in the introduction. Suppose a given cable towing a given float. 



(a) Obtain R and F cable-resistance curves against speed either by new tests 

 or by using the data of Wieselsberger for R and of Kempf for F, for a smooth cable 

 (see Appendix II). Recalling that r = R/w and p 8 = ^4r* + 1 + 2r, determine 



f = F/W and c = 2 f/p as functions of speed. 



(b) From Fig. 3 and table 3, compute T.. , T. sin <J> , T. cos <Ji for speeds and 

 f values corresponding to C = .02, .04, ......14, as illustrated in table 4. As in 



table 4, the additional displacement T.. sin $ will vary slowly with the drag of the 

 float, T- cos <J> . Choose a mean value of T. sin $ in the neighborhood of the esti- 

 mated drag T.. cos $ . Plot these mean values of T.. sin <j> against speed. 



(c) Using these mean values of the corrected displacement corresponding to 

 each speed, obtain the resistance curve of the float. This is equivalent to ob- 

 taining T 1 cos 4> and hence also T.. for each speed. T a is then calculable from (13). 



Let us illustrate the procedure in (a), (b), and (c) with the example from 

 which table 4 was computed. 



(a') Suppose that the cable-resistance curves have been obtained, as 



