I. 



w 



ire A 



weight 1*5 lb. 



II. 





, A 



„ 05 .. 



III. 



, 



. A 



„ l-*5 „ 



IV. 





, A 



„ 0-5 ., 



V. 





. B 



» i.-s „ 



VI. 



, 



, B 



„ 0-5 ,. 



VII. 





, B 



„ 1-5 „ 



nii. 





, B 



„ 0-5 „ 



New Form of Catenary. 459 



The two weights taken are 0*5 lb. and 1*5 lb., and the two 

 speeds 60 and 80 m.p.h. 



The eight cases are therefore : — 



W 



speed 60 m.p.h. For this p= '267 3^2 =323 



„ 60 ., „ „ =-267 =10-8 



„ 30 „ „ ,. =-15 =18-3 



„ 80 ,. „ „ =-15 = 6-1 



„' 60 „ ,. „ =0916 =967 



„ 60 ., „ ,, =-0916 =32-3 



„ 80 „ „ „ =-0517 =545 



„ 80 „ „ „ =-0517 =18-2 



For the purpose of plotting the results of equation XT. it 

 is to be noticed that dsjdO is the radius of curvature. This 

 has been calculated below at intervals of 5° and for the mid- 

 point of these intervals, i. e. } 2°*5, 7°*5. ... To construct the 

 curve it is merely necessary, starting from the lowest point 

 (# = 0), to describe an arc of 5° of a circle with radius first 

 value of ds/dO obtained. Joining this another 5° arc with a 

 radius = second value of ds/d6. Towards the upper part of 

 the curve, however, where the radii of curvature are incon- 

 veniently large it is more convenient to multiply these by 

 *087 (5° expressed in radians^, thus giving the actual value 

 of ds, and set this out as a straight line at the appropriate 

 angle. 



Again referring to equation VI., it will be seen that 

 increases from to a, at which angle ds/dd = r -o , and the 

 curve becomes a straight line. 



It is therefore in each case unnecessary to carry the calcu- 

 lations beyond the value of 6 given by the real root of 



eos 2 0— j» sin = 0. 



The form of this equation which does not involve W shows 

 that the ultimate shape of the aerial is independent of the 

 value of the weight on the end which only affects the shape 

 of the lower portion. 



These asymptotic angles have been calculated, and are as 

 follows : — 



Cases I. and 



» in. „ 

 v 



„ VII. „ 



II. 



61° 16' 



IV. 



67° 40' 



VI. 



70° 24' 



III. 



76° 54' 



