﻿114 Mr. A. E. Young on the Form of a Suspended 

 Therefore 



ft* sin* f I sin^r , ^ 8 sin 8 r , , 9 ^ 



"* + 12c 2 + 360c 4 + 20i60c 6 ~*~ ' " ' [ '^ 



We have thus obtained a development for I the length of 



the tape in terms of k the chord, f the angle the chord makes 



T 

 with the vertical, and c = — where T is the unknown 



horizontal component of the tension. 



By reversing the series we obtain the following develop- 

 ment of k 2 in terms of P : 



„_«[% sin^P /sin 8 ? sin 6 £W 4 



/ 5Bin»r sin^f sin«f\^ 1 



\ 1728 "~ 864 + 20160/ c 6 + * * J " K } 



And by extracting the square root we obtain after reducing 



/-/f~i sin4 ^ 2 sin 6 ^ 4 / 3 7cos 2 g \ 

 ' L 24c 2 + c 4 V640 1152 / 



_ sin 8 gZ «/ 5 43 cos 2 g 11 cos 4 g V 1 



c 6 \1728 23040 + 9216 / + " * ' J * ' 



If we make £ = 90° so that the chord is horizontal we have 



*= l i l ~ A + m? - rm? + • • •] =2c sillh_1 i ■ (25) 



T 2 fF2 72 



as it ought to be : and if we write for c 2 its value -4- = —. ; — T 

 fe ' w? 2 ic 2 4 



where T is the end tension, and then carry the correction due 



I 2 

 to j to the next terms we find 



L 24T 2 ~1920T 4 • • 'J • • • v-°; 



which is the series given in Mr. Adams' paper quoted above 

 and which can be written approximately 



r «,2/2 10/^ 2 / 2 \ 2 - 



f wH 2 10/ic 2 / 2 \ 2 l 

 L 24T 2 3\24TvJ' 



Returning to the general case we have to find an expression 

 for c in terms of the end tension and the angle of slope. 

 ( Considering the portion of tape let V 2 and Vi be the vertical 



