New Form of Catenary. 457 



compared with its area it will hang approximately vertical, 

 and for the lowest element of wire we shall have — 0, 

 W = weight of lead weight. 



L n 



i 



-) 



.\ from II., 



w 



(S).= K - 



.'. from X., 





W 



Hence full first integral of equation III. is 



Is _ W l+ps\n0 Vj 



/# ~~ Kv 2 cos 2 # — psin# 



The case in which the weight does not hang approximately 

 vertical will be considered later. 

 To integrate XL further we have 



7 W .. l+^sin-0 



Kv 2 («-sin0)(£ + sin0)' 



which by a process exactly similar to the previous one can 

 he reduced to' the form 



2K# 2 L^ — sin # + sin0J 



' ' 2 K v 2 J L « - sin 6 + /5 + sin J ' 



which is in standard form and integrable. 

 The integral is of the form 



S + 1J -2Kv>L y/p 1Q S tan 0/2-1-^/2+ y> 



^(l-^Otan- 1 -^;^^]. . . XII. 



This result is interesting hut far too complicated to be of: 

 any practical value. 



Equation XT. is, however, easy of evaluation, and from it, 

 it is quite a simple matter to construct any required curve 

 hy tangents. 



