New Form of Catenary, 455 



i. e. integrating 



logC + log(^ = f co ^^ + si "^ , ..... V- 

 . a w i cos 2 6/ — ;>sm 6 ' 



where C is some constant. 



Now cos 2 6 —p sin = 1 — p sin 6 — sin 2 (9. 



Let the roots of this be a and —ft, so that 



l_p sin (9- sin 2 = (a -sin 0)(£ + sin 0), 



where uft = l, oi—ft=:—p; .... VI. 



p 4- sin A B 



cos 2 0— _p sin ~~ a — sin0 /34-sin#' 



where A£ + Ba=p, A-B = l. . . . VII. 



.'. from V., 



, « , , N , f A cos 0d0 - fB cos 0d0 



log 4" log (z)* = \ 7—pr 4" I -r r— . 



to &v ' J a — sin0 J /3 + sin0 



= - Alog (a — sin 0) +Blog (/34-sin 0) 



(fti- sin 0) B 

 Og (a-sin0) A ' 



. 2 ^_(^+sin_0p 

 •* ^ d0~~( a -sin0) 2A ' 



where a, ft, A, B have to be evaluated in terms of p. 



Before proceeding to do this it will be advisable to in- 

 vestigate the quantity p a little more closely. 



w 



K is a constant depending on the wire diameter and on the 

 density of the air, and is of the form 



Ns, where 5 is wire diameter, and by Laplace's formula, 

 when the barometer at sea-level is at 30 inches, 



Nat sea-level ='0027 ") 

 N „ 6,000 ft. = -00214 f- 

 N „ 10,000 ft. = -00179 ) 



Now w, the weight per unit length, is proportional to the 

 square of the wire diameter, K to its first power ; therefore 



