On the Catenary Curve. 5 1 



y z=a X log. ^ "^ ^-' + "'- ^ (2) 



a;= y/z^ + a* — a; 



which are the usual e(|uations of the catenary. 



Let <p denote the angle between the curve and the ordinate 

 y; thus 



— — - — r = cos <p, and ~j r' = sin 4) ; and, on accoimt of 



the equation (1), 



a =fcos <p {S) 



X = f sin <p. 

 If these values be substituted in the expressions of y and x: 



then y =z f X cos $ log. — t-i"i^ = / x Q 



•J -> TO COS ip -^ « J . 



.r =/ X (1-cos $) = 2/ sin »i<^, ^ 



the symbol Q being put for cos 4> log. . 



Take the fluxion of Q ; thus 



■—^ = — sin $ log. i±±IL^ J_ 1 : and because 



rf p ^ ° cos ip 



' + ""? = _ L±^hl. = VH^ we have 



dQ sin (p , 1 + sin t) , , 



-; = Off. ■ + 1 5 



dp 2 ° 1 — sin <> ' 



and, by expanding the logarithms, 



■^ = 1 — sin' A — - sin* ^ sin^ ^ — &c. (5) 



Now Q is equal to zero, both when <^ — Q and c^ = 00': for, 

 in the latter case, although log. — — is infinite, yet cos (f) 



log. _ is evanescent. As the angle <^ increases from to 



90", -'- — is at first positive till sin <J) acquires a certain value, 



determined by the efjuation. 



1 = sin* 4) + — sin-* <?> + - sin^' $: -j- &c. ; (5) 



it then vanishes, and afterwards becomes negative. The func- 

 tion Q is tlurefore susceptible of a marimuni, which it at- 

 tains wiien sin (p satisfies cMjuation (^y). Now, y being constant 

 in the ecpiation y = f x Q,/" will decrease as Q increases; 

 an<l the first quantity will be a mivhnnm when the second is a 

 viaximiim. The minimum retjuired is therefore determined by 

 equation (f)). 



G 2 From 



