C32 Properties of the Catenarian Curve 



or by substituting its value for a from Equation A 

 N° 1, and dividing by z-^x. 



Equation B ..,...?/ = a x /* L. ^ + ^ 



z—a 



Thus far see Dr. Hutton. Vince. Mac Laurin, Sfc. 



Now, it is obvious, as there are not any arbitrary quantities, 

 that all catenaries must agree in specie, differing in magnitude 

 alone ; and since two Equations only can be deduced from the 

 general properties of the curve, and there are four unknown 

 quantities, no one of them can be exhibited in terms of any 

 other, unless some new Equation is introduced ; as in the case 

 of a maximum or minimum, or of an assumed relation in 

 magnitude between either two of the four quantities. 



The maximum, with reference to the subject of this inquiry, 

 will evidently take place, when the force of suspension at P 

 acquires a -rate of proportional increase equal to that of y, or 



if b represent this force, when — i^ But 6^ = a- + z^ r: 



h y 



a^ + 2«x + a,-" Eq. A N° 2. 



b =1 a-Yx '.' h T^ X And ^^ — consequently 



a-\-x y 



X \y : \ a-\-x : y But 



X \ y ', '. z la' therefore 



n-]-x _ ^ ^ z + a 

 y ^ax But y = axh L. — consequentlv 



z z—a ^ 



rt ii /j^ 7 -4- '7* 



-=1 h L, Or substituting for z from Equa. A N° 2 



a-^x v2aa;4-^''+ar 



zz:. h L. f- — and therefore 



'^lax-Vx' ' \flax-\-x'—x 



'^1ax-^x''-\-x 



^^ax^x" X h L. -.x~az=:f^ 



^2ax-i-x'^—x 



The expression may now be simplified by assuming a = I 



Then V2x-{-x'' x h L. . ~ — ,r — 1 - From 



^2x-^x''^x 



