234 Properties of the Catenarian Curve 



r= I y-^V-^f -\-x' By using this value for z"" 



m 



a--- ^^ Equ. AN°3 



zy 



a z=z -^ And since 6^ = a^+z^ 



Now assign to x and y their respective values 25 and 280 

 feet, as they are given for the Menai Bridge ; the quantities 

 will then be found, 

 a =1580 Feet 

 Xt:= 25 

 y = 280 

 z = 282.2 



b — 1605 or about 5.7 X by |^ the weight of the chains, 

 bridge, ^c. or three times their weight nearly. 

 The angle of suspension 10° 8'. 



If X be now doubled, or x and y are taken in the proportion 



of 50 to 280, the quantities will be, 



a = 808 



0;== 50 



y = 280 



z = 288.0 



6 = 8^8 



The angle of suspension 19° 39'. 



In this case the values of a and 6, representing the strains 



at the apex of the curve and at the point of suspension, are 



very nearly one-half of the former. And from the equations 



zy , , a/4F+w^ . 



a s= ~ and b ^ z x — ^-- it appears, that a, and conse- 



Zx Zx 



quently 6, must increase or diminish in the reciprocal proportion 



to x, as y is supposed constant, and z is found to differ, when x 



is 25 or 50 by no more than a few feet. If these relations of 



X and y are taken as the bases of calculation by the strict 



form&, the results will remain substantially the same, and 



this general conclusion may safely be deduced from the whole. 



