374 ON SUSPENSION BRIDGES. 



Prob. 7. To find the form of the curve when 

 the weights of the chain, road-way, and suspend- 

 ing rods are all considered. We have here the 

 general equation (11) in which 



adx z=. hzdy + cydy + edy/xdy. 



To integrate this, we shall assume the weight 

 bz of the curve to be the same as it would have 

 been if the curve were a common catenary from 

 the same span, versed sine, and thickness of 

 chain : an assumption which may be admitted, 

 as the difference of the weights of the two 

 curves, their lengths being so nearly alike, could 

 have had very little influence on the form of 

 the curve sought, when in the bridge. Let 

 then a^dx — hzdy be the equation of the common 

 catenary above, in which a^ is the tension at 

 the vertex, and if we substitute for bzdy in the 

 general equation above its value a'dx from this, 

 we have 



adx z= a'dx + cydy + edyfxdy, or 

 (a — a^) dx — cydy + edyfxdy. 



Differentiating, making dy constant, and 

 dividing by rf/ gives 



d\ 

 (a - a' J -— = c 4- e^ . ) 



