FLEXII XG. 



.mpl-; required the form of tlic curve when its 

 weight is the only force acting on it, and the area of the. 

 section at any point is proportional to the tci that 



point. Here Y is constant and may be denoted by -</, the 

 axis of y being vertically upwards. And T varies as / 

 that T= \m where X is a constant. Thus from (2) and (.",) 

 WC obtain 



therefore 



1 + 



therefore tan' 1 -~ = - -f constant. 



ax a 



The constant vanishes if we suppose the origin at the lowest 

 point of the curve ; therefore 



dy x 



-/ = tan - ; 

 dx a 



therefore ^ = lo cos - . ,. (4). 



a a 



Sinee in this case the area of the section at any point is pro- 

 portional to the tension at that point, the curve determined by 

 (4) is called the Catenary of equal strength. 



Since T=\m = mag, we have the following result: the 

 tension at any point is equal to tl of a length a of 



a uniform string of the same area and density as the string 

 actually has at the point considered. 



191. The equations (1) of the preceding Article may be 

 written 



