the tension of overhead vnres supported by equidistant poles. 199 



We must now make use of a well-known property of the elastic 

 catenary. (See, for instance, Routh's Analytical Statics, p. 373.) 

 Let X be the horizontal distance of any point of this curve 

 measured from its lowest point while .Sj is the unstretched length 

 of the intervening portion. Let further T^ be the horizontal com- 

 ponent of the tension ; let w be the natural weight of the material 

 per unit length and let E be the elastic modulus. 

 We liave then 



Treating our wire as perfectly flexible we may identify I\ with the 

 horizontal tension in any section and 2a; with the total distance 

 apart of the extremities, while 2.§i will be the natural length of the 

 portion of wire. We have accordingly 



L + H {Tn+i + ^/i-i — 22n) 

 If we expand this logarithm in powers of -~, neglecting those 



-^ n 



powers beyond the third (as is permissible if the sag of the wire 

 be not too great), we get 



L-\-H{ 1\^, + Tn-, -rrn) = ^' Tn + 2s,-^^, 



At a great distance from the break the tension in a section will be 

 practically the same as it was before the break occurred and may 

 accordingly be denoted by T^,. Our problem is thus reduced to 

 solving the above equation subject to the conditions 



^0 = 0, 



say, where T is a finite quantity. 



The equation may be written briefly 



-^ n—i H" -^ n+i = a + ZOl n — 7p~^ ■ 



J- n 



Putting n = (Xi we get 



~-2{b-l)T = a. 



In order to solve the equation we shall assume that T„ may be 

 expressed in the form of a series of a particular type and shall 



VOL. XV. PT. III. 14 



