198 Mr Bobh, Discmsion of a difference equation relating to 



DiscKsi^ion of a difference equation relating to the tension of 

 overhead wires supported hy equidistant poles. By A, A. RoBB, 

 M.A., St John's College. 



[Beceived 22 May 1909.] 



Section I. On the Form of the Solution. 



When overhead wires are suspended from poles for electrical 

 purposes, certain precautions must be taken in adjusting the 

 tensions, since otherwise the snapping of the wire may lead 

 to the breakage of a large number of poles. 



The mathematical problem thence arising has already been con- 

 sidered by several writers and among others b}' Messrs Hawthorne 

 and Morton in the Philosophical Maga:ine, Vols. xi. and xii., 1906. 



Their result is somewhat vitiated through taking the quantity 

 denoted by '"/'" a^ a constant : whereas it generally varies through a 

 wide range (Vol. XI., p. Go-iV 



The attention of the writer was drawn to the problem by 

 Mr W. H. Logeman who has himself investigated it by a gfaphi- 

 eal method which is hovvever quite different from the following. 



In practice the wire is stretched between "anchor poles" which 

 are often long distances apart and may be regarded as rigid; while 

 between these it is supported by ordinary poles which in normal 

 circumstances are not subject to transverse stress. When a break 

 occurs in the wire, the ordinary poles are subject to tbrces tending 

 to detlect them and the detiecting force on a pole is equal to the 

 difference between the horizontal components of the tensions in 

 the two adjoining sections of wire. 



If these forces lie within the limits of safety the deflections of 

 the poles will be proportional to them and we may assume a con- 

 stant of proportionality H, such that the horizontal displacement 

 of the point of attachment of the wire = JT (detiecting force). 



Let jfj, To, Ts, ... be the horizontal tensions in successive 

 sections of the wire counting from the break and let L be the 

 common distance between successive poles. 



If X.„ be the horizontal displacement of the top of the nth pole 

 counting from the break, we have 



Thus \„+, - X„ = H{Tn+, + r„_, - 2T,,). 



The total distance apart of the extremities of the «th section of the 

 wire will accordingly be 



L + \„+, - X, = L + H{T,^, + r„_i - -2T,). 



