382 J. H. ROHRS, ESQ. ON THE 



dy s . 



— = - in the catenary 



doc c 



s d*V 1 dV „ 

 »'---T7 + -- r + M 1 (2). 



c dir c ds 



s dV 2 



[uds = — + _(F- M ) + M x t ; 



c ds c 



M 1 and M being arbitrary functions of t only introduced by the integration. 



M will be so assumed that V may vanish when s = a, or V a = 0. 



v = —— + Z,,, Z,, an arbitrary function of t, will be assumed to have such a value that 



it s 



v may = when s = a. 



dV 

 fvds = — - + L x 8 + L , (L a being another arbitrary function of t). 



(18 



« dv dy . , . 



JNow — — = 0, — — = when 3 = 0, by hypothesis ; 

 ds ds 



dV , , .„ , dF 



.*. = h L , and if L = must = 0. 



a« s _ ds s=0 



Eliminating w by integration from our two fundamental equations, we have 



~d?v dw fd?u dy (dv dw du dy\ 



rd'v dw /-d'u dy (dv da du dy\ 



J de Is ~ J df * dl = Us Ts ~ Ts Ts) 



I s\ d?V 



Hence 

 d* nd 

 d~¥ 



d< rtdV _ \ c s> dV s . ._, ,,, __.-i 



d 3 r 



d^' 



s 1 



The greatest value of - will, throughout this investigation, be supposed to be - , as it is 



c ^ 



nearly in most suspension bridges of wide span, consequently powers of - above the second 

 will be rejected as too small to materially modify the result, we shall then have 

 d : dV d* T - / s 2 \ 2s d* ,„ „, 



-7Je* meg { 1 + &)d? (3) - 



2s d 2 d V 



If, as a first approximation we neglect the term -? -— , P, and assume for — — a sum 



° c 2 dr ds 



