ON SUSPENSION BRIDGES. 379 



— (dw- cdy) = ^a* -f- to*) dy^ which transposed leaves 

 , _ a* dw o« dw 



putting n for 1 -»- -1-, as it will often occur. 

 If we multiply both sides of equation (1) by 



b d^ii dw 



-;» and a\y we obtain (b -tc)dy=. ^^^ ^^ - 



The integral of this is a circular arc (b + c) y, 

 whose radius is aw*, and tangent w: or if w^e 

 reduce these quantities to radius unity we have 



a ( w \ 

 y=-^.arc(^tan. = -,;- - (8). 



To find 6\ — Suppose the quantities a, h and c 

 are given; we shall, assuming different values for 

 tr, be enabled to find, by equation (8) and a 

 table of natural tangents, y in terms of w; and 

 thence obtain the values of 5, since s = w^cy. 



To find X, — Since from equation 1, Art. 4, 

 adx = wdy; substituting in this for dy its value 

 derived from equation (7) above, we have, 



a 2 wdw 



26 c^n^'io'^ ^ 



whence «=~log. {^n^-w^) + Const. 

 3 b2 



