ARCHES. 623 
From (7) & (12) 
ier ape LR tor fo Td 
waft + sobes( a)tf X45 +) (13) 
= F(wW) a known function of w. 
D being the arbitrary constant of integration 
or e(satr)ar (14) 
of which the complete solution is — 
a ey (15) 
w= FReosyp +Ssinpteoos pf conte 
SS) 
wu is then known from (5) & (10) 
uw, v, are thus known throughout ; the 6 arbitrary con- 
ainate are determined by the efid conditions at each 
abutment. 
fag Se 
eS ae ae 
which furnish six equations for the arbitrary constants. 
=) (16) 
This is the formal solution of the problem. 
To apply this to the elliptic arch of uniform thickness, 
observe that we have 
w = gp(h—y) (17) 
where p is the height from the major axis to the roadway 
(or to an imaginary line representing the partial load 
treated as of the same density as the filling above the 
arch) p is the density of the material, 
and k = S A-° where A = ./1—e?si* # and ¢ is the 
2 
eccentric angle of P measured from the minor axis 
