CHAP. XIV.] 



THE BEAOED ARCH. 



263 



hyperbola, which has for asymptotes the vertical through the 

 abutment and a straight line which cuts the axis of symmetry 

 of the arch at the point b [PL 24, Fig. 93], h under the crown, 

 the tangent at the crown at f a from the crown, and the chord 

 of the arc at 6 a from the centre. The centre of the hyper- 

 bola is at <?, ^ h below the horizontal through the crown. The 

 two hyperbolas osculate at the point y vertically below the 

 crown. [See Fig, 93.] 



As an aid to the construction of these hyperbolas, we give the 

 following table : 



From the table it is easy to construct the hyperbola for any 

 given case. We have, of course, a perfectly similar hyperbola 

 for the other half, its centre e being similarly situated with 

 respect to the crown, to the right of c. We have then simply 

 to draw a line through the intersection m of the weight P 

 [Fig. 93] with the line i k, at -5- h above c, tangent to the hyper- 

 bola, and we have at once the direction of the resultant. This 

 tangent may be drawn by eye, or geometrically constructed if 



