Cor. -2. The horizontal pressure is represented by O X, and is the same 

 at each joint. 



Cor. 3. The pressures at the joints are represented by O T, O T &c. 



i 



and are therefore as the secants of the / which the joints make with 

 the vertical. If be the / of any joint with the vertical, and H the ho- 

 rizontal pressure, H sec. Q is the pressure at that joint. 



Cor. 4. The line X T will represent the whole weight of the mass be- 



2 

 tween D E and P Q, and similarly for any other joint ; hence H tan. 6 is 



2 2 

 the weight of any portion. 



2. The intrudes being a circle, 

 with the joints in the direction 

 of the radii, to find the extra- 

 dos, so that the voussoirs may F ~ V ~/r&' 

 be in equilibrium. 



Let P be any point of the in- 

 trados, O its centre, put D O P O T R 



r* - ft + (A* P) sec.2 6. 



Hence we have the following construction. Make O R horizontal, 

 R F - O E, F G horizontal. Let O P meet F G in S, draw S T vertical, 

 und take O Q = ET ; the locus of Q will be the extrados. 



Cor. 1. The extrados has F G for an asymptote. 



Cor. 2. To find the equation to the extrados. Let O be the origin of 

 the coordinates, x and y corresponding coordinates to the point Q, O D 

 ?, O F = a ; then the Equation to the curve is 



The extrados, in the case of a circular arch, is therefore a curve of the 

 4th order, very much resembling the conchoid of Nicomedes. It has an 

 asymptote F G and also a point of contrary flexure, so that it coincides 

 very nearly with the curve in which a road is usually carried over a 

 bridge. 



3. In an elliptic arch, or one of which the intrudos is a semi-ellipse, it 

 *2 be the span of the arch or the major axis of the ellipse, and f> tin 

 height of the arch or the semi-conjugate axis ; then if from any point in 

 23 B 2 



