EQUILIBRIUM OF THE ARCH. 299 



or, zpgcoscp - pgsivKp + xgsm{(i + 7)sec7{2a^ -r—secy.A^} 



=pgK sin ((p + /3) +gsm{fi + 7)8607 {aA^ ^-— r — ^sec7 . A^}. 



Differentiating which equation with regard to the arbitrar}' constant 

 A, we obtain 



A = x, 



whence by elimination and reduction, 



^^l( ^- ^ ] ( seey.sm{f3 + y) \ ^ 



^ \ pk J \ COS(p j ' 



_ /_a\ [ sec 7. sin (^ + 7) ] ^ 

 \pj \ cos(p /■ 



+ tan (f) . X 



_^ sin((/> + /3) 



COS0 



The above is the equation to the line of pressure. It indicates a 

 point of contranj flexure corresponding to 



ah 



X =■ 7 cos <h. 



a — h ^ 



The curve is concave to the axis of x, between the origin and this 

 point. It is afterwards continually convex. 



A minimum value of % is determined by the equation 



\Jta\ f / ip\^ sin>cos^7 ) 



^ \a-AV^ ^ ^ Ui •sinMi8 + 7)r * 



It will be observed that since all the forces applied to the system 

 may be supposed to act in the same plane, the two conditions. 



First, " That in every position of the intersecting plane, the forces 

 shall have a single resultant," and Secondly, "That the consecutive re- 

 sultants shall intersect," are necessarily satisfied. 

 Vol. V. Paet III. Re 



