478 PROFESSOR MOSELEY, ON THE THEORY OF 



Whence substituting for y^ its value 2 tan a^, and differentiating twice 

 in respect to s, a differential equation will be obtained, the solution of 

 which will determine the required relation of y^ and x. If a., = 0, or the 

 intrados be a vertical plane, we obtain, by the first differentiation in re- 

 spect to z, 



- Jy,<lyi = z' — Z'; .-. -{y\ - A') = ss'^ — Z'\ 



if A be the breadth of the embankment on the level of the surface of 

 the fluid ; 



.•. y? = « {r - (2^ - i^^^)} (41). 



The equation to an hyperbola, whose center is in the inner edge of 

 the embankment A, the ratio of whose axes is y/fx, and whose semi-axis 



is (Z' - - A'). 



9. The Arch. 



The plane of intersection has hitherto been supposed, in its successive 

 changes of position, to remain always parallel to itself. Let this hypothesis 

 now be discarded, and as the simplest case of a variable inclination of the 

 plane, let it be supposed to revolve about a given horizontal line or axis 

 within itself. Let moreover the extrados and the intrados be supposed 

 to be cylindrical surfaces, having this line for their common axis; and 

 suppose this arch to have a load uniformly distributed along the extrados 

 in a line parallel to the axis and at a horizontal distance from it equal 

 to ;r. - • 



Let ABB (Fig. 15), represent any section of this mass perpendicu- 

 lar to the axis C, and X the corresponding load. Let the horizontal 

 force P be applied in AD at a vertical distance p from C; and let CT 

 be any position of the intersecting plane, intersected by the resultant 

 of the forces P and X, and the weight of the mass ASTD in R, 



PC A = H. PCR = e, cr=R, CS =r, CR = p. 



