THE EQUILIBRIUM OF BODIES IN CONTACT. 477 



(see Fig. 13) ; <i> will then become ~ + a,, and P will vary as x ; let it 

 equal mx. Substituting in equation (.35), and integrating, 

 X (y'l - yl) - |m (s' - Z') sec a, + luz (a^ - Z') cos a, 



y " ;:; (37). 



^i (y. -yO f^» - /^(^^ - Z'^) sina. 



From this equation the line of resistance may be determined for any 

 given inclination of the internal face or form of the external face of the 

 embankment; or conversely, these circumstances may themselves be de- 

 termined according to a given equation to the line of resistance. 



If the section of the embankment be of the form of a trapezoid 

 (Fig. 14), by the integration (6) we have 



^^_^Mi(tan^a.-tan'a,)-M (|sec«,-cosa,)j+ff:;-tana, +g(a=-^Z^'cosa,)+|MZseca ., 



s»{tan a, -tan a, - „ sin a.J + 'i.az+p.Z' sin a, ...(38). 



If Z = or the fluid extend to the edge of the embankment, this 

 becomes the equation to an hyperbola. 



The point of rupture of the extrados, or the greatest possible height 

 of the embankment, may be determined as before, by assuming 



y = a + z tan a,, 

 whence 



^ s^{(tan a, - tan a,)(2tan a, + tan a,) - ,. [3 cos (n, - a,) sec a, - 2sec a,]J +flr{2tan <„ -tan a,-^.sina,j 



The supposition now about to be made, with regard to the line of 

 resistance, is, that it traverses the center of the embankment. We have 

 thence, by equation (37), 



1 („ + ,/.\ = -^(y- - y') dz- §^{&'- Z') seca, + ^x(x''-Z')cosa, 

 ^ 2/(y, -y,)dz-f. {z' - Z') sin a, ' 



whence observing that 



(yi + y-^ Ky^ y^) dz = //(y, - y,) dz . d (y, + y,) + /(yj - ^d dz 

 ffiV' - y.) (f^ ■ rf(y, + J/.) 



= M {(r' - Z') [z cos a, + I (y, + y,) sin «J - f {z' - Z') sec a,} (40). 



.3 r 2 



