Curtis — Pressure of Earth against a Retaining Wall. 17 



Through C (fig. 6) draw the line CT inclined to the vertical 

 line CA at the angle ^ ; then 



Cr = S" sec (/,, and -4r.= {Zr^ + ^Z^)^ 



If, then, we cut off 



•.P = ~ (CT-AT)\ 



TV=AT, pJ^CV; 



s'o that, if we construct the locus of V, the force on the portion of 

 the wall, measured by any line A 0, will be obtained by drawing 

 through C SL line inclined to the vertical at the angle to cut the 

 locus of V, and the force P will be proportional to the square on 

 CV, which may be denoted by s. 



This locus is easily obtained, as follows:— let LB be axis of Y, 

 and i(7 axis of X; let cc and y be the co-ordinates of point V; then 



QT = X tan^ = jUir, 

 and 



A' + \y + iixf = AT"" - TV^ = X- sec- ^ - (1 + ^a") x" ; 



.'. X' - if - ^jxxy = /r, 

 an equilateral hyperbola whose asymptotes are defined by 



x^ - y"^ - 2fxxy = 0, 



or , . 



2/ + tan f ^ + 2 j ^ = 0, 



and 



y-tan(^- ^]x 



We are concerned, however, only with the portion of the hyper- 

 bola which lies within the trapezium. 



To determine the point of application of the force P, extended 

 to the entire height of the wall measured loj AO; let be taken 

 for origin, and the axes of X and ¥ vertical and horizontal ; then, 

 if Xi denote the abscissa of the point of application of P, while x 

 represents the abscissa of any point U, and dP the elementary 

 pressure on a portion of the waU nieasured by dx, then affecting 

 with a suffix each quantity relating to the point 0, Pi Xi = jxdP 



SCIEN. PKOC, K.D.S. — VOLi IV. PT. I. C 



