to support the mass of earth HXYF, whose weight is W, upon the inclined plane XY. Produce XB and YF to meet in A, and let AX=x, AH=c, AXY=1, "=weight of each cubic foot of the earth, the natural slope of its surface FE. Now it may be shown, precisely by the same reasoning as before, that the actual pressure of the earth upon the portion BX of the wall is represented by that value of P which is a maximum in respect to the variable; moreover, that the relation of P and is expressed by the function P W cot. (+); where W, (area HXYF)=",(AXY— AHF)=('tan. —ccot. );

.....

..P=¿a ̧(x2 tan. —c2 cot. ) cot. (1+0) . . . . . (444). Expanding cot. (+),

Pu

(x2 tan. —c2 cot. ) (1-tan. tan. ❤)

tan. +tan.

To facilitate the differentiation of this function, let tan. +tan. be represented by 2, and let it be observed that whatever conditions determine the maximum value of P in respect to z determine also its maximum value in respect * Then tan. =2-tan. ; therefore 1-tan. ɩ tan. ❤= -2 tan. +tan. '=-2 tan. +sec. 2. Also, a tan. — ccot. =x'z- (tan. +c cot. ).

to .

Substituting these values in the preceding expression for P, and reducing,

The first condition of a maximum is therefore satisfied by the equation

or, solving this equation in respect to 2, and reducing, it is satisfied by the equation

Now the second condition of a maximum is evidently satisfied by any positive value of z, and therefore by the positive root of this equation. Taking, therefore, the posi tive sign, substituting for z its value, and transposing,

which equation determines the tangent of the inclination AXY to the vertical, of the base XY of that wedge-like mass of earth HXYF, whose pressure is borne by the sur face BX of the wall. To determine the actual pressure upon the wall, this value of tan. must be substituted in the expression for P (equation 445). Now the two first terms of the expression within the brackets in the second member of that equation may be placed under the form

But it appears by equation (446) that the two terms which compose this expression are equal, so that the expression is equivalent to -22.c tan. ; or, substituting for the value of

2, to

-2x tan. (sec. 2p+;

c2

cosec. ), or to

(x2 tan. 1 +c2)1. Substituting in equation (445),

Pu,{-2x sec. ¤(x2 tan. 3¤+c3) * + (æ2 tan. 2¤+c3) +x2 sec. 1} ..P, sec. -(x2 tan. '+c2) * } '. . . . . (448);

....

by which expression is determined the actual pressure upon a portion of the wall, the distance of whose lowest point from A is represented by a.

329. The conditions necessary that a revetement wall carry ing a parapet may not be overthrown by the slipping of any course of stones on the subjacent course.

Let, represent the limiting angle of the resistance of the stones of the wall upon one another; and let OQ represent

the direction of the resultant pressure

on the course XZ. The proposed
conditions are then involved (Art.
141.) in the inequality e,>QOM, or
tan., tan. QOM, or tan. >
RS
Р

>
OS weight of BZ; or substituting
for P its value (equation 448), and
(2ax+x tan. a) for the weight of
BZ, it appears that the proposed
conditions are determined by the
inequality

330. The line of resistance in a revetement wall carrying a parapet.

Let OT be taken to represent the pressure P, and OS the weight of BZ. Complete the parallelogram ST, and produce its diagonal OR to Q; then will be a point in the line of resistance. Let AX=x, QX=y, AB=b, AP=X,

XM=λ, W = weight of BZt. By similar triangles,

RS

OS

; but QM=(y-λ), OM=x-X, RS=P, OS=W;

QM

OM

Now the value of λ is determined from equation (414), by

The influence, upon the equilibrium of the wall, of the small portion of

earth BHE is neglected in this and the subsequent computation.

The influence of the weight of the small mass of earth BEH which rests on the summit of the wall is here again neglected.

substituting in that equation (x-b) for c; whence we obtain,

observing that tan. a,=0, and substituting a for a,,

入二

(x—b)2 tan. 'a+a (x—b) tan. a+a2

(x-b) tan. a +2a

Also W=(-b) {(x-b) tan. a+2a}.

;

..Wλ=1μ(x—b) {}(x—b)3 tan. 3a+a(x—b) tan. a+a3}.

It remains, therefore, only to determine the value of the term P. X. Now it is evident (Art. 16.) that the product P. X is equal to the sum of the moments of the pressures upon the elementary surfaces which compose the whole surface BX. But the pressure upon any such elementary surface, whose distance from A is x, is evidently represented A*; its moment is therefore represented by

by

dP dx

dP

da xAx

and the sum of the moments of all such elementary pressures

Performing the actual multiplication of the factors in the

second member of this equation, observing that

'tan.'

(x2tan.3p+c2) and re

(x2tan. '+c2)+'

(x2 tan. +c) ducing we obtain

* P being a function of x, let it be represented by f(x); then will f(x) repre sent the pressure upon a portion of the surface BX terminated at the distance a from A, and f(x+Ax) that upon a portion terminated at the distance +z; therefore f(x+Ax)-fx will represent the pressure upon the small element A of the surface included between these two distances. But by Taylor's theorem, d'P (4x) dx 1.2

dP
dx

+, &c.; therefore, neglecting terms in

f(x+Ax)-fx= Δ. +

dP

volving powers of Ar above the first, pressure on element =

dP

x (sec. *p+tan. *p) — 2 sec. 4 (œ” tan. *p+c′)+ +

Multiplying this equation by x, and integrating between the limits band x,

P.X=",

(sec.+tan.') (x3 —b3) — § sec. o cot. ' {(x2 tan.'

+c)-( tan.+c)+c sec. cot." {(x2 tan. 2+c2)—(U2 tan.' +c2)1}

....

(452).

This value of P. X being substituted in equation (450), and the values of W, W, P, from equations (448) and (451), the line of resistance to the revetement wall will be determined, and thence all the conditions of its stability may be found as before.*

THE ARCH.

331. Each of the structures, the conditions of whose stability (considered as a system of bodies in contact), have hitherto been discussed, whatever may have been the pressures supposed to be insistent upon it, has been supposed to rest ultimately upon a single resisting surface, the resultant of the resistances on the different elements of which was at once determined in magnitude and direction by the resultant of the given insistent pressurest being equal and opposite to that resultant.

The arch is a system of bodies in contact which reposes ultimately upon two resisting surfaces called its abutments. The resistances of these surfaces are in equilibrium with the

*The limits which the author has in this work imposed upon himself do not leave him space to enter further upon the discussion of this case of the revetement wall, the application of which to the theory of fortification is so direct and obvious. The reader desirous of further information is referred to the treatise of M. Poncelet, entitled "Mémoire sur la Stabilité des Revete ments et de leurs Fondations." He will there find the subject developed in all its practical relations, and treated with the accustomed originality and power of that illustrious author. The above method of investigation has nothing in common with the method adopted by M. Poncelet except Coulomb's principie of the wedge of maximum pressure.

The weight of the structure itself is supposed to be included among these pressures.