^lE. JklACQUOEN EAJ^KINE ON THE STABILITY OF LOOSE EAETH. 
17 
Note. Equilibrium of Arched Ribs. 
Although foreign to the immediate subject, the fact may here be noted, that by 
making X=F(H)=0 in equations (27.) and (29.), they furnish the means of deter- 
mining the form of the intrados of any equilibrated arched rib loaded solely with its own 
weight, or proportionally to the weight of each of its parts, when the form of the extra- 
dos is either represented by equation (28.), or developed by means of the formulae (30.); 
care being taken, in the latter case, to include the entire rib within the limits 
If the rib be di-vided mto voussou’s, their joints ought theoretically to be normal to the 
series of curves given by equation (27.) or equation (29.). 
§ 9. Extrinsic and Intrinsic Vertical Pressures. 
The exception referred to at the beginning of the last section is as follows. Besides 
the vertical pressure 
Xh=F(H), 
w'hich is uniform for each surface of equal thrust, there may also be a vertical pressure, 
uniform in each vertical hne, and varying proportionally to an arbitrary function of the 
horizontal coordinate, viz. — 
X,=0(3/) (31.) 
This last pressure does not enter into the differential equation (25.), and its amount is 
therefore to be determined in each problem by conditions distinct from the general con- 
ditions of internal equilibrium. It may be called the Extrinsic Vertical Pressure ; and 
the vertical pressure depending on the horizontal thrust, the Intrinsic Vertical Pressure ; 
so that the surfaces determined in the last section are surfaces of uniform intrinsic vertical 
pressure, the extrinsic vertical pressru’e being independent of their form and position*. 
§ 10. Limiting Relations of the Pressures, at a point in a Mass of loose Eaidh. 
The seven preceding sections, from § 3 to § 9 inclusive, contain principles which are 
common to every case of the mternal equilibrium of a solid mass, in what manner soever 
constituted. 
In the present section, those common principles 'will be combined with the peculiar 
principle of the stability of an incoherent granular mass, explained in §§ 1 and 2. 
The symbol ^ ■will be used in the folio-wing sense : — 
afb-Vc 
means, “ a is not less than b—c, and not greater than b-Vc.” The symbols < and > will 
be used in their ordinary sense of “ not greater than ” and “ not less than.” 
(p being, as in ^ 2, the angle of repose of a given kind of earth, and 0, as in § 3, the 
* In the sequel, when vertical pressures are not expressly designated as extrinsic, they are to be under- 
stood as intrinsic. 
D 
MDCCCLVII. 
