EQUILIBRIUM OF THE ARCH. 297 



Also 



p = VM^VMr+~m, 



where M^, M.^, M-j are supposed to be taken throughout the tvhole 

 mass. 



Thus there are six equations of condition, which together with the 

 equation 



cos^ a + cos^ /3 + cos^ 7 = 1. 

 determine the seven quantities P', Xi, y^, %i\ a, /3, 7 in terms of the 

 forces (other than P') which compose the system, and the constants 

 which enter its equation. These fix the relations necessary to the equi- 

 librium of the mass considered as one continued geometrical solid. 



Before proceeding to the discussion of the additional conditions 

 requisite to the equilibrium when the mass passes from the invariable 

 form here supposed, to a variable form, it will be well to give an 

 example of the application of the principles which have been already 

 laid down to the actual determination of the line of pressure in a par- 

 ticular instance. 



5. Let then ABCD (fig. 1.) represent a heavy mass, bounded at its 

 extremities by parallel planes AB and CD, and laterally, by the planes 

 AC and BD inclined at any angle to one another. 



Let the mass be imagined to be intersected by an infinite number 

 of planes parallel to AB, of which one is mn, and to be supported 

 by forces acting at p and p' at angles cp and <f>' with the horizon. 



It is required under these circumstances to determine the form and 

 position of the line of pressure. 



Let the line P'G bisect AB and CD. Draw P'E horizontal and 

 PM vertical. 



Let P'M= A, CD = 2b, P'p = k, 

 AB = 2a, P'G = h, Gp' = k'. 

 Inclination of P'G to the horizon =7, - 

 AB = /3. 



