EQUILIBRIUM OF THE ARCH. 907 



Also, since it touches the surface in the point ar^ysSSj; 



dz-i i d-^x-iyiZj ^^ 



\ dXi J 



( d'^x^.yi%i \ 

 dF,%., V dz, ) ^ ^ 

 d%2 l d'^X2yi%i \ 



\ dyi I 



Eliminating x.^, y^, z.^ among these Jive equations two relations are 

 established between the force P'*, the co-ordinates of its point of ap- 

 plication, and the angles which fix its direction (see Art. 4) ; by elimi- 

 nation between which a further relation is established between six of 

 these seven quantities, and, finally, by the equations of condition 



COS^ a + COS^ /3 + COS* 7=1. 



a relation is obtained between four of them. 



Thus then we may obtain the value of P' in terms of three of 

 the quantities x^, y^, s, ; a, /3, 7. 



Its maximum and minimum values are then at once determined by 

 the known conditions of the maxima and minima of functions of 

 three variables. 



8. It is evident that the minimum value of P', being that which 

 just counteracts the tendency of the mass to revolve about the point 

 where the line of pressure touches its surface, is also precisely that 

 force which would be exerted there by another equal and similar mass, 

 acted upon by equal forces, under the same circumstances, but placed 

 in a contrary position, so that its line of pressure shall have, at P, 

 a common tangent with the line of pressure of the first mass. 



* The line of pressure is here supposed to commence at P', and the force P" to enter 

 among the other forces which determine its equation. 

 Vol. V. Part III. S s 



