CHAP. XV.] THE STONE ARCH. 325 



mine the stability and the joints of the abutments. "Wlien the 

 live load of the arch can be neglected with respect to its own 

 weight, and when the material of the arch possesses the usual 

 strength, and the pressure line lies within the inner third, then 

 the lower point of rupture lies so low that the back masonry 

 reaching from this point beyond the pressure line completely 

 encloses it. 



There is, therefore, nothing arbitrary, when the form of the 

 arch is given, except the depth. Since in an arch of less depth 

 than is allowable in practice a support line can still be in- 

 scribed, the graphical method is unable to determine the 

 proper depth. We must then leave to theory the development 

 of formulae by which this can be determined, and assume that 

 not only the form of the arch is given, but also its proper 

 depth and the lower joint of rupture. It is required to deter- 

 mine the stability of the abutments. 



The stability of the abutments can be regarded from two 

 points of view. We may consider it as a continuation of the 

 arch, as in many English and French bridges, in which the 

 arch is continued as such, clear to the foundation ; or we may 

 regard it as a wall whose moment about the joint of rupture 

 resists the rotation about this joint due to the thrust. Both 

 views are identical, as the entire theory of the support line rests 

 upon the investigation of the rotation. They differ only in the 

 method of expressing the safety of the abutment. 



If the arch is continued to the foundation, and the space be- 

 tween it and the road line filled up by spandrels ; or if the 

 thicknesH of the abutment increases from above as the support 

 line requires ; or, as is often the case in England, the abutment 

 consists of walls parallel to the crown, separated by hollow 

 spaces; still, in every case the abutment is not to be distin- 

 guished from the arch proper it is stable when the support 

 line lies in the interior. If the prolonged arch is separated 

 entirely from the adjacent masonry, there is no reason for not 

 giving the axis of the prolongation the form of the support 

 line itself. 



If, on the other hand, there is no separatioc of the a-rch and 

 abutment, as in the English hollow abutments, it is sufficient 

 that the support line lie in the inner third, and the abutment 

 will be certainly stable. 



The supposition that the resistance of the mortar is suffi 



