324 TTTR STONE ARCH. [CTTAl". XV. 



In the ordinary form of joint, as shown in Fig. 103, the sup- 

 port line varies from the pressure line, since the vertical through 

 the centre of gravity S does not coincide with the joint under 

 S. If, however, we should conceive the arch divided into ver- 

 tical laminae, then the support and pressure lines fall together. 

 This is precisely the assumption always made in the analytical 

 discussion of the theory of the arch. 



Thus we take the area A = / ydx, and this expression sup- 

 poses the arch divided into vertical laminae. 



The first to make clearly this distinction between the lines of 

 pressure and support, was Mosely (Civil Eng^). Other authors 

 have after him adopted this distinction, and then proved that 

 the two lines always coincide, without remarking that this coin- 

 cidence is only because of the adoption of the above integral. 

 The same assumption simplifies greatly the graphical construc- 

 tion also (the analytical treatment is without it well-nigh im- 

 possible). We shall therefore assume vertical laminae where 

 it is at all permissible. This is always permissible at the crown 

 of arches with horizontal tangent, because there the joints are 

 vertical, and over all, when the pressure line lies below the 

 axis of the arch ; for the support line lies always above the 

 pressure line, and therefore, in this case, the conditions of sta- 

 bility are more favorable for it than for the pressure line itself, 

 when considered as the line of support. 



Moreover, it is easy at any point of the pressure line con- 

 structed with vertical laminae to pass to that line for another 

 form of joint, and to the corresponding support line. Thus, if 

 for the point A (PI. 25, Fig. 104) we have found the pressure 

 Q, and if now we wish to pass to the joint A B C, we prolong 

 Q till it meets P, the weight of the voussoir A B C D, and re- 

 solve P and Q at this point into Q'. Then Q' is a side of the 

 new pressure line, and it cuts A B in a point of the support 

 line. 



In this way we can easily determine whether the error com- 

 mitted when we substitute the pressure line for vertical laminaa 

 for that for the actual joints, which is given by the segment of 

 the joint A B between Q t and the pressure line, can be disre- 

 garded. 



177. Dimensions of the Arcb. The object of the con- 

 struction of the pressure or support line in the arch is to deter- 



