THE EQUILIBRIUM OF BODIES IN CONTACT. 465 



One of these lines — the line of resistance — determining the j^oint of 

 application of the resultant of the pressures upon each of the surfaces of 

 contact of the system, and the other — the line of pressure — the direction 

 of that resultant, the determination of the two includes the whole theory 

 of the equilibrium of the system. 



In its application to the theory of the arch there belong to the line 

 of resistance all those properties treated of in my former paper which 

 have reference to the condition "that the voussoirs shall not turn upon 

 the angles of one another." 



It follows, therefore, on the principles established in that paper that 

 this line touches the intrados of the arch at certain points equidistant from 

 the crown, called points of rupture, and that the position of these points, 

 and, consequently, that of the point of application of the resultant of 

 the pressures upon the key-stone, are subject to the condition that this 

 resultant is a minimum ; and this condition being supposed, all the cir- 

 cumstances which connect themselves with the equilibrium of the circular 

 arch, as a complete segment, and a broken or gothic arch, subjected to 

 any variety of loading, are discussed and determined in the eleventh 

 section of the following paper. 



The condition, however, that the resultant pressure upon the key-stone 

 is subjected in respect to the position of its point of application to the 

 condition of a minimum, is dependant upon Jnjpothetical qualities of the 

 masonry. It supposes an unyielding material for the arch-stones, and 

 a mathematical adjustment of their surfaces. These have no existence 

 in practice. On the striking of the centers the arch invariably sinks at 

 the crown, its voussoirs slightly opening there upon their lower edges, 

 and thus pressing upon one another exclusively by their upper edges. 

 Practically, the line of resistance then touches the extrados at the crown ; 

 whilst the condition of the minimum is satisfied by its contact with 

 the intrados at the points of rupture in the haunches. This condition 

 being assumed, all consideration of the yielding quality of the material 

 of the arch or of its abutments is eliminated. It is thus discussed as a 

 practical question in the twelfth section of this paper. 



