EQUILIBRIUM OF THE ARCH. 



295 



Further, let it be supposed that the resultant of the forces upon 

 the portion of the mass, cut off by the plane, in each of its positions, 

 intersects with the resultant similarly taken in its immediately previovis 

 position — an hypothesis which introduces a new condition into the 

 question and establishes a second relation between the quantities 

 M„ M„ M,; A, B, C. 



That relation is determined as follows. 



Since x, y, as are to be considered as the co-ordinates of a point of 

 intersection of two consecutive resultants; we may differentiate the 

 equations (3) with respect to the arbitrary constants A and B, consi- 

 dering X and y as constant. From this differentiation, the following 

 equations are obtained: 



= ss 



= » 



KS../(t).J,KSL/_i)., 



dA 



■ dAv 



dB 



.dB 



+ 





dA 



dA 



dB 



dA+-^dB{ 

 dB 



■(4), 



whence, eliminating z 



^M^^A^-^dB 



dA 



dB 



dA + r^ft— aJ? I 



dA 



dB 



dA 



d 



dA + 



N, 

 M, 



dB 



dA 



dB 



dBi 



^^^^clA+-^dB{ 



= 0...(5). 



This last equation determines the relation between A and B ne- 

 cessary to the continual intersection of the consecutive resultants ; and 

 the elimination of these quantities between equations (3) and (4), 

 produces two equations in x, y, % which are those to the locus of 

 that intersection. That is, they are the equations to the line of 



PRESSURE. 



