470 PROFESSOR MOSELEY, ON THE THEORY OP 



This equation being of three dimensions in x, it follows that for certain 

 values of y there are three possible values of is Tlie curve has therefore 

 a point of contrary flexure, and is somewhat of the form shewn in the 

 figure. 



The T^iiNE OF Pressure in a trapezoidal mass has been determined 

 in my former paper. It is there shewn to be of three dimensions in x, 

 and to have, like the line of resistance, a point of contrary flexure. 



The Points of Rupture being those where the line of resistance 

 meets the Infrados or the Extrados of the mass may be determined by 

 assuming in equation (7), y = a + x^ tan a, and y = z, tan a., ; whence 

 there is obtained 



3 3 a 5 „ sec esP sin <1> — sec tan u, 2P cos <1> - i d' 



V* _i_ ^ -4- o -~— — — X 



"' tan a, - tan 02 ' ' {tanu, — tana,} ^tanoj — 2 tana, — tan B} 



P sece {S + P/c cos<b — «2Pcos<t)i 



{tan a, — tana4 {tan aa — 2 tana, — tan 0j 



3« , „sec GSPsin "f - sec e tan a„ SPcos * + i«'^ ,. 



fe + ^ + o — z: 



■^ tan a, - tan 02 ' ^ {tan a, - tan a,} {tan a, - 2 tan a^ — tan Q\ ' 



^ g sec 62 + Pk cos <!■ ^ ^ 



{tana, — tan uj} {tan a, - 2tana2 - tan0} ^ 



If tan aj — 2 tan a, = tan there is but one point of rupture in the 

 Extrados. 



If tan a, — 2 tan a, = tan there is but one point of rupture in the 

 Intrados. 



These single points of rupture are determined in the two cases by the 

 equations 



_ ^ ± Pk cos <l> — alP cos <I) . 



^' ~ tana,2Pcoscl) - SPsin* + l«=cos0 ^ '' 



1 ± Pk cos <t> . 



^' ~ tana.SPcos* -2Psin$ - ia^cosG ^ ^' 



