THE EQUILIBRIUM OF BODIES IN CONTACT 469 



The equation to the line of resistance is determined by eUminating 

 C between the equations (1) and (2), and tiiat to the line of pressure 

 by eliminating it between (2) and (3). 



If the first elimination be made, and it be observed, that 



- » /(yi - ?A) <J^ + )~ (y. - y^) d^ = - jjil/i - y.) d:i', 



there will be obtained the following general equation to the line of 

 resistance, 



_ :y S P sin 1> + ^ cos e /Cyi° - ;/,') <7as - sin 9 ffjy^ - y.) </g' + S J= PJe cos tt> 

 ■^ " 2P cos O + cos e j (y, - y,) r/s; ••••(.)■ 



The second elimination is greatly simplified in the case in which 

 P, <I>, k are independent of C. Since in this case, equation (3) gives 



y - i(y. + y-^ + (« - C) tan e = o (5). 



If the intersections be supposed to be made horizontally, (Fig. 2,) 

 o must be assumed = 0. If they be made vertically, (Fig. 3) = -. 

 In the latter case, equation (5) gives C = as. 



The elimination of C between (3) and (2) is therefore the same as 

 that between (1) and (2), and the line of pressure in this case, coin- 

 cides with the line of resistance. 



3. Let the mass be a trapezoidal form. fFig. 4.) 



Let AB and CD be inclined to the axis of s at angles a,, a.,, and 

 assume CA = a; .-. yi = a + « tan a, , y-i = z tan a-.. 



.(6). 



Hence f{y] - i/V) rlz = f/z{a + z tan a,) + ^z' (tan" a, - tan-' a,) 



/C'/i - yO d^ = «« + iss' (tan a, - tan a^) 



ff(?/i - y^) d^' = i «£' + i a" (tan a, - tan a,) 



Therefore by substitution in equation (4) we have for the equation 

 to the line of resistance 



_ cg^{tang|-tana,|{tanai+t3nai,- tan9} + ^a2'|tana,-tane} + z{sec 9I)Psin<l) + ^a'} + sec6SJrP^cos<l> „ 

 •''" ^i^ltan a, - taiT^}^ az + secBZP cos <Sf ■U)- 



.So 2 



