52 SECTIONS SUSTAINING PRESSURES 



68. Again, in the case of the sixth problem, where the given paral- 

 lelogram is divided by a line drawn parallel to the diagonal ; we find, 

 that the pressure on the triangle cut off by the line of division, is 

 expressed by bx*s(3l x) sin.0 -H 6 /, and consequently, by sub- 

 traction, that on the remaining portion is expressed by bssm .<f> 

 { 3 l a x* (3 I #) } x ~ 6 / ; now, these expressions, by the condi- 

 tions of the problem, are equal to one another ; but in the present 

 case, they are to be reduced in the ratio of m to n ; for which purpose 



we have 



a* (3 / x) : 3 / 8 x*(3 I x) : : m : n ; 



therefore, by equating the products of the extreme and mean terms, 

 we get 



nx*(3l x) 3ml 3 mx*(3l a?) ; 

 and from this, by transposition, we shall obtain 



(m + ) (3 lx* x s ) = 3 m I s ; 



therefore, by dividing and transposing the terms, we have 



3ml 9 



(28). 



Now, in order to reduce the above equation, there must be substi- 

 tuted the numbers which express the given ratio, together with the 

 length of the parallelogram, and then, the value of a; will be obtained 

 by any of the rules for resolving cubic equations. 



69. In like manner as above, by referring to the eighth problem, 

 where the given parallelogram is divided by a line drawn from one of 

 the upper angles, and terminating in the lower side ; we find, that the 

 pressure on the triangle cut off by the line of division, is expressed by 

 ^/ 8 #ssin.0, and consequently, by subtraction, the pressure on the 

 remaining portion is expressed by Fssm.(f>(3b 2#); and these 

 expressions, according to the conditions of the problem, are equal to 

 one another ; but in the present instance, they are to be reduced in 

 the ratio of m to n ; hence, we have 



consequently, by equating the products of the extreme and mean 

 terms, we get 



2nx nr 3 bm 2 war, 



from which, by transposition, we obtain 



2(m + n)x = 3bm, 

 and finally, by division, we have 



36m 

 r -2(m-fn)' (29). 



