SUSTAINING EQUAL PRESSURES. 51 



and meeting one another in the point m ; then is m the required place 

 of the centre of gravity. 



66. In computing numerically the values of the rectangular co- 

 ordinates mr and ms, we have supposed, that DF the base of the 

 applied triangle, is determiriable by the application of equation (20) ; 

 this supposition however is perfectly unnecessary, for the base of the 

 triangle is always equal to the difference between the parallel sides of 

 the given trapezoid ; and moreover, the equation (20), applies only to 

 the particular case for which it has been deduced, viz. when the 

 pressure on the applied triangle and that on the trapezoid to which it 

 is applied are equal to one another. 



9. WHEN THE PARALLELOGRAM IS SO DIVIDED, THAT THE PRESSURES 

 ON THE TWO PARTS ARE TO ONE ANOTHER IN ANY RATIO WHATEVER. 



67. In the sixth, seventh and eighth problems preceding, we have 

 supposed the given rectangular parallelogram to be divided into two 

 parts, such, that the pressures upon them shall be equal between 

 themselves, and the investigation has accordingly been limited to that 

 particular case ; but in order to render the solution general, we shall 

 consider the division to be so effected, that the pressures on the two 

 parts may be to one another in any ratio whatever, such as that of 

 m to n, 



For which purpose then, by referring to the fifth problem, where the 

 given parallelogram is divided horizontally, we find, that the pressure 

 on the upper portion is expressed by J6a; 2 ssin.^, and that on the 

 lower portion, by { J (I x)*-\-x(l ,z) } 6ssin.0 ; but these ex- 

 pressions in their present state are equal to one another, and they 

 are now required to be reduced in the ratio of m to n ; consequently, 

 we have 



1*': \ k(l x )* + x(l x}} ::m:n, 

 and this, by expanding the second term, becomes 



x* : I* a 8 : : m : n; 

 or by equating the products of the extremes and means, we obtain 



n a; 2 nz m / 2 ma? \ 

 therefore, by transposition, we get 



(m -\- n} o^zr mZ% 

 and finally, by division and evolution, we have 



E2 



