40 Statics. 



hence 



A'D : CD : : r : p, 

 and consequently 



p x A'D = r X CD'. 



In like manner from the parallels DD, EE f , BB, we ob- 

 tain, 



DE : BE :: DE : B'E'; 



and from the equation p' X DE = q X BE, we have the propor- 

 tion 



DE : BE : : q : Q' ; 

 therefore 



DE' : B'E' :: q : p' 



:: q : p + r ; 

 whence 



(p + r) X DE' = q x B'E. 



Let us now take in the intersection ZF of the two planes, a fixed 

 point F, and seek the distance FE of this point from the point 

 E', corresponding to E, through which the resultant passes. It 

 is clear that 



A'D = FD FA 1 , CD = FC FD, 

 DE = FE FD, BE = FB' FE. 



Substituting these values for their equals in the two equations, 



p x A'D' =rx CD, (p + r)xD'E' = qX B'E', 

 -we shall have 



p x FD p x FA' = r x FC r x FD, 



(p + r ) x FE' (p + r)FD'=qxFB'qX FE'. 



The first of these two equations gives 



(p + r ) x FD =p X FA' + r X FC '; 



substituting this value for its equal in the second equation, we 

 obtain 



(p + r ) X FE' p X FA' r X FC = q X FB' q x FE'- 9 

 or, the terms multiplied by FE' being collected into one factor, 



(p + q + r ) X FE' = P xFA' + qxFB' + rX FC, 

 which gives 



= P X FA> + 9 X FB' + r X FC' 

 P + <l + r 



