Resultant of two Pressures on a Fixed Point. 355 



through G from I), will, produced, determine the point Q, the 

 place of the resultant of p and q, which point Q may be, if we 

 please, stationary, while the points L and P will vary as r varies. 

 If the units of linear and force magnitude are the same, and p, 

 q, r are considered, two and two, sides of parallelograms, we shall 

 have (by common geometry), if the directions of the diagonals and 



resultants coincide, -r-^ = -, &c. ; if not, we shall have 

 Ai^ q 



BQ_pe BL_W AP_re^ 



AQ~ q DL~ q DP" p ] 



e, e ! j e" being coefficients to be determined ; but since (by common 



geometry) the product of the three alternate segments LD, AP, 



B Q equals that of the alternates B L, D P, A Q, we have at 



e' 

 once e= T/ . if e' and e" have each different values, while r 



changes, let L be a point where e' has a niaximum'value, and P 

 a point where c" has a minimum value ; then if Q, and therefore 



e' 



e, and therefore -^ are constant while r varies, it follows that 



neither e' nor e n can, independently of each other, change; nor 

 can they, both together, become greater or less ; therefore each 

 is constant for every value of r, and therefore unity, since we 

 know the one is so when r equals q, and the other when r equals 

 p ; therefore whatever the ratio of^> to q, and whatever the angle 



between them, we have -r^r =-• 

 AQ q 



That the diagonal and resultant are equal in magnitude is 



obvious, if we admit that if three forces in the same plane, acting 



on a point, are in equilibrium, either, reversed, is the resultant 



of the other two; for if one is changed in magnitude to make it 



a diagonal, no other can afterwards, with that as a side, be made 



a diagonal without a change of direction. 



Second Demonstration. 



The construction and the distribution of the forces being the 

 same, and s s' being drawn, through G, parallel to B D, we have, 

 by common geometry, 



BQ_B^ BD 



AQ *" SA ' DI ; W 



we have also, as before, 



BQ_BL DP 



aq"dl'ap ( ; 



