8 PARALLELOGRAM OF FORCES. 



Bat, by th< <>sis, p and n acting at C ha 



sultaiit b >n CO; therefore p and may )> 



red by their resultant and its point .t' Application trans- 

 ferred to (y. And wi has also been transferred to G. Hmee 

 by this process we have removed the forces which 

 at A to the point O without alteri; tl, rt. \Ve may 



infer then (sec Art. vi) that # is a point in tin* direct! 

 resultant of p and m-f n at A; that is, the resultair 

 /i + n acts in the direction of the diagonal AG, provided 

 ypothcsis is correct Hut the hypothesis is corre< 

 equal forces, asp, p, I i'"iv it i> true for t'mv 



consequently forp, 3/?, and so on: li.-m-e it is true for j>. 



Hence it is true for/?, r.y>. ami p, r.p. an<l 



.nd so on; and it is finally true for s.p and / .//. 

 where r and * are positive integers. 



We have still to shew that the Proposition is true for 

 incommensurable forces. 



This may be inferred from the fact that when two mag- 

 nitudes are incommensurable, HO that the ratio of one to the 

 . cannot be expressed exactly },y a Iracti'-n, w- can still 

 find a fraction which differs from the true r;<ti-> l>y a <V 

 lest than any assigned fraction. Or it may be established 

 indirectly thus. 



J/?, AC represent two such forces. Complete the 

 parallelogram BC. Then if their 

 resultant do not act along Al> 

 pose it to act along AE\ drav 

 parallel to BD. Divide AC into a 



Iwr of equal portions, eacli 

 than J>K; mark off from CD por- 



cqual to these, and 1 t A' be _ 

 the last division; this evident! 

 falls between D and E; draw GK parallel to AC. Then 

 two forces represented by AC, AG have a resultant in the. 

 AKi because they are comincnsural.le; therefore the 

 forces AC and AB are equivalent to JA t .- ther with a 

 force equal to GB applied at A alon^ AB. And we may 

 as obvious that the resultant of these forces must lie 



