KBOHANICAJL PHILOSOPHY. STATICS. [ROBERVAL'S BALANCE. 



The tix pivots A, B, C, D. 



that the bars may more, freely shout thora, with as little 

 friction u possible, in the vertical plane while no lateral 



Fig. 110. 



motion is permitted. From this construction it is 

 evident that whatever position the framework A B C D 

 may assume, when weights are suspended from the arms 

 K 1, .-iii.l M X, the four bars A B, CD, AC, and BD, 

 will always form a pftnUlalognun, and the arms K L and 

 M N retain a horizontal position. 



It is a peculiar property of this machine, that if two 

 weights P and Q balance i-;-h other when suspended 

 from two points S and T in K L and M N, they will also 

 balance from whatever point* in K L and M N they are 



leil. 



To show the properties on which this peculiar statical 

 paradox depends, we shall neglect the weights of the 

 various parts of the machine ; or, which will come to 

 the same thing, suppose the weights of its various parts 

 so chosen that it shall be in equilibrium for every position 

 in which the framework A K C D can be placed, when 

 no weights are suspended from the arms K L and .M N. 

 Fig. 111. 



Also lot the distances A K and i ' F be represented by 

 a, E II and I- 1 D by ' K 1, and AC are rigidly 



connected, the pressure of P on S will be transmitted by 

 the bars K L and AC to the pivots A and C, 

 where it will produce two pressures I', and P a . 



These pressures are indeterminate, both in magni- 

 tude and < 



Let Pj, the unknown pressure on A, be repre- 

 sented in magnitude and direction by A P. ; and P. 

 that on C by C P,,. 



As the machine is supposed to be in a state of 

 equilibrium, P, will be counteracted by the reaction 

 It, acting in the direction AR,(Fig. Ill'), All 

 and A P, being in the same line, and R, <!',. 



Similarly 1'.. will be counteracted by the equal 

 and opposite reaction R 2 , represented in magni- 

 tude and direction by C R.j. 



Again, because the bars BD ami M N' are rigidly 

 connected, the pressure of Q on T will be trans- 

 mitted, and produce two pressures, P 3 and P 4 , at B 

 and D, which will be counteracted by equal and oppo- 

 site reactions R 3 and R,. 



The bars K L and A C are kept in equilibrium by the 

 three forces P, K,, and R n , acting at S, A and C in the 



:nus SP, A U, and C'R 2 . 



Resolving the forces represented by AR, and CR. 

 into the vertical and horizontal forces X,, Y,, and 

 X.,, Y,, represented by A X^ AY,, CY 2 , and ( 

 (1% 111!). 

 We sliall have by Prop. XVIII. 



P-Y.-f-YjandX.-X,. 



The couple X, -AC, whose tendency is to twist the 

 rod A C, U entirely destroyed by the reaction of the 

 pivots at A and C, and the forces Y, and Y,, will be 

 equal and opposite to two forces which will exert a pros- _ 

 sure on the extremities A and C of the levers A B and 

 CD. 



In a similar manner, by resolving the forces R, 

 and R 4 into the horizontal and vertical forces X 3) 

 Y,, X., and Y.,'we shall have 



QiY+Y.w.dX s -X,. 



The forces Y s and Y 4 being equal and opposite to 

 two vertical pressures acting on the extremities B and 

 D of the levers A U and (.' D. 



Finally, wo have the lever A B resting on the ful- 

 crum , kept in equilibrium by the forces Y, and 



.ilso neglect the friction of the six pivots A, B, C, D, 

 Let two weights P and Q, represented in magnitude and 



. Fig. lit. 



direction by S P and TQ (Fig. Ill), be suspended from 

 the arms K L and M N at the points 8 and T, and let us 

 snppose that thoy balance each other 



Hence Y, -o-Yj-6. 

 Also for the lever C D we have 

 Y 3 . -a-Y -6. 



Hence (Y , +Y, ) o= (Y, + Y 4 ) 6 ; 

 But Y,+Y 4 -P, and Y a +Y 4 -Q ; 

 Therefore P-o-Q- 6. 



Provided therefore that the pivots are strong enough 



to resist the lateral strains acting on them, the ronditii.ii 



of equilibrium for the machine is, that P multiplied by 



a shall be equal to Q multiplied by 6 : a result entirely 



independent of the quantities c and d, or of the di- 



,ud M T. 



