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134 PHILOSOPHICAL TRANSACTIONS. [aNNO 1733. 



larger that body is, the greater will be the moving force which it receives from 

 the spring. 



Having now clearly proved, that the moving forces are not proportional to the 

 squares of the velocities. Dr. J. proceeds next to demonstrate, that they are pro- 

 portional to the velocities themselves: and, in oider to this, he makes use of no 

 other principles or axioms, than such as are admitted on both sides, or at least 

 have never yet been controverted a priori by either party. 



Axiom 4. — Springs of unequal lengths, when bent alike, have equal pressures. 



We speak here of springs equal in all respects, except the length only; and, 

 by being bent alike, we mean, that they are so compressed, as that the lengths 

 they are now reduced to, are exactly proportional to their natural lengths, or to 

 the lengths they are of when no way compressed. 



In this condition, if one be directly opposed to the other, they will n)utually 

 sustain each others pressure, so as to maintain a perfect equilibrium ; or, if 

 each be placed separately in a vertical situation, they will sustain equal weights. 

 And in one or the other of these cases, it is evident that they must exercise 

 equal pressures. 



Axiom 5. — Equal pressures in equal times produce equal moving forces. 



Frop. 1. — Moving forces are proportional to the masses and velocities jointly. 



Demonslr. — Let there be two springs, of the lengths 1 and 2, but equal in 

 all other respects, and bent alike; and, in unbending themselves, let the spring 



1 drive before it a body whose mass is 2 ; and the spring 1 another body of the 

 mass 1. 



Now, by corol. 1 of his general theorem concerning the action of springs, 

 these two springs will unbend themselves exactly in the same time; and conse- 

 quently the spring 2 will unbend itself with a velocity double of that of the 

 spring 1; and, by corol. 12 of the same theorem, it will give to the body 1 a 

 velocity double of that which the body 2 will receive from the spring 1. 



Also, as the two springs were supposed to be bent alike at first, and the spring 



2 unbends itself with a velocity double to that of the spring 1, it is manifest, 

 that during the whole time of their expansion, they will be always bent alike, 

 one to the other. 



Therefore, by axiom 4, their pressures will be constantly equal to each other; 

 and hence, by axiom 5, the infinitely small moving forces produced by each of 

 these springs, in every infinitely small part of time, will be equal to each other. 

 Consequently the sums of those infinitely small moving forces, that is, the whole 

 moving forces produced by the two springs, will be equal to each other. And 

 the masses of the two bodies being 2 and 1, and their velocities being 1 and 2 

 respectively, it is plain, that the moving forces are proportional to the masses ami 

 velocities jointly, a. e. d. 



