VOL. XLIII.] PHILOSOPHICAL TRANSACTIONS. IQ 



Thus, if the spring cl, fig. 4, pi. 1, resting with the end l against any im- 

 moveable support, but otherwise lying in its natural situation, and at full liberty, 

 shall, by any force p, be pressed inwards, or from c towards l, through the 

 space of 1 inch, and can be there detained by that force p, the resistance of the 

 spring, and the force p, exactly counterbalancing each other ; then the force 2p 

 will bend the spring through the space of 2 inches, 3p through 3 inches, 4p 

 through 4 inches, &c. the space cl, fig. 5, through which the spring is bent, or 

 by which the end c is removed from its natural situation, being always propor- 

 tional to the force which will bend it so far, and will detain it so bent. 



And if one end l be fastened to an immoveable support, fig. 6, and the other 

 end c be drawn outwards to 1, and be there detained from returning back by any 

 fofee p, the space cl, through which it is so drawn outwards, will be always pro- 

 portional to the force p, which is able to detain it in that situation. The same 

 principle holds in all cases, where the spring is of any form whatever, and is, in 

 any manner whatever, forcibly removed from its natural situation. 



Here it may be noticed, that the elastic force of the air is a power of a dif- 

 ferent nature, and governed by different laws, from that of a spring. For sup- 

 posing the line lc, fig. 4, to represent a cylindrical volume of air, which, by 

 compression, is reduced to l1, fig. 5, or, by dilatation, is extended to l1, fig. 6, 

 its elastic force will be reciprocally as l1, fig. 5 and 6 ; whereas the force or resist- 

 ance of a spring will be directly as cl. 



Dr. J. now proceeds to his general proposition, and its corollaries; in which 

 he remarks, that if he happen at any time to express himself with less accuracy, 

 as in making weights, times, velocities, &c. to become promiscuously the sub- 

 jects of geometrical or arithmetical operations, he desires, once for all, to be 

 understood, not as speaking of those qualities themselves, but of lines, or num- 

 bers, proportional to them. 



Theorem. — If a spring of the strength p, and the length gl, fig. 7, lying at 

 full liberty on a horizontal plane, rest with one end l against an immoveable sup- 

 port; and a body of the weight m, moving with the velocity v, in the direction 

 of the axis of the spring, strike directly on the other end c, and thereby force 

 the spring inwards, or bend it through any space cb; and a middle proportional 

 CG, be taken between the line cl X -, and 2a, a being the height to which a 

 heavy body would ascend in vacuo with the velocity v ; and, on the radius r = cg, 

 be erected the quadrant of a circle gfa, it will be, 



1. When the spring is bent through any right sine of that quadrant, as cb, 

 the velocity v of the body M, is, to the original velocity v, as the co-sine to the 



radius : that is, v =: v X — • 



a 



2. The time / of bending the spring through the same sine cb, is to t itbe 



D2 



