336 Mr. J. J. Wateistou on the Integral of Gravitation, 



elements, matter, velocity, separately measurable. When united 

 the standard of measurement is not obvious. Is the velocity or 

 square of the velocity to be taken as that which takes equal part 

 with matter in the phjenomenon Avhich results from their union ? 

 This question, debated since the time of Newton*, seems now to 

 have settled down into a tacit admission of both. This was the 

 opinion of Dr. Thomas Young, who considered the fact that two 

 bodies while freely acting on each other obtained from their mu- 

 tual action velocities inversely proportional to their respective 

 masses, as conclusive of the simple ratio of the velocity being the 

 natural standard. If this proposition had been stated in other 

 terms as above, § 4, viz. that bodies while freely acting on each 

 other obtained from their mutual action powers of producing 

 work inversely proportional to their masses, no such necessity of 

 adopting the simple ratio of the velocity would have been felt ; 

 only a noteworthy characteristic of centripetal action. The po- 

 sition of the centre of gyration in a revolving body is clearly on 

 the side of the scpiare velocity being the natural standard. The 

 motion of impinging elastic bodies also requires the square 

 velocity. 



The following considerations as to impacts, it may be useful 

 to keep in view as accessory to the principle of the conservation 

 of force. 



§9- 

 Force of impact depends on the relative motion of the impinging 



bodies. 



Whatever be the molecular texture of bodies, we may bring 

 all impacts to the condition of perfect elasticity by inteqjosing 

 a spring. There are three stages in such pheenomena; — 1. The 

 motions before impact before the spring is acted upon. 2. The 

 motion at impact, or at the instant when the maximum com- 

 pression of the spring has been effected by the foi'ce of im- 

 pact. 3. The motions after recoil, after the spring has given out 

 again the force it had received. During all three the common 

 centre of gravity is unaffected in its state of motion or rest, and 

 the force of impact, as measured by the spring's tension, is depen- 

 dent, not on the absolute, but on the relative motion of the impin- 

 ging bodies. 



* This is discussed in the last part of the article on virtual velocities, 

 ' Penny Cyclopaedia.' It concludes as follows : — " But if pressure be 

 merely considered as the cause of motion, and called force, in that sense it 

 is very difficult to see why the cause, which is only known by the effect, is 

 to be measured by anything but the simple effect. Probably this discus- 

 sion gave rise to the ehajiter of the Mecanique Celeste, in which Laplace 

 speculates upon what the laws of motion would have been if force had 

 been as a function of the velocity instead of as the simple velocity. We 

 have never met with any one who could give us an intelligible account of the 

 meaning of this investigation." 



