80 REPORT— 1894. 



Hence the remaining differentials are connected by the relation 



8T ; 8T' , ^ 



Gp,, dp,: 



i.e., 

 or 



dPn gn ' 



Moreover, since the impulsive action takes place instantaneously, the 

 co-ordinates do not vary, and therefore 



dqi'=dqi, dq^'^dq-i . . . dq„_^'=dq„_^ ; 



^.^ d p^' . . . dp,l dq^' . . . dq„_/ _q„ _ Q_E.D. 



" dpi ... dp^dq^ . . . dq„_i q,,' 



This form of the Jacobian is applicable to the hypothetic law of 

 molecular force, often assumed in the Kinetic Theory, where, when two 

 molecules (regai'ded as material points) reach a certain distance, c, their 

 mutual attraction becomes infinite. Their directions of motion undergo 

 refraction towards the line joining them at the beginning of the encounter 

 and away from that line at the end of the encounter, and each refraction 

 must be treated separately, Watson's relation (22) being used for the motion 

 of the molecules between the two refractions. 



33. In the case of a collision unaccompanied by loss of kinetic energy, 

 such as occui's between perfectly smooth elastic bodies, the Jacobian 

 relation between the velocities or momenta just before and just after the 

 collision is easily found. For Burbury has shown ' that in a system or 

 pair of colliding systems with n degrees of freedom, n — 1 Linear functions 

 of the velocities, which he calls S^S.,, . . . S„_„ are unaltered by the 

 collision, and one linear function E, has its sign changed. Therefore, 



dS^'dS^' . . . dS'„_^dR'=-dS,dS. . . . dS,_,dR, 



and by the properties of Jacobians it follows at once that, since the 

 co-ordinates of the system are unaltered, the initial and final momenta^ 

 specified by any co-ordinates whatever, are connected by the relation 



^ (Pi',P2', • • • Pn' )_ 1 (.-yn^ 



<J{P\,P2, ■ • ■ 1\) 



In a collision between smooth bodies, R is the relative velocity of the 

 points which come into contact resolved along the common normal, and 

 Burbury has given examples of the functions S , , . . . S„_i, R in several 

 simple cases, viz., a pair of unequal smooth spheres, a sphere colliding 

 with a spheroid, and a system of two spheres loaded at one side of their 

 centres. 



The same argument could probably be extended to multiple collisions, 

 for if 01 — r linear functions of the velocities were unaltered and the remain- 

 ing r had their signs changed, the functional determinant would be equal 

 to ( — l)*". Again, for a collision between two ' perfectly rough bodies ' with 

 a coefficient of ' f rictional restitution ' equal to unity the three components 

 of the relative velocity of the points of contact would be reversed, so that 

 r have the value 3. 



' ' On the Collision of Elastic Bodies,' Phil. Trans. R.S., 1892, A, p. 408. 



