Equilibrium of Vis Viva. 169 



applying even to them ; but the proof of this is much more 

 difficult. ' For, if such collisions existed, it would follow that 

 F would depend upon c" in exactly the same manner as it 

 depends upon c, and there we should be forced to stop. In 

 this case we must consider more closely the internal motion 

 of the doublet. 



We can next prove without difficulty that in the distribu- 

 tion of velocities just found the values of c, a, /3 are altered 

 on the average in exactly the same manner and degree by 

 each impact as by its opposite one, for every set of values of 

 c", p", and u n . If, then, certain forms of central motion are 

 suddenly destroyed by a collision, the same forms are pro- 

 duced again elsewhere equally often by other collisions, and 

 consequently the same law of distribution of central motions 

 would exist even if all impacts were suddenly to cease. 



It is remarkable that just for the simple case imagined by 

 Lord Kelvin, in which the central force is taken proportional 

 to p, the calculation is exceedingly long. In order not entirely 

 to forget the sentence of De Morgan, previously quoted, I will 



assume some other law, e. g. ap + — dy or, indeed, any one in 



r 



which the angle between two consecutive apsidal lines bears no 

 rational proportion to it. 



The total energy of a doublet is 



mc 2 



L=^+~+<j>{ P ) > .... (11) 



where <£ is the potential-function. The doubled velocity of 

 the relative motion of shell and nucleus in the plane of their 

 central motion is 



K=/> Vc 2 sin 2 a + c" 2 sin 2 a" - 2cc" sin a sin a" cos 0, . (12) 



and the velocity of the centre of gravity of the doublet mul- 

 tiplied by m + m" is 



Gr= vWc 2 + ra"V 2 + 2mm'W(cosacos ex." + sin asina" cos j3) (13) 



and its component perpendicularly to the plane of central 



motion is 



TT cc" sin ct sin ex." sin B ^ . v 



H= — ; m , (11) 



Vc 2 sin 2 ol + c" 2 sin 2 a" — 2cc n sin a sin a" cos fi 



The number of doublets in unit of volume for which 

 K, L, Gr, H lie between the limits 



Kand K + dK, L and L + dL, G and Qt + dQt, H and H + dH, 



PHI. Mag. S. 5. Vol. 35. No. 214. March 1893. N 



