Life of Radioactive Substances. 561 



within a sufficiently short period. There are of course 

 other possible assumptions, but the above seems the simplest. 

 The particles whose position determines the atom's stability 

 may be taken to rotate or oscillate with the mean energy 

 E = /iz/, the energy of the a particle emitted. Each particle 

 passes through the critical region v times per second, so that 

 the probability of its being there within the time t is rv. 

 As the particles are considered to be independent, the 

 probability of N particles traversing the critical position 

 within the time r is (tv)™. If X atoms are considered, the 

 number which become unstable and explode in the time 

 dt is : 



dX=-X(rv)^dt ) 

 whence 



x=x *-("A 



The radioactive constant A is thus equal to (ti/) n . The 

 equation v= j leads to \=/-| E N , or introducing the 

 empirical formula for the range R, R = 1*35 . 10 8 E 32 , one finds 



log \ = N (log^*- + log 3-80 . 10- G ) +2/3N logR 



or 



log X=N(log 5-76 . 10 s0 + log t) + 2/3N log R. 



If, as was explained above, t is to be regarded as the time 

 taken by a strain to traverse the nucleus, it is possible to 

 determine its order of magnitude in terms of the charge on 

 the nucleus ne, n being the atomic number, the mass of the 

 nucleus M equal to the mass of the atom, and finally of the 

 radius of the nucleus r. The first essential is to calculate 

 the velocity of propagation of an elastic wave (sound wave) 

 in the nucleus. For this purpose we may obviously as a 

 first approximation treat the nucleus as a homogeneous, 

 positive volume charge of ne electrostatic units and density 



M 



p= — . The error introduced by neglecting to take tlie 



discontinuous structure into account cannot change the 

 order of magnitude, as the nucleus consists of such a large 

 number of particles. The velocity of an elastic wave may 



thus be roughly taken as a/ — , k being the compressibility 

 of the nucleus v . -y-, V being the volume -q-r 8 . The 

 Phil. Mag. S. 6. Vol. 30. No. 178. Oct. 1915. 2 



