the Excited Activity of Radium to the Cathode. 533 



which have encounters with air molecules on their way to the 

 sides of the cylindrical vessel. 



Consider a number of particles travelling a distance x 

 through air in which the average distance between molecules 

 is A, and the effective radii of molecules is o\ The distance 

 may be divided into N layers each of thickness X, in which 



each layer contains, on the average. -^ molecules per unit area, 



and the proportion, on the average, of encounters in each layer 



2 



is — y Hence the proportion of the particles which have 

 no encounters is ( 1— ^- ) = ( 1 — ^- j , and this for small 



_x ^3 



values of X is equal to e L , where L = -, and is the " mean 



free path " of the particles in air. 



Supposing now that the proportion of encounters effective 

 in displacing a negative electron be R, the above result will 

 be modified and the proportion of particles experiencing no 



_R 



" effective encounters " will be e L . 



In the experiments with which we are dealing, the central 

 rod on which the activity was deposited was shielded at 

 both ends, so that the calculation may be simplified by con- 

 sidering the problem as a two-dimensional one, that is by 

 taking the cylinder to be of unlimited length. It is con- 

 venient to calculate the proportion of particles which reach the 

 side of the cylinder without having any effective encounters, 

 and then to subtract the ratio from unity. 



Consider the number of particles which reach a small area 

 of magnitude A, containing a point 0. Use polar coordi- 

 nates ?% 0, cj), with as origin ; if P be any point of space 

 let 0P = r, and if Q be the projection of P on the right 

 section of the cylinder through 0, of which C is the centre, 

 let COQ = 0, and Q0P = <£. 



Let M be the number of particles projected by the emana- 

 tion per unit volume. The number of particles reaching the 

 area A from an element of volume 8Y at P is proportional to 



A 



the solid angle subtended by A at P, which is -^ cos COP. In 



particular, the number reaching A from P is 



* r MA 



2 cos 6 cos <£ SV. 



