Pleocliroic Tlalos. 631 



of tlie x ray towards the end of its range, as shown in the 

 Bragg curves, was confirmed ; and the identification o£ 

 thorium halos recorded. 



As the number of measurements multiplied it appeared as 

 if the calculated or theoretic values tended to range a very 

 little above those determined by observation. The source of 

 this has been found to lie in the mode in which Bragg's law 

 w r as being applied. The method of applying this law had in- 

 vited the comparison of the average square root of the atomic 

 weight of the mineral containing the halo with that of air. 

 Now Bragg and Kleeman's figures and results do not bring 

 air quite harmoniously into line with the solid substances 

 examined (Phil. Mag. Sept. 1905), and it seemed a more 

 correct procedure to calculate the range in the mineral by 

 comparison w r ith the square root of the atomic weight of a 

 substance more generally in agreement with the law and 

 similar to the mineral in physical state. Choosing aluminium 

 for the purpose, the table given further on for the ranges in 

 some important halo-bearing minerals, has been calculated 

 both for the a rays of RaC and RaA and for ThC and ThX. 

 The calculation is most readily made as follows. 



Bragg and Kleeman found that if the products of range 

 and density in a number of different substances are compared 

 with the corresponding products in the case of air, the several 

 quotients obtained stand in the same relation one with another 

 as the square roots of the atomic weights of the several sub- 

 stances. The quotient for aluminium has the value 1*23. 

 IE we assume the range in air to be one centimetre, then the 



,. . , . . . Ax 1-23 , 

 corresponding range in aluminium is r= ^ , where A 



and 8 are the densities of air and aluminium. Also we have 



or a , ^ .11- i 



s — = ; where d , r arc the densitv and range m any 



or a ° 



particular mineral ; a' is the average square root of the 

 atomic weight of the mineral, and a the square root of the 

 atomic weight of aluminium. Substituting the expression 

 for r and taking the density of air as 0'0012 ; and writing 



5 15 for a ; we have y'=0'000287^, ; which gives the range 



in the mineral corresponding to one centimetre in air. Thus 

 for certain biotites a 1 is found to be 4'5 and h' 2*8, hence 

 r =0'000461. Then for the rays of RaC, having a range 

 in air of 7'06 cm., we find the range in this biotite to be 

 0*0326 mm. The quantity a' may be determined either from 



