262 PROFESSOR TAIT ON THE 



particle. Its value will be more exactly indicated when the reason for its 

 introduction is pointed out. 



The last factor of the integrand depends on the fact that the particles are 

 emitted from moving layers : — involving the so-called Doppler, properly the 

 Komer, principle. 



We neglect, however, as insensible the difference between the absorption due 

 to slowly moving layers and that due to the same when stationary. 



Because a, the range of x, is small we may write with sufficient approxima- 

 tion 



n = n + n 'x, &c, &c. 



Introducing this notation, the expression above becomes 



2 Jo J o Jo °° °\ \% v e J J vcos0 



Now, to the degree of approximation adopted, 



/ edx = e x + e 'x 2 /2 . 

 Jo 



The second term of this must always be very small in comparison with the 

 first, even for an exceptionally long free path. But, if we were to make 



30 = -' e o/ e o 



the second term would become equal to the first. Hence a, the upper limit of 

 the x integration, must be made much smaller than this quantity. Thus we 

 may write 



g - sec O/'edx = 6 -e x sec 9(1 _ e^ggg 0/2 + . . . ) . 



We said, above, that 



/l 



CM = ah- 



lS a large number, say of the order 10 2 . It appears then at once that terms in 



g -, « =g -100 =1 0- 43 uear ly 



may be neglected. Such terms occur at the upper limit in the integration with 

 regard to x above, and what we have said shows, first why a had to be intro- 

 duced, second why it disappears from the result. 



Writing now only those factors of the above expression which are concerned 

 in the integration with respect to x, we have 



/ (l + ( j' + V f + %f)x + . . . Vl - e 'xhee 8/2 + . . . V «>*»ec »dx , 

 or 



l( C08 e+^(<+<) C0S w). 



e \ e \n V J ) 



