( 697 ) 



1 — e— P 1 »" 1 



— —■ 71 (l—e-t*) 



p > m*e—P m ' t f a, \ 



— - — 1 — e -i' er-P etc. 



2! ^ p ) 



from which is immediately evident that in many cases the three first 

 terms are sufficient, so that then the calculation of the coefficients 

 can be entirely avoided, or at most only a l must be taken into 

 account; for generally jt is small, so that already 



1— er~ p 



will be a small quantity. If q is not small the calculation becomes 

 rather tedious. 



8. To find expressions for the quantities V and V*, the mean 

 velocity and the mean square of the velocity independent of the sign, 

 we have to multiply (9) successively by R and R* and to integrate 

 between the limits co and which, with the well known fundamental 

 equation, leads to the expressions : 



1 =A 



f 2a, 2 4a, 2.4.6a, \ 



f 1 4. — 1 + — * 4- I + • • • 



A t Sn / 3a. 3.5a, 3.5.7.a, 



2 V p V 2p (2 P y ^ (2 P y 



2A f 4a, 4.6a, 4.6.8a, 



V — ( 1 H \4- ! + 8 + . . . 



\ 



(13) 



9. For the calculation of the frequency of the directions independent 

 of the velocity we have first to integrate (5) with respect to R between 

 the limits go and and then with respect to 6 between the desired 

 limits 6; the mean velocity as function of the direction is found by 

 the application of the same operation to (5) after multiplication by 

 R. It is then easy to give to a frequency-formula found in this way 

 the form of a Fourier series. For brevity we treat here only the 

 case that q = and the angle-limits are jt to 0. 



By putting 



V 



