( 696 ) 



by developing the last exponential factor and then by expressing the 

 powers and products of cosines in cosines of multiples. 



It is clear that when integrating (8) with respect to S from 2jt to 

 only those terms are left which are independent of 6 and which 

 appear with the common factor 2t. 



The expression to be found for the probability that a velocity lies 

 between the limits R and R + (IR then becomes : 



2 \/f^f .e-i' .e-pW{\ + a^ + aJV . . .) It dR , . . (9) 



where : 



a, = « s , 



a, = qJV + qsV2! cos 2 (?— 0) + *7(2/)% 



a, = 9 V/2 2 4- JfVS/ cos 2 (<p-il) + s fl /(3/) 3 . 



For Falmouth, where as was noticed above q can be put equal 

 to nought these coefficients become simply : 



« 2 » \(n. 



-= /u 



and farther 



s = p7^ , ft = p/2 s , v =p , <ƒ)=:« , X = ;>/t' cos (# - «)• (10) 



In practice it will frequently be only necessary to calculate a few 

 of these coefiicients ; if we put : 



q/p = €, 



the integration of (9) between the limits m and leads to the 

 expression : 



(i-^» 2 )-(i + - + -^4--V + - • • • 

 V P P P J 



pm'e-l™* (a x 2/a s N 



H — r + 



1/ \p p 



+ 



2/ V. P 2 

 As for m = 00 this expression must become equal to unity, we have: 



a, 2/a, 



1 + - + -r 



P p* y/i—s' 



or, for the case q — 0, (11) becomes : 



