292 



and therefore 

 4ttS 

 c 



Prof. Max Mason on the Flow of 



-M 



Cn 2 . 0,0, 1 



»V 





C^ 





We are principally interested in the £{me w^an S of the 

 vector S, which determines a field of steady flow. By inte- 

 grating over a period and dividing by the period, the following 

 values for the time means of C x 2 , OiC 2 , C 2 2 are obtained : 



("v-r"2- _ 



u n, , 



0A=^COS-(r 2 . 



.. a 1 Sir. , 



»"i) =9 cos srO'2-n), 



where X is the wave-length. Therefore 



rd") 



^?=k! 



a~c 



+ k ; 



1'2 



27T. 

 COS -— (?• 



■*0 



r^ 2 



By the aid of this expression the differential equation of 

 the lines of mean energy flow may be found, i. e. the differ- 

 ential equation of the curves which have the direction of S 

 at each point. Along such a curve r 2 may be considered as 

 a function of rj. If Si and S 2 denote the coefficients of k : 

 and k 2 in the above equation, it may readily be seen (fig. 2) 



Fig. 2. 



that along the curve in question 



dr 2 : dr ± — S 2 + Si cos 6 : S x + S 2 cos 0, 



where 6 is the angle between k x and k 2 . The differential 

 equation of the lines of mean energy flow is therefore 



dr 2 



9__ 



r 2 2 cos 6 + r x 2 -f yy^l + cos 6) cos T-( r 2"~ r i) 



v 2 



a—. 



-f r x 2 cos Q -f- ^(l + cos d) cos ^(r 2 — r 2 ) 



