30 Mr. T. K. Chinmayam on the Flow of 



In the neighbourhood of any point, D and e can be 

 regarded as constant and equation (32) integrated. 



The " mean lines of flow" are still given hj r0 = () ; but 

 the actual lines of flow which wind about them cut them 

 along the curves 



~2r0 2 + e = imr, (33) 



c 



instead of the curves- 2 r0 2 — mir. It may be noted that it is 



equation (33) that determines the position of the maxima 

 and minima of illumination when the effect of diffraction is 

 also taken into account, so that the points of intersection of 

 the lines of flow with the loci of maxima and minima of 

 illumination still lie on the " mean lines of flow." 



If the squares and products of k, k! are not neglected it 

 can be shown that the mean lines are given by the equation 



rd0(l + K ,2 + KK\/2) + 0dr = Q 



or r0 1+a = const,, 



where ol = k' 2 + ^2kk, and is hence very small. 



It will be interesting finally to deduce the results for the 

 case of diffraction by a straight edge from the above 

 investigation. Putting a = 0, k becomes zero also. We 

 have then directly from equation (29), 



20(,c' 2 -k' cos Q2r0 2 + ~y^ 

 tan <j) = 



l + tc' 2 -2fc'cos - 



(>^l) 



and -y- = 



ar r 



1 



2*'cos(^2rt? 2 +jJ 



an expression very similar to (21). The " mean lines of 

 flow'' are given by r6 = const., or in the rectangular co- 

 ordinates used, by #= const. They are straight lines parallel 

 to the ?/-axis ; the mean direction of energy- flow is apparently 

 not altered by the presence of the edge. The actual lines of 

 flow cut the successive loci of maxima and minima of 

 illumination (i. e. the curves 2r0 2 -h\/S — m\/2) at points, 

 which lie along the mean lines of flow. The shape of the 



