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



which on reduction becomes 





dO _ 

 dr 



e 1 - 



'1- 



-k cos — 2rd 2 

 c 



■2 k cos- 2r<9 2 

 c 



Equation (21) may 



also be 



written 





d(rO) 



K c 

 ~~20'n 



. cos -^rdyjr — 



lere ^= 



--2r6 2 . -. 

 c 



In the immediate n( 



(21) 



(22) 



point we can regard /c/@ = {ald(2r + ad)}* as constant, since 

 its variation with r and 6 will be small compared with the 

 periodic part. On integrating (22) we get 



t6 



(lOn* 81 ^" 00118 **' * " ' ^ 



which determines the shape of the lines of flow in the 

 immediate neighbourhood of a point. The points of inter- 

 section of these curves with the loci of maximum and 

 minimum of illumination (-\|r = m7r) are given by the equation 



rd = const (24) 



This gives the " mean lines of flow " about which energy 

 crinkles down. They are shown in fig. 4 (thin lines). We 

 find that so long as 6 is not very large, these curves are 

 inclined to the direct rays at angles smaller than those 

 corresponding to the maxima and minima loci. If we 

 imagine the latter set of curves as forming successive bright 

 and dark tubes, energy will flow down across the tubes, its 

 direction being periodically shifted such that it tends to 

 flow along the bright tubes and to cut across the dark tubes, 

 The shift in its direction goes through one complete cycle as 

 the energy passes from one dark tube to the next or from 

 one bright tube to the next, so that the " wave-length " of a 

 crinkle in the neighbourhood of a point (r, 6) may be deter- 

 mined by finding the distance along the curve rd = const, 

 between two successive points of intersection of that curve 

 sind the family of curves r6 2 = m\/2 (the loci of minima of 

 illumination) . Thus 



r0 2 = m\l2; r0 = C, giving 0=m\/2C 



or se=\/2C, 



