Potential Problems connected with the Circular Arc. 273 



In view of these facts it appears to me that Strutt's expe- 

 riments should be repeated before we recognize the scattering 

 of light by air molecules as demonstrated. It would be well 

 to employ sunlight and a glass lens, and look for the cone 

 with the eye. Its absence would prove that Strutt's results 

 were due to a cloud resulting from the action of the ultra- 

 violet rays on the air. 



XXVI L Some Two-Dimensional Potential Problems connected 

 with the Circular Arc. II. By W. G. Bickley, B.Sc* 



§ 1. TN a recent paper f the author has given the solution 

 X of some potential problems connected with the 

 circular arc, and interpreted the results in terms of elec- 

 tricity and hydrodynamics. In particular, the velocity 

 potential and stream functions for circulatory flow about an 

 infinitely long lamina in the form of a circular are, and for 

 the disturbance of a stream due to such a lamina, were 

 determined. Tt is now proposed to give drawings of the 

 stream-lines in the latter case, to examine the case of rota- 

 tion of the arc, and to give a brief discussion of the motion 

 of the arc when free to move, and acted upon by the con- 

 sequent fluid pressures. 



§ 2. The stream-lines were obtained by first mapping out 

 the ,2-plane by a system of orthogonal coordinates given by 

 the relation 



1 + 5/ , . 



*— *T+^ (1) 



where T = p + ia, and s is written, for brevity, for sin£. The 



a 



results of the preceding paper show that for circulatory flow 



W = (f> + iylr =— T , (2) 



so that the figure of the last paper is the requisite map, for 

 the particular case of the semicircle. Making the substitu- 

 tion (1) above, in equation (14) of the preceding paper, we 

 obtain for the case of flow past the arc 



iv — 2issinh (t — t/3), (3) 



giving yjr = 2s sinh p cos (<r — l3) (4) 



* Communicated by the Author. 



t Phil. Mag. [6] vol. xxxv. p. 396 (May 1918). 



Phil. Mag. S. 6. Vol. 36. No. 213. Sept. 1918. T 



