79] 



BOBYLEFF'S PROBLEM. 



113 



The definite integral which occurs in this expression can be calculated from 

 the formula 



where () =d/dt. logn (), is the function introduced and tabulated by 

 Gauss*. 



The normal pressure on either half is, by the method of Art. 77, 



sin nir 

 2a 2 



' TT sin a ' 



The resultant pressure in the direction of the stream is therefore 



4a 2 



(vii). 



of 



Hence, for any arbitrary velocity q Q of the stream, and any breadth b 

 either half of the lamina, the resultant pressure is 



4a 2 

 P=~.pq<>*1> ................................. (viii), 



where L stands for the numerical quantity (v). 



For a=^7r, we have L 2 + |TT, leading to the same result as in Art. 77 (10). 



In the following table, taken (with a slight modification) from Bobyleff's 

 paper, the second column gives the ratio P/P Q of the resultant pressure to 



* " Disquisitiones generates circa seriem infinitam," Werke, GSttingen, 187077, 

 t. iii., p. 161. 



L. 8 



