110 MOTION OF A LIQUID IX TWO DIMENSIONS. [CHAP. IV 



and we note that for the point / which represents the parts of the stream-line 

 ^ = for which </> = + oc , we now have 



= e~ l (TC ~ a \ z l =(ir a) ?', z. 2 = - cos a. 

 The remaining step is then given by 



(* 2 + COSa) 2 =--, 



leading to 



f, -cosa + 



- I)* + {(^ w)t - cosa) 2 -!}* .................. (i)* 



Along the surface of the lamina we have >// = () and real, so that the 

 corresponding values of <f> range between the limits given by 



The resultant pressure is to be found as in Art. 77 from the formula 



T ,. 1 l-/3cosa 



If we put 77Y- cosa= -=-*- . 



05 /3-COSa ' 



the limits of /3 are 1, and the above expression becomes 



sma_ 1 sma 



The relation between x and |8 for any point of the lamina is given by 





sin* a 



the origin being chosen so that x shall have equal and opposite values when 

 j8= 1| i-c- ^ is taken at the centre of the lamina. The breadth is therefore, 

 on the scale of our formulae, 



4+7rsina 



sin 4 a 



(iv). 



We infer from (ii) and (iv) that the resultant pressure (P ) on a lamina of 

 breadth , inclined at an angle a to the general direction of a stream of 

 velocity q Q , will be 



TT sin a 



* The solution was carried thus far by Kirchhoff (Crelle, I. c.); the subsequent 

 discussion is taken substantially from the paper by Lord Eayleigh. 



