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



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

 \I/ = Q for which &amp;lt;= oc , we now have 



= e~* (7r ~ a) , z l = - (IT - a) ? , z. 2 = - COS a. 

 The remaining step is then given by 



+ COSa) 2 =--, 



leading to 



Along the surface of the lamina we have \^ = and real, so that the 

 corresponding values of &amp;lt; range between the limits given by 



( - ) 



V W 



cosa= 1. 

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



-^ 



rf 1 1 /3 cos a 



If we put - cos a= -- , 



05 0-cosa 



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



The relation between A- and /3 for any point of the lamina is given by 



-^4 j (1 - 13 cos + sin (! - 



sin 



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

 j3= 1, i.e. it is taken at the centre of the lamina. The breadth is therefore, 

 on the scale of our formulae, 



4+rrsina 



(iV). 



sin a 



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

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

 velocity , will be 



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

 discussion is taken substantially from the paper by Lord Rayleigh. 



