440 Lord Rayleigh on the Resistance of Fluids. 



Now if the arbitrary constant be taken suitably, the complete 

 value of z is 



/8 2 cosa + 2/3 /Wl^+sin- 1 /? 

 sm a sur a 



The odd terms in z will contribute nothing to the integral ; and 

 therefore we may take for the moment of pressure about ^ = 0, 



i 



+ 1 9*/.1 _/Q2 



'2 s/ 1— ft' 2 /3 2 COS0i _ 7T cos a 



sur a sin a sin d a 4 sin a 



In this result the first factor represents the total pressure, and 



COS OL 



therefore -. — v—r- expresses the distance of the centre of pres- 

 4 sm ol L r 



sure from the point z = 0. With the same origin the value of 



COS (X, 



z for the middle of the lamina is . . ; and thus the displace- 



sin* a x 



ment of the centre of pressure from the middle of the lamina is 



3 cos « 



4 sin 4 a 



This distance must now be expressed in terms of I or 



4 + 7rsina-r- sin 3 a, 

 which gives as the final result, 



3 cos ex. . I 



4 4 + 7r sin a. 



The negative sign indicates that the centre of pressure is on 

 the upstream side of the middle point. 



As to the form of the surface of separation, its intrinsic 

 equation is given at once by the value of ? in terms of <w. The 

 real part of f is cos 6 (since p = l), where 6 is the angle be- 

 tween the tangent at any point and the plane of the lamina. 

 Along the surface of separation co is identical with cf>, and 



~- = l. Thus if * be the length of the arc of either branch 

 ds & 



measured from the point where it joins the lamina, the intrinsic 



equation is 



cos o = cos a ± , 



v s + c 



and the constant is to be determined by the condition that 



dx 

 s = () when cos 0= zt 1. Since cos 6 = -^-, the relation between 



as 



x and 8 is readily obtained on integration ; but the relation 



between y and s is more complicated. 



