4<°>8 Lord Rayleigh on the Resistance of Fluids. 



very considerable, if the experiments of Vince are to be at all 

 relied on. 



In the following analysis cf> and yfr are the potential and 

 stream functions, z = d' + ii/, co = cf> + iyfr ; and it is known that 

 the general conditions of fluid motion in two dimensions are 

 satisfied by taking z as an arbitrary function of co. If 



g = ?=jt) (cos0 + J S m0), ... (A) 



Kirchhoff shows that £ represents the velocity of the steam at 

 any point, with the exception that its modulus p is proportional 

 to the reciprocal of the yelocity instead of to the yelocity itself. 

 If the general yelocity of the stream be unity, the condition 

 to be satisfied along a surface of separation bounding a region 

 of dead water is p = l. The value of f must of course also 

 preserve a constant value along the same surface. 

 The form of £ applicable to the present problem is 





?= 



1 



cos a -1 h 



V co 



A/ ( cos « + 



JL) 2 _ 



s/cq/ 



1. . 



(B) 



When 



ft)=0O y 



cos a — i sin a. 











The 



surface 



of separation 



corresponds 



toyjr = 0, 



for 



which 



value off w becomes real ; and the point at which the stream 

 divides corresponds to co = 0, for which £= go. For tfr = 



and real values of cosa-1 less than unitv, p = l. This 



s/ (o J ' 



portion therefore corresponds to the surface of separation, for 



which the pressure is constant. When cos a -\ is real and 



s/co 



greater than unity, K is real, indicating that the direction of 

 motion is parallel to the axis of x. This part corresponds to 

 the anterior face of the lamina. 



The augmentation of pressure at any point is represented 



by x(l ^ )? ^ the density of the fluid be taken as unity; 



and thus the whole resistance is measured by the integral 



jK'-?)* 



if dl represents an element of the width of the lamina. Kirch- 

 hoff shows how to change the variable of integration from I 



to co. The velocity of the fluid is -—, or, since f is here zero, 



