These formulas are general. If, however, the motion is steady, then, the pressure 

 being taken as zero at points where the velocity q is zero, p = - pq^/'2, where p is the density 

 of the fluid and 



;;: = - p q^dy, Y = p q^dx.N = p q^{xdx^ ydy) [74d,e,f] 



It was shown by Blasius that these expressions could be transformed so as to contain 

 only integrals of a certain analytic function of s where z = x + iy., the methods of complex- 

 variable theory then become available for their evaluation. From Equation [74d,e], since 



2 2 2 



q = u^ + V , 



■^f 



X-iY = — p \ {u^ + v-^) {dx - i dy) 



Now (w^ + V ) (dx-idy) = (-u + iv) (-u-iv) (dx-idy). Since the path of integration is part of 

 streamline, the vector {dx, dy) is parallel to the velocity or to (u, v) at the same point; hence 

 dy/dx = v/u or v dx - udy. Thus 



(- M - iv) {dx - i dy) = - udx + iudy-iv dx-v dy - (-u + iv) (dx+i dy), 



and, using -u + iv - dw/dz, as in Equations [25i] and [34f], and dz = dx + idy. 



The torque requires a somewhat different artifice. Clearly 



q^ (xdx + ydy) = {R) [(w^ + v-^) {x + iy) {dx - idy)] 



where the symbol (/?) signifies that only the real part of what follows is to be taken. The 

 same changes as before can be made in this expression. It is then found that, since the real 

 part of any integral arises exclusively from the real part of the integrated expression, [74f] 

 may be written 



N = -^ p{R)\ 



dw\ 2 



, zdz [74h] 



dz 



165 



