If the motion is irrotational, substitution in the last equation from such equations as 

 [136e] gives the Laplace equation for the potential <^: 



8s Ss 



SX 



d\ \ Sfi 8iy 



d\ 



Ss 8s 



A Sfi (90 



)i' S\ &s dji 



8s ^ 8s g 

 A II 01 



du \ 8\ 8)1 Ss 



= 0. 

 [136g] 



In some problems the mass of fluid under consideration is actually bounded by a coor- 

 dinate surface. For example, let A be constant over the boundary. Then, provided the fluid 

 at infinity is at rest, formula [17c] for the kinetic energy T of the fluid can be written 



T = 



is 5s 

 5// 01^ 



(p q\ dn dv 



8\ 

 Ss\ 



Ss 8s 



Sm 



(h —^ dudvl [136h] 



Sv d\ 



where p is the density of the fluid and the surface integral extends over the entire finite 

 boundary. For, the element of area on the surface dS can be taken in the form of an elementary 

 rectangle with sides drawn in coordinate directions, so that along two opposite sides fi 

 changes by d^ = S^i, and along the other two p changes by du = 8v; see Figure 225. Thus dS 

 can be replaced by the area of this rectangle or 



8s 8s = {8s /Sfi) (Ss /8iy) dfidv. 



The normal component of the velocity \s q ~-q\ where q. is given by Equation [136e]. 

 The sign in this latter equation is necessarily the same over the coordinate surface; and the 

 absolute value of the integral is taken because T is necessarily positive. 



For axisymmetric flow, the angle oj around the axis of symmetry is usually employed 

 as one orthogonal coordinate: the other two, say A and /i, then function as two-dimensional 

 coordinates on any plane drawn through the axis. Any orthogonal coordinates may be used for 

 A and ^. The following relations between the A and ji components of the velocity and the axi- 

 symmetric stream function i// may be noted: 



_ 1 8n dip 



1 Sk difj 

 oT Ss^ d\ 



[136i,j] 



where oT denotes distance from the axis, which may be expressed in terms of A and (i. The 

 proper sign to use in these equations is easily chosen in a given case, or the following rule 

 may be used: at a given point, the upper sign is to be taken in both equations when the coor- 

 dinate direction for A is carried into that for ^ by a rotation of 90 deg in the direction from the 

 assumed positive end of the axis of symmetry toward the point as in Figure 227: otherv/ise. 



344 



