Furthermore, let I, m, n be the direction cosines of the coordinate direction for A, which is 

 the direction of the displacement 8sy Then 



Sx^ - Ihsy, hys = m Ss^, 8z^ - n&Sy 

 Hence 



8\ dx 8X dy 8X dz ^ ^ 



/ = — , m = -i , 71 = . [136b, c,d] 



8s\ dX Ss^ dk 8s^ d\ 



If the coordinates are orthogonal, I, m, n are also the direction cosines of the normal to the 

 surface A = constant. The ratio Ssv/SA is easily calculated from the formulas connecting the 

 coordinates with a;, ?/, z. 



For cartesian coordinates this ratio is unity. For spherical polar coordinates ;■, 6, w 

 as defined in Section 7, 8s^/8r = 1, 8sg/86 = r, bs /Sw = r sin 6; for, increasing « by 8cj, for 

 example, displaces the point (r, 6, cj) through a distance r sin d8<x) along a circle of radius r 

 sin 6 whose axis is the polar axis. For cylindrical coordinates x, To, co as defined in Section 7, 

 8s /8x - 1, 8s— '/8a - 1, 8s /Scj = oT, since the variation Sco produces a displacement 7^001 

 along a circle of radius tTT. 



In a flow having a velocity potential <;6, the component of the velocity in the coordinate 

 direction of any coordinate A can be written, from Equation [6f], 



1 Ox = = , [136eJ 



'^ ds 8s ^ d\ 



since in this direction ds = Ss^, c?0 = SA d<f)/dk\ 



see Figure 225. If the three coordinates A, ^, v are orthogonal, the magnitude of the velocity 



q is given by 



2222 



? = ?A +?/.+?!, • 



A general form of the Laplace equation may be obtained by expressing the continuity 

 equation for an incompressible fluid in terms of the orthogonal curvilinear coordinates A, fx, v. 

 Consider the element of volume bounded by the six surfaces that are defined by the^following 

 equations: 



A = Aj /i = /^i V = v^ 



A = A. + 5A /i = jUj + §/i V = Vy + 8v 



where A,, /i., v. refer to any given point in space. If SA, 5/i, 8v are small, the element is 

 sensibly rectangular in shape, as illustrated in Figure 226. Since the fluid is assumed to be 

 incompressible, as much fluid must enter this element as leaves it. 



342 



