the same origin, the fixed plane as the xy-pla.ne and the fixed line as the a;-axis. It is evident 

 that 



X = r cos d, y - J" sin d cos w, s = r sin d sin co [6h, i, j] 



Any point P can be displaced in such a direction that only one of the coordinates r, 9, a> varies. 

 The three mutually perpendicular directions thus defined may be called coordinate directions; 

 the corresponding elements of distance ds are dr, rdd along a circle through Ox, and r sin Odw 

 along a circle of radius r sin 9 and in a plane normal to Ox. From [6f] the components of the 

 velocity in these three directions are 



1r 



dr ' 



1 ^'P 



(90 



r sin 9 dco 



1 



[6k, 1, m] 



In other cases, cylindrical coordinates x, oT, cu are useful. Here o7 denotes distance 

 from thea;-axis and ty denotes angular distance around this axis measured from a fixed plane 

 drawn through it; see Figure 8. If the Cartesian axes are drawn so that co is measured from 

 the xy plane, 



y - oTcos &), 2 = oTsin a> [6n, o] 



The elements of distance in the coordinate directions are now dx, (f&Tand codoi, and the com- 

 ponents of the particle velocity in these directions are 



1. = 



dx 



17X=-' 



1 ^ 



[6p, q, r] 



In all of these equations connecting the velocity potential with the velocity, the sign 

 is that of recent textbooks on hydrodynamics. An older usage must be noted in which cA is 

 defined as <^ = jt q ds. Then all differences between values of 6 are reversed in sign and 

 the signs in equations equivalent to [6b, c, d], [6f, g], [6k, 1, m], and [6p, q, r] are all positive. 



Figure 8 — Illustrating cylindrical coordinates 



11 



