the direction cosines of its normal at any point x, y, z can be found as 



df df df 



I ^ k , m - k , n - k 



dx dy dz 



[135f,g,h] 



where 



k = 



dfV df 



dx 



dy 



\dz 



1/2 



[135i] 



in order to make I + m^ + n"^ ~ 1. The sign of k must be determined by inspection. 



For, if the point is displaced over the given surface through an elementary distance 

 ds, whose components are dx, dy, dz, in the direction of a tangent whose direction cosines 

 are I', m', n', from Equation [135e] 



df df df 



df = dx + dy + dz = 0. 



dx dy dz 



Substituting here dx = I'ds, dy = m'ds, dz = n'ds and multiplying through by k/ds, 



, df , df 



I k — + m k 



dx dy 



df 



n'k — ^ = 0. 



dz 



Now a line can certainly be drawn through {x, y, z) whose direction cosines are /, m, n as 

 defined by Equations [135f,g,h]. Then, from the last equation, I'l + m'm + n'n - 0, so that 

 the line thus drawn is perpendicular to the tangent whose direction is {V, m', n'). Since the 

 latter may be any tangent to the surface at {x, y, z), the line (/, m, n) must be the normal to 

 the surface. 



136. GENERAL FORMULAS FOR ORTHOGONAL CURVILINEAR COORDINATES 



It is convenient at this point to generalize certain ideas and formulas so that they may 

 be used with any type of orthogonal coordinates. 



A coordinate system may be regarded as set up by means of three families of coordi- 

 nate surfaces. The surfaces of any one family do not cut each other, and are numbered with 

 the values of one of the coordinates. Usually the three surfaces that intersect at a given 

 point meet there orthogonally. 



For example, for Cartesian coordinates the surfaces consist of three sets of parallel 

 planes. For the polar coordinates defined in Section 7, the surfaces are concentric spheres 



340 



