for r, cones for 9 and planes through the axis for co. For cylindrical coordinates as defined in 

 Section 7 the surfaces are planes perpendicular to the axis for x, cylinders for &r, and planes 

 through the axis for w. 



At any point there are three coordinate directions, in each of which one coordinate in- 

 creases while the other two remain constant; see Figure 225. These directions are tangent 

 to the curves of intersection of the three coordinate surfaces through the point. If the coor- 

 dinates are orthogonal, the three coordinate directions at a given point are mutually perpen- 

 dicular, and they are also perpendicular to the corresponding coordinate surfaces. 



Figure 225 — Diagram illustrating coordinate directions. See Section 136. 



When the coordinate A of a point is given an elementary increase 8k while the other two 

 coordinates remain fixed, the variables x, y, and z receive certain elementary increments which 

 can be written 



dx dy dz 

 \ = 5A, §V), = SA, 8z\ - SA. 



(9A ' "'^ ^A 

 The total displacement of the point is then 



d\ 



8s^ = (Sx^ + 8yl + 8zl) 



and 



8s, 



8k 



dx_Y 



dk ) \ dk 



dy 



dz y 

 dk I 



1/2 



[136a] 



341 



