VARIABLES AND FUNCTIONS. 



[INT. ii. 



Polar Coordinates. If we call the point-functions of 

 Examples 2, 3, and 4, of 14, r, 6, <, 

 we obtain the system of spherical, or polar 

 coordinates. 6 and <f> may be called the 

 co-latitude and longitude. The level sur- 

 faces of r being spheres, the normal coin- 

 cides with r. Accordingly 



dr _ dr , _ 



FIG. 5. 



The level surface of 6 is a circular cone 

 of angular opening 0, (Fig. 5), and 



, Q W d6 I , I 

 an = rau, ^- = 7^ = - , /i 9 = . 

 on rdu r r 



The level surfaces of <f> are meridian planes 

 through the axis of the above cones, (Fig. 6), and 



d<b dd) 1 



FIG. 



For 



= r sin 



r* sin 



r sin ' 



"-m / 



rsm# 

 any function /(r, 0, <), the partial parameters are 



dr 



P fl = 



The total parameter, the resultant of these, is given by 

 p 2 = /c)A 2 1 /9A 2 __1 idf y 



18. Cylindrical, or Semi-polar Coordinates. If we take 

 the rectangular coordinate z, the perpendicular distance from the 

 ^-axis, p, and o> the longitude, or angle made by the plane includ- 

 ing the point M and the ^-axis and a fixed plane through that 

 axis, we have the system of semi-polar, cylindrical, or columnar 

 coordinates, for which we have immediately, 



