334 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



or the parameter of a homogeneous function is inversely proportional 

 to the perpendicular from the origin to the tangent plane to the 

 level surface. For example, if n = 1, 



7 = ax -r by -f cz, 



The level surfaces are parallel planes, and the parameter is 

 constant, 



V is proportional to the distance of the level surface from the origin. 

 If n = 2, 



a s 







P=2 y + + ii 



For the surface, F= 1, 



a familiar result of analytic geometry. 



111. Polar Coordinates. If we call the point functions of 

 Examples 2, 3, and 4, of 108, r, #, <p, we obtain the system of 

 spherical, or polar coordinates. # and y may be 

 called the co- latitude and longitude. The level 

 surfaces of r being spheres, the normal coincides 

 with r. Accordingly 



= '-= 1 h 



dn dr ' 



1. 



Pig. 116. 



The level surface of # is a circular cone of 

 angular opening #, (Fig. 116), and 



89 d l 1 



The level surfaces of cp are meridian planes through the axis of 

 the above cones, (Fig. 117), and 



