338 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



114. Infinitesimal Arc, Area and Volume. If we have any 

 three point -functions g 1? g 2 , g 3 forming an orthogonal system of co- 

 ordinates j since their parameters are 



7 __ V l h ~L __ V */3 



cn^ 2 dn^ 3 dn s ' 



the normal distance between two 

 consecutive level surfaces q 1 and 



q l + dq is dn = -~i consequently 

 if we take six surfaces 



&*%,& 



Fig. 118.* 



the edges of the infinitesimal curvi- 

 linear rectangular parallelepiped 

 whose edges are the intersections 

 of the surfaces are 



and since the edges are mutually perpendicular, the diagonal, or 

 element of arc is 



v r ft# 



" 



dq 3 



the elements of area of the surfaces q lf q 2 , q s are respectively 

 and the element of volume is 



Examples. Rectangular coordinates x f y, z. 



Polar coordinates r f -9-, cp, 



r sin -9-' 



dS r = r^sin&d&dcp, element of area of sphere, 



12) dS#=r sin#dr dcp, element of area of cone, 



dS(p=rdrd& , element of area of plane, 



dr = 



