330 Vm. NEWTONIAN POTENTIAL FUNCTION. 



As AS moves continuously M describes a line, and this line in 

 its motion describes a surface, for every point of which V = c. Such 

 a surface is called a level surface of the function V. A level surface 

 divides space into two parts, for one of which V is greater, and for 

 the other less, than in the surface. 



As examples of point -functions we may take (1) the length of 

 a line drawn from the point M parallel to a given line until it cuts 

 a given plane. Its level surfaces are planes parallel to the given 

 plane. (2) The distance of M from a fixed point 0. The level sur- 

 faces are spheres with centers at 0. (3) The angle that the radius 

 vector OM makes with a fixed line OX. The level surfaces are right 

 circular cones with OX as axis. (4) The dihedral angle made by 

 the plane MOX with a fixed plane through OX. The level surfaces 

 are planes through OX. 



109. Coordinates. If a point is restricted to lie on a given 

 surface S, the intersections of that surface with the level surfaces of 

 a function V are the level lines of the function on the surface $; 

 e. g. in examples (3) and (4) above, if S is a sphere with as center, 

 the level lines are parallels and meridians respectively. 



A function f(V ly F 2 , . . .) of several point -functions is itself a 

 point -function. If it is a function of one V only, its level surfaces 

 are the same as those of F, for when V is constant, f(V) is also 

 constant. 



Let # 1? # 2 > q 3 be three uniform point -functions. Each has a level 

 surface passing through the point M. If these three level surfaces 

 do not coincide or intersect in a common curve, they determine the 

 point M, and we may regard the point -functions q lf q 2 , q^ as the 

 coordinates of the point M. The level surfaces of q ly q 2 , q B are the 

 coordinate surfaces, and the intersections of pairs (q 1 q 2 ) ) G&fe), (Q'sQ'iX 

 are the coordinate lines. The tangents to the coordinate lines at M 

 are called the coordinate axes at M. If at every point M the co- 

 ordinate axes are mutually perpendicular, the system is said to be 

 an orthogonal system. 



110. Differential Parameter. The consideration of point- 

 functions leads to the introduction of a particular sort of derivative. 

 If F is a uniform point -function, continuous at a point M, and 

 possessing there the value F, and at a point M' the value F', in virtue 

 of continuity, when the distance MM' is infinitesimal, F' -V=JV 

 is also. The ratio y , _ y ^ y 



is finite, and as MM' As approaches 0, the direction of MM f 

 being given, the limit 



