12 16] VARIABLES AND FUNCTIONS. 21 



As AB 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 surfaces are spheres with centers at 0. (3) The angle that 

 the radius vector OM makes with a fixed line OX. The level sur- 

 faces 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. 



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

 surface S, the intersection 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 x , 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 F is constant, /(F) is also 

 constant. 



Let q lt q 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 

 #i> <?2> & a s the coordinates of the point M. The level surfaces of 

 qi, <?2> #s are the coordinate surfaces, and the intersections of pairs 

 ((MaX (MS)* (<Mi)> are tne coordinate lines. The tangents to the 

 coordinate lines at M are called the coordinate axes at M. If at 

 every point M the coordinate axes are mutually perpendicular, the 

 system is said to be an orthogonal system. 



16. 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 



