107, 108] POINT -FUNCTION. LEVEL SURFACE. 329 



CHAPTER VIII. 



NEWTONIAN POTENTIAL FUNCTION. 



107. Point -Function. If for every position of a point in a 

 region of space t a quantity has one or more definite values assigned, 

 it is said to be a function of the point, or point -function. This 

 term was introduced by Lame. If at every point it has a single value, 

 it is a uniform function. Functions of the two or three rectangular 

 coordinates of the point are point -functions. A point -function is 

 continuous at a point A if we can find corresponding to any posi- 

 tive , however small, a value d such that when _B is any point 

 inside a sphere of radius < d, 



\f(B)-f(A)\<s. 



We may have vector as well as scalar point -functions, the length 

 and direction of the vector being given for every point. A vector 

 point -function is continuous if its components along the coordinate 

 axes are continuous point -functions. 



108. Level Surface of Scalar Point -Function. If V is a 



uniform function of the point M, continuous and without maximum 

 or minimum in a portion of space r, through 

 any point M in the region t we may construct 

 a surface having the property that for every 

 point on it V has the same value. 



For let the value of V at M be c. Then 

 since c is neither a maximum nor minimum, 

 we can find in the neighbourhood of M two 

 points A and J5, such that at A, V is less, 

 and at _B, greater than c, and that in moving 



along a line AS through M, V continually increases. If the line 

 AMB is displaced to the position A'M'B', so that 



\V(A)-V(A')\<c-V(A) 

 and 



then V(A) < c < 7(5'), therefore there is a point M' on the line 

 A'B' for which 7=c. 



