20 VARIABLES AND FUNCTIONS. [iNT. II. 



It is evident that 



8V 



if/ and its first derivatives are continuous functions of x and y. 



The definitions here given may be extended to functions of any 

 number of variables. 



13. Point-Function. If a quantity has for every position 

 of a point in a region of space T 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 corre- 

 sponding to any positive e, however small, a value 5 such that 

 when B is any point inside a sphere of radius < 8, 



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



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 co- 

 ordinate axes are continuous point-functions. 



14. 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 r 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 B, such that at A, V is less, and at B, greater 

 than c, and that in moving along a line AB through .&, V con- 

 tinually increases. If the line AMB is displaced to the position 

 A'M'B', so that 



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

 and | V(B)-V(B') \<V(B)-c, 



then V(A')< c< V(B'\ therefore there is a point M on the line 

 A'B' for which F=c. 



