48 DEFINITE INTEGRALS. [INT. III. 



can be equal only if the point functions 



h l h. 2 h s f(q 1) q 2) q 3 ) = g^g 9 $ (p^ p 2 , p s \ 



everywhere in the volume r for the same point M, denoted by 

 <?i> #2> #3 CT P\> P-2, PS in the respective coordinates. 



29. Calculus of Variations. We shall frequently in what 

 follows have to make use of the calculus of variations, which, 

 since we shall use it always in connection with definite integrals, 

 is introduced here. 



In the differential calculus, we have to consider questions of 

 maxima and minima of functions. A function of one variable has 

 a maximum or minimum value at a certain value of the variable if 

 the change in the function is of the same sign for any change in 

 the variable, provided the latter change is small enough. Since if 

 f(x) is continuous at x t 



If h is small enough, the expression on the right will have the 

 sign of the first term, which will change sign with h. Accordingly 

 the condition for a maximum or minimum is 



/'<*) = o. 



Suppose that we change the form of the function such a 

 change may be made to take place gradually. For instance 

 suppose we have a curve given in any way, e.g. 



a! = F l (t) t y = F a (t), z = F 3 (t), 



where the F's are any uniform and continuous functions of an 

 independent variable t. If we change the form of the F's we shall 

 change the curve suppose we change to 



x = G 1 (t), y=G,(t), z=G t (t). 



To every value of t corresponds one point on each curve, con- 

 sequently to each point on one curve corresponds a definite point 

 on the other. Such a change from one curve to the other is called 

 a transformation of the curve. The change may be made gradually, 

 e.g. 



