124 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



(where V is the functional symbol, showing how the variables are connected to- 

 gether), maximum or minimum, as might be the case? Will it be a straight line, 

 y = mx -(- n, or a parabola y^ = 2px + c, or a cycloid 



y = a cos \\ 2ax -x t 



a ^ 



At any rate the object is always the same, to find the form of a function such that the 

 value of a certain integral will be maximum or minimum. 



Here are a few examples : — 



I. Find the shortest curve between two points in a plane (which, of course, will 

 be a straight line). Here the integral to be made minimum is simply — 





I -I- ( -r ) dx 



dx, 



The integrand involves only the first derivative dy/dx. 



2. Find the form of a body of revolution such that, immersed in water, it will 

 offer the least possible head resistance to motion (Newton's problem). Here the 

 problem hinges on the minimum value of the following integral — 



X'i 3 



s=j-yyL-jx 



(where y' is the simplified notation for dy/dx). So that the integrand here in- 

 volves only the function itself and its first derivative. 



3. Find the shortest path between two points on a given curved surface. It 

 can be shown that here the integral to be made minimum is 



Xt . 

 x\ ^ 



2 2 



\-\-y +z' dx 



so that here the integrand contains the derivatives of two unknown functions of x. 



The method of the Calculus of Variations is based upon the following viewpoint, 

 rather broad but not unnatural. Instead of limiting ourselves to just one definite 

 curve, let us imagine that any curve (or function), including the one we wish to find, 

 is only a special case of a whole infinite family of adjoining curves, all controlled 

 by a certain parameter, the continuous changing of which gradually alters one curve 

 into another. Such a parameter, the existence of which we force, as it were, into 

 the conception of our function, must possess only one property : it must obligingly 

 disappear, whenever, in the continuous changing of curves from one to another, we 

 pass through the form, such asy = f{x), in which we are interested. But it may 

 be in evidence on all other occasions ; and, indeed, it can be shown that an infinite 

 variety of curves, such as y^^{x,t), can be written down, such that they gradu- 

 ally change one into another, owing to the continuous change of the parameter, t ; 

 provided, however, that for some special value of t, say = to, the function ^ (x,to) 

 becomes simply =:/ (x), the function we have under direct consideration. 



