ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 125 



Then the differential of the function 4>(.i', f) with the specific value, t = to, sub- 

 stituted in it, is called the variation of the function y and is denoted Sy. These 

 variations are functions of x, of a perfectly arbitrary nature (since we did not in 

 any way limit ourselves to the manner in which these curves deform into each other). 

 Herein lies the difference between variations and differentials: — the latter (such as 

 dy) are also functions of x. but are not arbitrary, since they are found in a definite 

 way, according to the known rules of calculus. Now, if we want a maximum or a 

 minimum of a definite integral, say S, of a certain expression, involving y or its de- 

 rivatives, etc., let us again imagine that we deal with a broader conception of y, in- 

 volving not only x but also the parameter t (the latter, by the way, will be supposed 

 to be such that it merely drops out when y is what we want it to be, in view of our 

 conditions). Then the first derivative of the integral as to the parameter', t, in other 

 words, the variation ^S, must be = o for maximum or minimum of the value of 

 the integral, precisely as dy/dx = o is the maximum or minimum condition in ordi- 

 nary calculus. 



In other words, when y is what we want it to be, then the integral 5" does not 

 change for any small increments of the parameter t, whichever way these might take 

 place. Take, for instance, the problem of the shortest distance between two points 

 in a plane or even in space. It makes absolutely no difference what family of curves 

 we assume to deform into each other, as soon as the required minimum is reached 

 (straight line), the curve becomes inert, so to speak, to the changes of the parameter, 

 t. Similarly, in the simple problem of calculus, where the greatest (or least) ordi- 

 nate is found, ymax, we know that in the immediate vicinity of the maximum or 

 minimum value the function refuses to respond to the infinitely small increments of 

 a-, and we simply say dy/dx = o. It is not our purpose to give the working methods 

 of the Calculus of Variations, but we will mention the fact that equating to zero of 

 a certain variation of a definite integral (which means maximum or minimum value 

 of said integral) leads to certain differential equations, involving various orders or 

 derivatives of 3; as to x, as well (generally) as 3; and x themselves, and constants. 

 Integrating these differential equations, we obtain the required connection between 

 the variables, dependent and independent, a functional relation required, such as 

 y = f (x), clearly establishing the fact that the required curve is a circle or a cate- 

 nary, etc. 



It is well to remember that, in general, the object of differential equations is pre- 

 cisely the research of functional interdependence of variables, from expressions in- 

 volving their derivatives. From this standpoint the Calculus of Variations supplies 

 means for forming such equations, with a certain object in view — compliance with 

 certain requirements. 



2. Energy. — In dynamics we are concerned with two forms of energy : — Poten- 

 tial Energy, otherwise known as Latent Energy, or Energy of Position, usually de- 

 noted V, and Kinetic Energy, T, otherwise known as Active Energy or Energy of 

 Motion. The sum of these, as is well known, is constant, so that both forms can 

 only transform into each other (dissipation of energy through friction, etc., will not 



