ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 123 



('/) The result will be again an easy differential equation of second order. 

 Integrating it we have a better approximation for the curve. Placing x ^ o in the 

 result we again have an equation of the end amplitude of the form y" = — b'y. 

 Hence the desired frequency n =^ b/zn. 



8. Slocwn's Formula. — Those interested in the subject of statics of beams of 

 variable cross-section will find much valuable information in Professor Slocum's 

 paper "A General Formula for the Shearing Deflection of Beams of Arbitrary 

 Cross-section, Either Variable or Constant."* Many practical examples are given 

 as illustrations and Professor Fraenkel's formula for flexural deflection is derived 

 from the principle of least work by applying Castigliano's theorem. 



III. A FEW GENERAL PRINCIPLES. 



I. Calculus of Variations. — The reader will greatly profit by studying Pro- 

 fessor Byerly's little pamphlet, "Introduction to the Calculus of Variations,"! un- 

 less, of course, already familiar with the subject. For our immediate purpose only 

 a few words will be said regarding this vast and most interesting branch of mathe- 

 matical analysis. 



In ordinary calculus we are often confronted with the problem of finding max- 

 imum or minimum values of a certain quantity, function of another quantity ; for in- 

 stance the greatest ordinate of a given curve (and, of course, the corresponding ab- 

 scissa) ; the greatest volume of a body, whose surface is given, etc. This is done 

 simply by equating to zero the first derivative, of the function, as to the independent 

 variable (generally abscissa), and of finding from such an equation the corresponding 

 value of the independent variable (say x„). Then the value of the function, with that 

 x„ in it, will be either maximum or minimum; in order to determine which it will 

 be, we must find the second derivative. The negative value of the latter will mean 

 maximum, and inversely. 



But there are problems in which we are to find the form of a function, such that 

 it will answer certain requirements, consisting mostly in some such condition as ren- 

 dering maximum or minimum the value of a certain definite integral, involving the 

 independent variable, the function itself, and often some of its derivatives, as well as 

 constants. In other words, the problem of plain calculus is as follows : — Being given 

 a certain function, y =^f (x), of the independent variable x, find such a value of x 

 that 3; will have the greatest (or the least) of all adjoining values. The answer is 

 a certain x„ and the corresponding y„„^, or ymi„. 



The problem of variations is quite different : — What is that form of a function, 

 / (connecting the dependent and the independent variables in some such manner as 

 y^=f {x), which will render a certain definite integral — 



S = fvix,/,-f,...)dx, or simply = f V{x, y, y', y", . . .) dx, 



*JoHrnal of the Franklin Institute, April, 1911. 

 tHarvard University Press, 1917. 



