132 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



which nothing has been said so far ; they will temporarily play the part of independent 

 variables in completing our problem, and, according to the ordinary rules of finding 

 maximum and minimum of functions with many variables, we must find derivatives 

 of the given expression with respect to each of these independent variables, and put 

 it = o; so that we shall have (according to the choice of n) — 



dS ^,dS ^. dS ^. 



from these equations we can find the actual values of the coefficients ai, a^, . . . a„, 

 which solves our problem, because we thus have everything we need to make the ten- 

 tative solution ( I ) the required actual solution of the problem. As a matter of fact, 

 working out ithe variation of the original integral would not have led us to a solution, 

 but only to a differential equation, which we would then have to solve ; whereas, by 

 applying Ritz's method, we both avoid the method of variations and at the same 

 time find the solution of the problem itself. 



"A mere engineer's method!" exclaimed M. Poincare, with a touch of some- 

 thing that might have been interpreted as the usual contempt of a man of pure 

 science for a practical man ; "except," added he, "that Ritz has supplied an actual 

 rigid proof of the fact that, when n is large, the result can approach the exact solu- 

 tion as close as we please." This proof, by the way, is the main difficulty of the sub- 

 ject, as well as the most brilliant part of Ritz's investigations. If the reader is suffi- 

 ciently advanced in variations and in the theory of functions, he will greatly enjoy 

 Ritz's original memoirs. We shall liinit ourselves to practical problems. 



2. Completion of the Problem Begun under "Virtual Work." — In our attempt 

 to find the solution of this problem, that is, an expression giving the deflection, y, in 

 terms of the abscissa, x (or ^, since the variable was changed, for convenience, ac- 

 cording to the condition .r = ^/), we arrived at the following definite integral: — 



of which we were to find the maximum or minimum value — a typical problem of cal- 

 culus of variations. However, in accordance with Ritz's principle, let us try a tenta- 

 tive substitution: — 



where \J/i, i/'2, etc., will be certain functions of ^, the choice of which is more or less 

 arbitrary, as long as they answer the same end conditions as y itself would; the 

 proper form will be suggested presently. Now the coefficients «i, ^2, etc., constant as 

 they are, will be chosen so that the value of the integral S„, with y„ substituted in it 

 (for y or its derivatives), will be a maximum or a minimum. Since the integral S„, 

 with the new value, y„, substituted in it, is no longer a function of x, but only of the 

 limits, the only variables we have, by which to adjust the value of .S",,, are the coeffi- 

 cients ai, ^2, etc., which, once properly selected, become constants, once and for all, 

 in the solution-to-be-derived, y„. But they are independent variables, in the process 



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