ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 133 



of finding the conditions under which the integral S„ becomes a maximum or a 

 minimum. Hence all partial derivatives of S„, with respect to a^, a-2, etc., must be 

 put = o. 



dS„ ^. dS„ ^, dS„ 



aai a a 2 da„ 



From these equations the coefficients themselves, ui, 02, . . . a„, will then be found, 

 by method of determinants or in any desired manner. The functions i/'i, t^2, etc., can 

 be chosen in a great variety of manners. In the present case we shall take them in 

 form of polynomials: — 



or, limiting ourselves to n = 3, this will be — 

 or, in a different form, 



Indeed, it is easy enough to see that on the right end of the beam, where ^ = i, 

 this expression satisfies the condition of both y and dy/ d ^ being = o. 



On the left end the condition EJ--^ = o (the bending moment equals zero), 



(mOC 



is automatically fulfilled by the fact that the moment of inertia, /, itself vanishes at 

 1=0, owing to the triangular nature of the beam profile. 



Substituting this value of ys into the integral S, and differentiating with respect 

 to Oi, 02 and as, we have the following system of linear equations : — 



\ 30/ \ 5 105/ \ 5 280/ 12 



5 105/ \ 5 280/ \ 7 630/ 30 



^ + -f ) + .. (A + J^) + ,3 (^ + _^) =^ 

 5 280/ \ 7 630/ \35 1260/ 60 



Assuming y = 10 we have the following expressions of the constants ai, a^, 

 03 in terms of /I : — 



Ui = .05869 X; 02 ^ .0013 A; ^2 = — .002164 Jl ; 

 and if y is taken = 100, then 



Oi = .Oi25i;i; 02 = . 0212 ;i; 03 = . 00079^1. 

 Had we taken n = 2, that is, had we limited ourselves to two terms in the expression 

 of y, ^2 = ai + 02^, the results would have been as follows : — 



li y ^ 10, oi = .0589 A ; az ^ .00946 Jl. 



If y=ioo, Oi = .oi24X; 02 = .02i8;i. 



Had this elastic line been established according to exact methods, the discrep- 

 ancies would have been proven not to exceed i to 2 per cent, for both the deflection 



