136 ON VIBRATIONS OF BEAMS OF VARIABLE CROSS-SECTION. 



The first approximation may be obtained by limiting ourselves to the first term 

 of the expression 3;", that is, by making n= i. Then 



yi = «i (I - I )^ and -^ = 2 fli. 

 Substituting these values into the integral S„ we have, on integration — 



— ■ s "1 



73 1 30 



Here there is only one derivative to be taken for establishing that value of Oi which 

 will make the integral maximum or minimum. 



From -—^ = o we find — 

 dai 



This result dififers only by about 3 per cent from the exact answer, worked out by 

 other methods, more difficult and not applicable to more complicated problems of 

 this nature, where Ritz's method is of especially great value. 



By way of second approximation let us make n^2, that is, assume 



whence — 



2^ = 2 («l-2«2 + 3«20- 



ai. 



Substituting into the integral we have, carrying out the integration — 



y 3^ L 5 J \30 105 280/ 



Equating to zero the partial derivative of this expression as to a^ and 122, that is, 



— — = = o and - — - = o , we have — 

 dtti da^ 



y ' 3I* 30/ ' V 5 y 3^' 105 



\ 5 7 3^ 105/ V 5 y 3^ 280/ 



Such a system of equations can have the values of Oi and 02 different from zero 

 only upon the condition that the determinant of the coefficient is = o, that is, if — 



/ Es- b' f\ ( 2 E^ _^_^\ _ /A^ ^_iLV=o 



\ r 3^' 30/ V 5 y ■ 3^* 280/ V 5 y • 3^' 105/ 



whence we have two values of />^ The smaller value of p, corresponding to the fun- 

 damental tone, is 



