Modes of a Vibrating System. 437 



In order to determine the normal functions, we assume that 

 y —■ u cos pt, where u is a function of x. We thus obtain 



dSi 



dx* 



B ^r4-^ 2 "=0; (7) 



from which u is found as the sum of four distinct functions of a? 

 (which it is not necessary to write down), each multiplied by a 

 coefficient, which is up to this point arbitrary. The four equa- 

 tions expressing the terminal conditions determine the ratios of 

 the four coefficients, and cannot be harmonized without ascribing 

 special value to^. The condition expressing the compatibility 

 of the four equations limits p to certain definite values which 

 appear as the roots of a transcendental equation. Correspond- 

 ing to each admissible value of p there is thus a normal function 

 in which every thing is definite except a constant multiplier. 



Since the most general motion may be compounded of the 

 normal component vibrations, it follows that the most general 

 value of y may be expressed in the series 



y = 1 M 1 + <£ a tt 8 + ...; (8) 



where u l} w 2 , &c. are the normal functions, and (f) l} cp^, &c. arbi- 

 trary coefficients. The determination of cf> l &c. is effected by 

 means of the characteristic property of the normal functions, 

 that the product of any two of them vanishes when integrated 

 over the length of the rod. From this it follows that 



I yu r dx=(t> r \ 

 Jo Jo 



u\dx (9) 



The fact that the normal functions are conjugate (which is the 

 name given to functions possessing the property above stated) 

 may be proved from the form of the functions, or better from 

 the differential equation and terminal conditions which define 

 them. The last course is that which has been followed by the 

 distinguished mathematicians who have treated of the present 

 and similar questions. 



If u and v be two normal functions corresponding to p&ndp 1 , 

 we have from (7), 



(ys-p^'m^j^g -v d ^)d* 



d 3 v d s u dv d 2 u du d 2 v ,, , 



~ U dx~ 3 ~ V d^ + TxloP ~ TxJtf* ' ' ' ( j 



as may be found by integrating by parts. The integrated terms 

 arc to be taken between the limits, and vanish in each case, 



