30 MEASUREMENT OF OSCILLATIONS. 



This expression shows that, to the degree of approximation, the 

 disturbed motion, for a very small force proportional to the square 

 of the deflection, may be regarded as the superposition of two 

 motions : one, represented by the first term, is identical with that 

 v;hich we have investigated, and takes place about a certain position 

 of equilibrium; the other, represented by the second term, is a 

 periodical displacement of this very position of equilibrium itself 

 considered as a moving body by the fact of the perturbation. This 

 latter attains its first elongation at the time / = / , and the following 



at the time / = / + . The period of this motion is therefore 



less than half that of the first, so that at each half oscillation the 

 effect of the disturbance again becomes the same. The extreme 

 values of the bracket are i and 2. It follows therefore that the 

 minimum and the maximum are to each other as 1:2. They 

 correspond very nearly to the periods of the oscillation and to 

 those of the medium of the oscillation.* 



685. APERIODIC MOTION. If the roots of equation (15) are 

 real that is to say, if we have e 2 - 2 >0, the motion represented 

 by equation (14) is no longer periodic. The system, moved out 

 of its position of rest and left to itself, returns to it gradually, only 

 attaining it after an infinitely long time. 



M. Dubois-Raymond has investigated this motion, which he 

 calls aperiodic.\ We may first of all study the conditions in which 

 the system reverts practically to its position of equilibrium, without 

 oscillation, and in a relatively short time. 



Putting now y 2 = 2 -# 2 ; equation (14) becomes 



(23) x 

 which gives 



(24) g= - 



To determine the constants, we will assume that at the time /=0 

 we leave the system to itself at a distance a from its position of 



* CORNU and BAILLE. Comptes rendus de V Academic des Sciences. 

 Vol. LXXXVI., p. iooi. 1878. 



t DUBOIS-RAYMOND. Monatsberichte der Konig. Preuss. Akad. der Wissen., 

 1869, 2870, 1873. 



