2^8 



NATURE 



IJan. 24, 1878 



succeeding values of a function, with finite sums in- 

 stead of with integrals or with infinite series. Of 

 course for Fourier's series as well as for the developments 

 of Laplace by means of spherical harmonic functions the 

 proof for the correctness of their values can also be fur- 

 nished in the case of continuous functions. For a large 

 number of other functions which are given by differential 

 equations of the second degree this proof results, under 

 certain suppositions regarding the continuity of the func- 

 tions and the limit conditions, from the theorems of Sturm 

 and Liouville, which Lord Rayleigh explains when speak- 

 ing of the vibrations of strings of unequal thickness. Yet 

 in mathematical physics we are still compelled to employ 

 a great number of series-developments of functions which 

 do not belong to this class ; and even the vibrations of 

 rods and plates are cases in point. In this respect the 

 treatment of the problems mentioned with a finite but 

 arbitrarily large number of degrees of freedom of motion 

 is interesting also with regard to analysis. 



For vibrating systems of one degree of freedom, the 

 oscillations of which are subjected to damping, the 

 doctrine of the laws of resonance is developed in the 

 third chapter. The author calls the vibrations which are 

 continuously maintained by the influence of a periodical 

 force acting externally, forced vibrations. In all cases 

 their intensity is greatest when their period of vibration, 

 which equals the period in which the force changes, is 

 also equal to the period of the system vibrating freely 

 and without friction. For the relations between the in- 

 tensity and the phase of the co-vibration, between the 

 breadth of the co-vibration in case of small alterations 

 in the pitch and the degree of damping, which I had 

 myself proved for certain instances and used for certain 

 observations, the general proof is given here. The 

 author has further employed these chapters to set up 

 certain general maxims respecting the direction and 

 magnitude of the corrections which must be made in 

 cases where one cannot completely solve an acoustic 

 problem, but can only find the solution for a somewhat | 

 altered vibrating system. These are like the outlines of a 

 " theor>' of perturbations " applied to acoustic problems. 

 The author illustrates these maxims by many various 

 examples. Thus, for instance, he replaces a string by 

 an imponderable stretched thread which carries weights 

 either in the middle only or at certain distances from 

 each other'; or a tuning-fork by two imponderable springs 

 with weights at the ends. 



For vibrations of very small amplitude, the forces 

 which tend to lead the moving points back to their posi- 

 tion of equilibrium may always be considered propor- 

 tional to the magnitude of their distance from the position 

 of equilibrium. As long as this law holds good, the 

 motions belonging to different tones are superposed, with- 

 out disturbing one another. But when the vibrations 

 become more extensive, so that the law of proportion- 

 ality just named no longer applies, then perturbations 

 occur which become manifest by the appearance of new 

 tones, the combination tones. In my book on acoustic 

 sensations (" Die Lehre von den Tonempfindungen ") I 

 have myself explained this manner of origin of the com- 

 bination tones, only for the motion of but a single material 

 point. In Lord Rayleigh's book this explanation is given 

 with reference to any compound vibrating system of one 



degree of freedom, and it is further amplified with regard 

 to the manner in which the forces deviate with the dis. 

 placements from the law of proportionality. 



Certain laws of reciprocity, of which I had given single 

 instances in my investigations on the vibration of the air in 

 organ pipes, may be proved in a general way for all kinds of 

 vibrating elastic systems. If on the one hand at point A 

 an impulse is given, and the motion at point B is deter- 

 mined after the time / has elapsed, and if on the other 

 hand an impulse is given at point B in the direction of the 

 motion, which occurred there, and, after the time /, the 

 motion-component falling into the direction of the first 

 impulse is examined at point A, then the two motions in 

 question are equal if the impulses were equal. 



Chapters VI. to X. of Lord Rayleigh's book treat of the 

 vibrations of strings, rods, membranes, and plates. The 

 vibrations of strings have played an important part in 

 acoustics ; their laws are simple, and the physical condi 

 tions which the theory demands are fulfilled with com- 

 parative facility, different modes of producing the tones 

 may be employed, and a number of various motions may 

 thus be produced. It is just because the physical pheno- 

 mena in connection with strings were well known, that 

 the observation of the way in which the ear is affected by 

 their various modes of vibration has materially facilitated 

 the solution of the problems of physiological acoustics. 

 The musical importance of strings rests on the circum- 

 stance that the series of their proper tones corresponds 

 to that of the harmonics, the vibration-numbers of which 

 are entire multiples of those of the fundamental tone. 

 For this reason, if the motions of many proper tones are 

 superposed on one string, a periodical motion again 

 results, and this is the cause why on strings we can pro- 

 duce notes of the most varied quality. We need only 

 remember how differently the same string sounds ac 

 cording to whether it is plucked with the finger or with a 

 metallic point, whether a violin bow is drawn across it or 

 whether it is caused to vibrate by means of a tuning-fork 



In this chapter less new work remained to the author ; 

 however, this example shows how much easier it is to 

 understand all these separate problems if they are not 

 treated separately but developed in coherent representa- 

 tion, after the most general principles, the validity of 

 which is independent of the special peculiarity of the 

 case, have been first explained. 



The short chapter VII. gives the laws for the longi- 

 tudinal and torsional vibrations of rods ; the laws are 

 simple and resemble those of the open and stopped organ 

 pipes. The lateral vibrations of rods, during which these 

 bend, give more complicated analytical expressions ; their 

 proper tones do not form a harmonic series, but are given 

 by the roots of a transcendental equation. The tones are 

 different according to whether one or both ends of the rod 

 are free to rotate and to move, or free to rotate, but hindered 

 from moving (supported), or hindered from rotating and 

 moving (damped). With this more complicated problem 

 the advantage of first treating of the general principles 

 becomes clearly apparent. The forms of the simplest 

 vibrations are calculated and represented graphically. 

 The mode of vibration of a stretched rod, for which 

 Seebeck and Donkin have already given the solution, is 

 also treated here in order to determine the influence of 

 rigidity upon the vibrations of strings. 



