108 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



of air vibrations in tubes with open ends" (1859) we find researches 

 analagous to those in hydrodynamics already referred to. The question 

 is raised in what way plane sound waves, excited within cylindrical 

 tubes and corresponding to simple tones, are modified upon passing 

 out into the free air. 



This inquiry prepared the way to determine the form of vibration 

 which finally results when the cause exciting the vibration operates 

 regularly and continuously. The most important of the general laws 

 of the j)otential function were found to be applicable to sound waves; 

 for it was shown that when sound waves are excited at a point within 

 a space filled with air their velocity-potential at any other position is 

 the same as this quantity would be at the first point were waves of the 

 same intensity excited at the second, and from this it follows that the 

 phase difference is the same in both cases. Assuming certain restric- 

 tions in the dimensions of the opening, Helmholtz obtained the relation 

 between the plane waves within the tube and the semispherical diverg- 

 ing waves in the free space at a distance, and thus was able to answer 

 the inquiry with regard to the influence of the open end on the plane 

 waves. Further investigation gave the positions of the maximum 

 and minimum amplitude of vibration and the pitch of the tone of 

 strongest resonance. He treated the difficult question for which of a 

 series of forms of tubes the motion of the air in the orifice is charac- 

 terized by the greatest wave length. In a later memoir it was shown 

 that the results of calculation agree better with experiment when the 

 interior friction of the air is taken into consideration. 



All these results of his acoustical investigations, among which I have 

 presented only those of the greatest mathematical interest, are contained 

 in connected order in his famous work, "The study of tone perceptions 

 as the physiological basis for the theory of music." 



Among the many results important for the science of music, it may 

 be mentioned that he distinguished in exact mathematical way between 

 melody as the basis of music and harmony which serves only to 

 increase the effect of melody, and that he found a mathematical foun- 

 dation for the observation that for an harmonious union of several 

 tones the rates of vibration must stand in a simple ratio, in the fact 

 that the partial tones accompanying the fundamentals are disagreeable 

 to the ear when their relative vibration numbers are not in a small, 

 simple multiple of the ratio of the fundamentals. 



Before proceeding to a short account of the much later aerodynamic 

 investigations of Helmholtz, so far as they are of interest to mathe- 

 maticians, some consideration is due a memoir on the border between 

 hydrodynamics and aerodynamics which appeared in 1873 with the 

 title, "On a theorem concerning geometrically similar motions of fluid 

 bodies, together with an application to the problem of governing air 

 balloons." Hydrodynamic equations were here employed to enable the 

 production of results of observation obtained by the use of apparatus 



