HERMANN VON HELMHOLTZ 



in optics, if one point can be seen from a second, the 

 second can also be seen from the first. Sound shadows 

 may thus be produced, but they are only partial, in 

 consequence of the wave-lengths of sound being great 

 in comparison with the sizes of ordinary obstacles. In 

 connection with this subject, in 1860, he published an 

 important paper, in which he examined the movements 

 of the air in open organ pipes, and extended Green's 

 well-known theorem, the chief application of which 

 belongs to Statical Electricity. 1 He gave the 

 correct theory for the open organ pipe. Lagrange, 

 Daniel Bernoulli, and Euler assumed that at the open- 

 ing the pressure could not vary with that of the 

 surrounding atmosphere ; but Helmholtz showed that 

 in ordinary cases the inertia of the air outside the pipe 

 has the effect of practically diminishing its length. 

 He also took into account the friction of the air 

 against the walls of the tube, a point omitted by his 

 predecessors. He also investigated some of the con- 

 ditions under which the hammer of the pianoforte 

 strikes the strings. He invented a vibration micro- 

 scope for the examination of vibrating points on any 

 body, say a violin string, so as to see the well-known 

 figures of Lissajous. He established the mathematical 

 relation of velocity and density in the propagation 

 of small disturbances in gas or air. 



As already mentioned, he examined mathematically 

 the laws that regulate the actions of resonators of various 



1 Rayleigh, op, cit. t vol. ii., p. 145. 



1 66 



