C. GENERAL PHYSICS. 157 



in number) having the same axis as the given curve, and the 

 lengths of whose recurrent periods are 1. -?t, ^, i, etc., of the 

 leno'th of the oriven curve. 



The above theorem is the statement of a mathematical 

 possibility, and it does not necessarily follow that it can be 

 immediately translated into the language of dynamics with- 

 out experimental confirmation ; for, as Helmholtz remarks, 

 " That mode of decomposition of vibratory forms, such as 

 the theorem of Fourier describes and renders possible is it 

 only a mathematical fiction, admirable because it renc^.ers 

 computation facile, but not corresponding necessarily to any 

 thing in reality? Why consider the pendulous vibration as 

 the inevitable element of all vibratory motion? We can im- 

 agine a whole divided in a multitude of diflTerent ways : in 

 a calculation we may find it convenient to replace the num- 

 ber 12 by 8 + 4, in order to bring 8 into view; but it does 

 not follow that 12 should be always and necessarily consid- 

 ered as the sum of 8 + 4. In other cases it may be more ad- 

 vantacreous to. consider the number as the sum of 7 + 5. 



" The mathematical possibility, established by Fourier, of 

 decomposing any sonorous motion into simple vibrations, can 

 not authorize us to conclude that this is the only admissible 

 mode of decomposition, if we can not prove that it lias a sig- 

 nification essentially real. The fact that the ear eflfects that 

 decomposition, induces one, nevertheless, to believe that this 

 analysis has a signification independent of all hypothesis in 

 the exterior w^orld. This opinion is also confirmed precisely 

 by the fact stated above, that this mode of decomposition is 

 more advantageous than any other in mathematical research- 

 es. For the methods of demonstration which agree with the 

 intimate nature of things are naturally those which lead to 

 theoretic results the most convenient and the most clear." 



But although Helmholtz thus states the importance of an 

 experimental confirmation of this theorem, yet he did not at- 

 tempt to test its truth by a course of rigorous experiments. 

 This Professor Mayer has succeeded in doing by the aid of his 

 new method of sonorous analysis. 



It is well known that if a surface advance regularly under 

 a point of a body having a pendulous vibration in a plane 

 parallel to the surface, this point will describe on the surface 

 a sinusoidal, or, as it is now more generally called, a harmonic 



