Instruments as Resonators. 121 



are gradually further and further apart, dividing their respec- 

 tive ventral segments more and more unequally, until at the 

 apex of the cone is a node common to all the notes. It fol- 

 lows from this that the centre of a ventral segment in a cone 

 is not the centre of the length between its nodes, and, con- 

 versely, that as the diameters of the two ends of the ventral 

 segment approach equality, so does the position of the node 

 become more central, until the condition of vibration existing 

 in an open cylindrical tube is reached ; and such a tube may 

 evidently be considered as a portion of a cone whose apex is at 

 an infinite distance. It is to be noticed that in the cone the 

 number of J wave-lengths, or semi- ventral segments, is not 

 directly proportional to the vibrational number as in the open 

 tube, but, with the exception of the fundamental note, is always 

 in excess. Thus let 



N = number of \ wave-lengths, 



n = relative vibrational number ; 

 then 



N = n + (n-l) = 2n-l. 



Instances. — Note 1 (fundamental) N = 1 + ( 1 — 1) = 1, 



Note 4 (double octave) N = 4 + (4 — 1) = 7. 



The velocity of the portion of wave or waves in the cone there- 

 fore differs with the pitch of the note, and is in no case the 

 same as the velocity in free space. Assuming this latter to 

 be 1120 feet per second, we should have in the cone the fol- 

 lowing velocities : — 



Note. n. N. Feet P er 



second. 



c 128 1 1 2240 



c' 256 2 3 1493-4 



g' 384 3 5 1344 



c"512 4 7 1280 



and the space traversed by the waves of the different notes in 

 one second, measuring from the apex of the cone to, say, the 

 ear of an observer: — 



c 128 1122-1875 feet. 



d 256 1121-0937 „ 



g' 384 1120-7292 „ 



c" 512 1120-5468 „ 



The method I used to find the positions of the nodal points 

 in the cone, and which is applicable to wind instruments or 

 tubes of any varying section, may be illustrated by a conic 

 frustum open at both ends. Holding a vibrating fork over 

 one end (in this case c 512), gradually sink the tube in water : 



