246 Mr. Willis on the Vowel Sounds, 



pipe, length = a, will also be produced by applying to the same, 

 a pipe, length = 2na + s, n being any whole number. 



Let now the secoiidary interval = 2 a— s, a similar process will 

 give the following series. 



No. 7. 



A B a C b D a c E b d F 



1 1 2 1 2 1 



This after the four first is exactly similar to the former in the 

 order of the intervals of the pulses, and the alternation of con- 

 densation and rarefaction, and only differs from it by the little 

 groups of pulses increasing in intensity in this case, and dimi- 

 nishing in the other, a difference scarcely worth noticing, so that 

 we may now say, that whatever effect be produced by applying 

 a pipe ( = s) to reed ( = a), the same will be produced by a pipe 

 2na ± s. 



No. 8. 



a e a o o a e a o 



Hence if a given effect a be produced by a pipe whose length 

 is ea (No. 8.) the same will be produced by a jiipe whose length 

 is ee—ea, ee + ea, &c. (taking ee=2a). Again, if another effect 

 be produced by a pipe = eo, the same will be produced by a pipe 

 = ee — eo, ee + eo, &c. and so on, and in this way it appears that 

 if upon gradually lengthening the pipe we find a certain series 

 of effects produced at the beginning, we shall have the same 

 .series in inverse and direct order alternately, the centers of each 

 set of effects being separated by an interval ( = ee) the length of 

 the open pijie in unison with the reed. So far we find a perfect 

 agreement with experiment, and it is plainly impossible to ex- 

 tend our reasoning much further with respect to the effect which 



