XI. On the Theory of the Small Vibratory Motions 

 of Elastic Fluids. 



By J. CHALLIS, MA. 



FELLOW OF TRINITY COLLEGE, AND OF THE CAJIBRIDGE 

 PHILOSOPHICAL SOCIETY. 



[Read March 30, 1829.] 



1. Any one that has given much attention to the mathe- 

 matical theory of sound, will be aware that notwithstanding 

 the labours of the most eminent mathematicians, great obscurity 

 is .still attached to it. Much of this obscurity, I have been 

 led to think, is owing to the manner in which discontinuous 

 functions have been introduced into the subject; and as geo- 

 meters of late have been more engaged in the use of them 

 than in scrutinizing the evidence on which they rest, I will 

 endeavour to state, as briefly as possible, the nature of this 

 evidence. It depends, I believe, almost entirely on the au- 

 thority of Lagrange, and on his two dissertations contained in 

 the first and .second volumes of- the Mi.scellanea Taurinensia. 

 His first Researches, however admirable in other respects, cannot 

 be adduced in reference to the point before us, because that 

 part of them which bears upon it, contains a step in the proof 

 which can by no means -be admitted. In fact, it mainly depend.^ 

 on the sum of the .series cos 9 + cos 29 + cos 36 + &c. adinfinitum, 

 which he determines to be always equal to - ^. And in trnth, 

 if the exponential expres.sions be put for the cosines, and the 



