Small Vibratory Motions of Elastic Fluids. 271 



but discontinuous per se, and by this property of discontinuity 

 distinguished from every other. This definition has been admitted 

 by all subsequent writers. But it deserves to be considered 

 in what sense, and to what extent an investigation of this nature 

 can demonstrate any property of functions. The science of 

 quantity is a perfect science; it needs not the aid of any other, 

 and exists prior to its applications to questions of nature, and 

 independently of them. When in the applications, any form 

 or property of functions is arrived at by the operations that are 

 performed, it will always be possible to arrive at the same, 

 by abstracting from the physical question, and performing the 

 same operations by pure analytical reasoning. For in the ap- 

 I)lications we are, in general, concerned about time, space, force, 

 and matter, — ideas of a totally dissimilar kind, but possessing 

 this in common, that we can conceive of them as consisting of 

 parts, and in virtue of this common quality, after establishing 

 a unit for each, we are able to express their observed relations 

 numerically, or by lines or letters the representatives of numbers. 

 All subsequent reasoning is then conducted according to the 

 rules of analysis, and cannot possess a greater generality in regard 

 to the modes of expressing quantity, than the operations con- 

 ducted by those rules admit of. If an attempt be made to 

 prove the existence of discontinuous functions by pure analysis, 

 it will be impossible to succeed, because, as Lagrange says, " the 

 principles of the Ditferential and Integral Calculus, depend on 

 the consideration of variable algebraical functions, and it does 

 not appear, that we can give more extent to the conclusions 

 drawn from these principles, than the nature of these functions 

 allows of But no person doubts that in algebraic functions, all 

 the different values are connected together by the law of con- 

 tinuity." (Misc. Taur. Tom. I. p. 21.) Accordingly, no dis- 

 continuous function can be exhibited to view. The inference to 



