and the Observed Velocity of Sound in Air and Gases. 7 



second head, originating in the different physical interpretations 

 of the mathematical processes and their results, are of a more 

 intractable character. Here we plunge into the quicksands of 

 equations of partial differentials, of discontinuous functions, and 

 of integrals containing arbitrary functions', the arbitrariness of 

 which has a signification in the applications of the functions to 

 physical questions*!. Many of the most interesting and important 

 dynamical problems involve the consideration of the true signi- 

 fication of mathematical results which are known to have been 

 reached by processes which are not rigorously exact. Many of 

 the equations are utterly unmanageable and incapable of integra- \ 

 tion unless certain assumptions are made. Hence questions in 

 relation to the warrantableness of such assumptions in particular 

 cases are perpetually arising among the most eminent mathema- 

 ticians. Such difficulties in the mathematical theory of sound 

 have been sources of perplexity and controversy from the time 

 of Lagrange and Euler to the present period. It is very ques- 

 tionable whether the vast amount of intellectual energy and ana- 

 lytical ingenuity recently displayed in the discussions of the 

 various points bearing on this problem by Challis, Airy, Stokes, 

 Moon, Rankine, Haughton, Potter, Earnshaw, and others (how- 

 ever instructive and important in other respects) has made any 

 substantial contribution towards a clearer reconciliation of the 

 physical with the mathematical aspects of the questions at issue. 

 It is not my purpose to venture upon ground rendered historical 

 by the labours of the greatest geometers of the present century. 

 But I must insist that purely mathematical questions should be 

 kept quite distinct from the physical considerations, — and that, 

 in problems of this character, no deduction from analysis is worthy 

 of confidence which does not admit of a rational physical interpre- 

 tation, capable of being tested by observation or experiment. 



Judged by this criterion, many of the anomalous and contra- 

 dictory results which mathematicians have deduced from the 

 discussion of the theory of sound must be placed in the domain 

 of the hypothetical, until physical facts shall be produced in cor- 

 roboration of their reality. As an illustration of this, I would 

 call attention to the startling conclusion to which Mr. Robert 

 Moon has been led by his analytical investigations, namely, 

 "that waves of rarefaction are transmitted more rapidly than 

 waves of condensation," — and that, as in the production of sounds 

 both kinds of waves are usually generated in immediate succes- 

 sion, " we should hear sounds double .... if both kinds of dis- 

 turbance were capable of affecting the human ear." Inasmuch 

 as such a result is contradicted by experience, the difficulty is 

 summarily removed by assuming " that the sensation of sound is 

 produced by aerial rarefactions alone-," that the waves of conden- 



