c^= 



ON WAVES. 333 



Having ascertained that no one had succeeded in predicting the phsenome- 

 non which I have ventured to call the wave of translation, or wave of the 

 first order, to distinguish it from the waves of oscillation of the second order, 

 it was not to be supposed that after its existence had been discovered and its 

 pheenoraena determined, endeavours would not be made to reconcile it with 

 previously existing theory, or in other words, to show how it ought to have 

 been predicted from the known general equations of fluid motion. In other 

 words, it now remained to the mathematician to predict the discovery after 

 it had happened, i. e. to give an a priori demonstration a posteriori. 



Theoretical Results subsequent to the publication of the Author s Investiga- 

 tions. — Since the publication of my former observations on the wave of the 

 first order, two attempts have been made to elicit from the wave theory, as 

 developed by Poisson, &c., results capable of such physical interpretations as 

 should represent the phaenoraena of that order. 



The first of these investigations is that of Mr. Kellano in the Edinburgh 

 Philosophical Transactions. This valuable and elegant investigation deduces 

 theoretically, from the general equations of fluid motion, on the hypothesis 

 of parallel sections, and of oscillations of the general form of the curve of 

 sines, the following value for the velocity of a wave, 



a * c'''' + e-"* I gaA + e-aAj' • • • • {\.\ 



s being the semi-elevation, h the depth in repose, X the length of the wave, 



c the velocity of transmission. 



This expression gives values for the velocity of the wave which Mr. Kel- 



land has himself compared with my experiments as follows : — 



Theoretical value when h=3'97 and 2e=0-53, is c=2'8693 



Observed value c=3"38, 



1 2 

 showing the error in defect = or of the whole theoretical velocity. 



least accurate and the least valuable, and these are the ojily observations employed by MM. 

 Weber in their larger wave observations. Further, as they did not recognise at all the possi- 

 bility of the existence of the solitary wave of the first order, nor the difference of its phaeno- 

 mena from the negative waves, nor the distinction of waves into separate first and second orders, 

 they have mingled together the observations and phasnomena of both. Thus have they failed 

 to recognise the existence of the law of the velocity which I have elicited. 



Nevertheless, their observations are very valuable, and furnish interesting information to one 

 already master of my observations. In their very deviations from the laws exhibited by my 

 observations, they become instructive as manifesting and enabling us to measure the amount of 

 those interfering influences which diminished the value of their experiments when taken by 

 themselves. For this purpose I have taken some of their experiments and placed them beside 

 the results of mine ; the effects of adhesion to the sides, and of more or less perfect fluidity, 

 are well manifested in the difference of the results. It is however to be remembered that in 

 point of accuracy and precision, and also of weight, the shortness of period and path in their 

 observations diminish their value. 



These remarks, which I make with perfect deference, are designed to apply only to the large 

 class of waves to which chiefly I have directed my attention ; the observations on dropping 

 waves, and all those made with reference to the phenomena of light and sound, are to be 

 exempted from these remarks. I desire that my experiments should enhance rather than 

 derogate from the value of those of my estimable predecessors, and I wish rather by these 

 statements to make an apology to them for having arrived at different conclusions, by showing 

 that the methods I chanced to light upon, and the circumstances in which 1 observed, were 

 more favourable than those which they happened to employ. I only aspire to having brought 

 to a more favourable conclusion what they had most meritoriously begun under circumstances 

 less propitious; my having arrived at different conclusions is probably more owing to the 

 chance of my being ignorant of their methods when I began, and alighting by chance upon 

 better; for had I known of their elegant apparatus at first, it is not improbable that I should 

 have been satisfied to adopt what so much ingenuity had contrived, and so failed to extend the 

 subject beyond the conclusions they had attained. 



