336 



Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



A mere inspection of this table, the fifth column of which gives the proportion of the 

 error of theory to the whole velocity, will enable the reader to judge whether the theory 

 advanced in the preceding pages is borne out by experiment. I am not aware what degree 

 of accuracy Mr. Russell is disposed to ascribe to his observations, but I imagine he will hardly 

 maintain that the velocity of a wave could in any case be observed with greater accuracy 

 than the fortieth part of the whole. I am therefore inclined to pronounce that the coincidence 

 of theory and experiment is exact. 



The last column of the above table, though not necessary for the comparison of theory 

 with experiment, is added for the purpose of shewing that there was a considerable degree of 

 variation in the circumstances which characterized the several waves that are here selected as 

 tests of theory. It may also serve as a further test of theory, if ever the experiment should be 

 repeated. 



It is worthy of remark that the ratio — depends entirely on the value of the ratio — , and 



fj k 



not at all on the absolute value of either h or k. 



It also appears that there is no means of determining the absolute values of \ and /3, from 

 those of h and k: consequently we must consider either X or /3 a necessary element in the 

 experimental determination of a wave of the kind we have been considering. 



If h k and /3 are observed, then all the circumstances of the wave can be calculated; i.e., the 

 path, velocity, and position at a given moment, of each particle ; and the place, form and velocity 

 of the wave. 



We must now advert to a circumstance of considerable importance. The equation whith 

 we have found for the pressure at any point within the moving portion of fluid is 



p = - \{c - uy — gy + i n^y' + constant. 

 Now for those particles of the wave which are immediately in contact with the quiescent 

 portion of the fluid, i« = ; and consequently p= constant — gy + ^ny" ; which varies partly 

 as the depth and partly as the square of the depth of any particle below the quiescent surface. 



