Mr. EARNSHAW, ON THE MATHEMATICAL THEORY, ETC. 327 



made before we can advance a single step towards their integration ; and by the aid of it we may 

 only advance to a certain point, and no farther. If it be asked why we are thus stopped, the 

 answer seems to be this ; the results obtained up to that point are still of too general a character, 

 embracing every variety of fluid motion which is compatible with the hypothesis on which we 

 started. Now among the large class of such motions, there may be some varieties which cannot 

 be analytically expressed by the same final formulae ; and consequently these require to be sifted 

 from the others and from one another, by additional hypotheses ; each hypothesis pointing at the 

 variety or subdivision to which it belongs, and to no one else. Nothing in fact can be more 

 clear than this, that if there be varieties of fluid motion the laws of which do not admit of being 

 expressed in the same analytical forms, those varieties must be separately treated by the 

 mathematician ; and to the oversight of this necessity I attribute the insignificance of the progress 

 which has hitherto been made in this subject, 



I have thought it necessary to introduce these remarks, because some persons, especially 

 among such as have not made Hydrodynamics a special object of study, ai'e apt to depreciate 

 investigations which set out upon a set of hypotheses which manifestly limit the range of the 

 results obtained. They prefer investigations which set out with fewer and broader hypotheses, 

 because they have the appearance of greater generality ; and this character they continue to 

 ascribe to such investigations, thougii it is found that in carrying them out it may have been 

 found necessary to introduce a system of approximations by the neglect of certain terras. I am 

 persuaded that this view is utterly fallacious in the majority of cases of any importance in nature : 

 and that the wiser and better course when possible is, to consult experiment and thence obtain 

 authority for a set of hypotheses to start with, and to carry out these hypotheses to the end 

 without the introduction of analytical approximations. Our results will then be as comprehensive 

 as our hypotheses, and as far as they go may be relied upon with unlimited confidence. This 

 is the course which has been adopted in the following investigations. The experiments which 

 I have taken as a guide in framing my hypotheses are those of Mr. Scott Russell which are 

 printed in his " Report on Waves''' in the " Report of the Fourteenth Meeting of the British 

 Association^ These experiments were conducted with well-contrived apparatus and great care, 

 and are as worthy of confidence as experiments on wave motion can be : and there seems to be 

 but one circumstance in them to be regretted, which is, that Mr. Russell having been led by his 

 results to adopt a certain empirical formula for the velocity of transmission of a wave, his 

 expei'iments seem in a great measure to have degenerated into an effort to establish the truth 

 of that formula, in which he appears to have overlooked or forgotten the probability that after 

 all it might only be an approximate result, and that the exact mathematical form might contain 

 elements not recorded in his tables, because not required in his formula. The consequence of 

 this oversight is that he has not recorded one element, very easy of observation, and of essential 

 importance ; viz. the distance through which each particle was transferred in space by a wave in 

 passing it. Had this element been recorded, the experiments would have been much more 

 complete : and without it they are certainly defective as accurate tests of theory. It is true 

 Mr. Russell has given a rule for calculating this element ; but he has not furnished us with the 

 requisite data. These are the volume of the fluid which is elevated above the general level, and 

 the breadth and depth of his canal. Tiie last two are given, but the first is not given in any 

 one instance. He has indeed stated the volume of fluid originally put in motion, and seems to 

 have supposed that this would supply all that was wanted ; entirely overlooking a fact, which 

 must have forced itself upon his attention in the very first stages of his experimental researches, 

 viz. that a single wave could never be generated alone, and that consequently all the fluid 

 originally displaced did not go to form the single wave of observation ; which besides, us the 

 experiments themselves shewed, and as we siiall prove theoretically, was continually wasting away, 

 and thereby rendering the data still more inaccurate as the experiment proceeded. 



