XXIV. The Mathematical Theory of the two great Solitary Waves of the First Order. 

 By S. Earnshaw, M.A., of St. John's College, Cambridge. 



QRead December 8, 1845.] 



Though it is now about a hundred years since the general equations of fluid motion, expressed 

 in partial differential coefficients, were first given to the world, I am not aware that any 

 important case of fluid motion has hitherto been rigorously extracted from them. This however 

 has not arisen from want of effort, for the subject on account of its importance has successively 

 occupied the attention of the first mathematicians from the days of D'Alembert to the present 

 time ; but rather from the peculiarly rebellious character of the equations themselves, which 

 resist every attack, except it have reference to some case of a very simple and uninteresting nature. 

 This want of success I am inclined to attribute chiefly to our experimental ignorance of the 

 peculiar and distinctive characters of different species of fluid motion. In this matter indeed 

 there was a tendency to ignorance produced by that little success which had attended mathematical 

 research ; for as it was found that fluid motions of every sort, providing they are continuous, 

 are all expressible by the same partial differential equations, it was thought that those equations 

 ought to admit of being integrated in some general forms which should consequently include the 

 properties of every possible kind of continuous fluid motion. The natural consequence of this 

 idea has been that much effort has been unsuccessfully expended in attempts to obtain general 

 integrals. Two ways of approximation however are open to research ; — the one, in which the 

 approximations are made by neglecting certain terms on account of their supposed smallness in 

 comparison with the terms retained ; and the other, in which ab initio hypotheses are made 

 as to the paths or velocities or some other character of the motions of the particles. With 

 regard to both these methods, it is evident tliat they must first be authorized by experiment, 

 before they are used in verifying or predicting results. The former however is peculiarly 

 liable to error, from our being uncertain in many cases, whether with the neglected terms, we 

 may not have discarded some of the peculiar and essential properties of the motion we are 

 investigating. And with respect to the latter method, recourse must be had to experiment 

 to ascertain what are the really distinctive characters of the various kinds of fluid motion. 

 Hence nothing seemed more likely to conduce to the advancement of the Theory of Hydro- 

 dynamics than the appointment of a Commission, by the British Association for the Advancement 

 of Science, the object of which was the discovery of the " Varieties, Phcenomena, and Laws of 

 Waves :" for if there be varieties of waves differing in their phasnomena and laws, it was too 

 much to expect the mathematician (considering the exceedingly intractable nature of the 

 equations with which he has to deal) to discover what are the precise hypotheses which lead to 

 each variety. He must at least be allowed to know something of the peculiar phasnomena of 

 each variety, before he proceeds to the integration of his equations ; and there is no way in 

 which he could gain this knowledge except through the medium of experiments such as the 

 Commission, just alluded to, were directed to institute. The differential equations of motion 

 are too comprehensive to admit of general management. An hypothesis is in fact necessary to be 



