328 Mb. EARNSHAW, ON THE MATHEMATICAL THEORY OF 



And in fact Mr. Russell tells us he found it necessary to wait awhile after the completion 

 of the process of generating a wave till the main wave had separated itself from the residuary 

 waves, which always accompanied its genesis. To generate a single wave required, as we shall 

 see, the exertion of a peculiar law of pressure ; and as no attempt was made to secure the 

 observance of this law in Mr. Russell's experiments, the inevitable consequence was the genesis of 

 residuary waves. We shall also see from our theory, that the nature of the motions given to the 

 particles of the fluid in this kind of wave produces a natural tendency in the wave to generate 

 and cast off irregular disturbances from itself, working its own destruction as it proceeds. While 

 therefore I look upon these experiments as very valuable additions to our knowledge, I still regard 

 them as imperfect even to the extent to which they profess to have been carried. It is in)possiblc 

 indeed to read the Synopsis which Mr. Russell has given in page 343 of his Report without 

 perceiving that he was too eager to adopt as results of experiments certain geometrical analogies, 

 of which there seemed to be some faint shadowings indicated in his observations. 



In his Report Mr. Russell conceives that his observations authorized him to consider waves 

 as divisible into four distinct species : the first of which he has denominated " The great 

 solitary wave." It is found to comprehend two varieties, the positive and the tiegntive wave, 

 which though agreeing in some general characters differ in others. The object of the present 

 paper is to furnish the mathematical theory of this species. But how are we to sift this from 

 the other species.'' I have examined the phaenomena which Mr. Russell has recorded, and fixed 

 upon such as belonged to this species alone ; and these I have made the basis of my calculations. 

 But it is obviously desirable that the phaenomena thus selected should be of such a character 

 as admitted of easy and accurate observation. That the reader may judge in this matter I will 

 here propound them with Mr. Russell's statement of the method by which he obtained the one on 

 which there might possibly be a doubt : merely premising that I suppose the wave to be 

 transmitted in a horizontal canal of uniform breadth and depth, and that the fluid is incompressible. 



1st. The velocity of transmission of a wave is uniform. 



2nd. The horizontal velocity of all particles, which are situated in a vertical plane, 

 intersecting the axis of the canal at right angles, is the same. 



By a contrivance of peculiar ingenuity Mr. Russell was enabled to obtain the velocity of 

 transmission with great exactness; and the result at which he arrived, and which we shall assume 

 to be accurately true is, that abstracting from friction and the cohesion of particles, the velocity 

 of transmission is uniform and the wave is permanent. We shall in the end shew that this 

 hypothesis is not strictly accurate. 



With respect to the verification of the other principle which I have assumed, Mr. Russell 

 thus writes: — "The methods I had employed for such observations were the observation of the 

 motion of small particles visible in the water of the same, or nearly the same specific gravity with 

 water, or small globules of wax connected to very slender stems, so as to float at required deptlis. 

 The motions of these were observed, from above on a minutely divided surface on the bottom of 

 the channel; and from the side, through glass windows, themselves accurately graduated, the 

 side of the channel opposite the windows being covered with lines at distances precisely equal 

 to those on the window, and similarly situated. These methods are the only methods of 

 observation I have found it useful to employ, but I have now increased the number and variety 

 of the observations sufficiently to enable me to adduce the conclusions hereinafter following, as 

 representing the pha-nomena as far as their nature will admit of accurate observation." " If the 

 floating spherules before mentioned be arranged in repose in one vertical plane at right angles to 

 the direction of transmission of a wave, and carefully observed during transmission, it will be 

 noticed that the particles remain in the same plane during the transmission, and repose in the 

 same plane after transmission. It is further found, as might be anticipated from the foregoing 



