PROCEEDINGS 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



February 1846. 



The Mathematical Theory of the two great Solitary Waves of the 

 first Order. By S. Earnshaw, M.A. 



The nomenclature of this paper is adopted from a Report on 

 Waves by Mr. J. S. Russell, printed in the Proceedings of the 

 British Association. From the extreme comprehensiveness of the 

 equations of fluid motion, the author infers a necessity of appealing 

 to experiments for the suggestion of data which may be used in mo- 

 difying the generality of those equations so as to suit actual cases of 

 known fluid motion. With this view he has made use of the expe- 

 riments recorded in Mr. Russell's report, and thence selected the two 

 following properties : — 1st. The velocity of transmission of a wave in 

 a uniform canal is constant. 2nd. The horizontal velocity is the same 

 for all particles situated in a vertical plane, cutting the axis of the 

 canal at right angles. By reference to Mr. Russell's report, it will 

 be seen that these two properties, selected on account of their sim- 

 plicity and ready experimental examination, are distinguishing cha- 

 racteristics of what he has denominated the two great solitary waves 

 of the first order. By the aid of them the equations of motion take 

 such modified forms as to admit of exact integration ; so that 

 without employing any analytical approximations the author is en- 

 abled to obtain theoretical expressions for all the circumstances of 

 the two solitary waves. The results are tested by a comparison of 

 the velocities of transmission of various waves given by theory and 

 by experiment. The greatest difference of these in the case of the 

 positive wave is not found to exceed -g-Vth part of the whole velocity ; 

 but in the case of the negative wave it is found to be much greater, 

 and to amount in one instance to as much as -g-th of the whole velo- 

 city. The reason of this discrepancy is conjectured ; and the agree- 

 ment in the case of the positive wave is considered to be exact. 



It is found in the course of the investigation that one of the ne- 

 cessary conditions of fluid motion is not satisfied ; and it is shown 

 that it cannot be satisfied as long as the two principles, adopted from 

 Mr. Russell's report, are supposed to coexist. They are proved iu 

 fact to be incompatible with each other. But as the second prin- 



No. III. — Proceedings of the Cambridge Phil. Soo. 



!90C 



