500 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



processes employed in the body of the work. In two of these, the 16th and 20th, 

 we find solutions of the question now before us, but in both of them the approxi- 

 mations are used which I have before alluded to. The results will be found in 

 equation (141) of note 16, and equation {Q6) of note 20. They agree closely with 

 those obtained by M. Poisson. This memoir of M. Cauchy is of itself a large 

 and abstruse work, and consequently, the brief notice taken of it must be under- 

 stood to have reference only to those portions which belong to our present por- 

 tion of the problem. To bring the history of the subject down to the present 

 time, I have only to mention the names of Mr Challis,* Mr Earnshaw,! and 

 Mr Green ; | the two former of whom solve problems nearly connected with our 

 own, and the latter takes a very limited case of the actual problem, viz. that 

 solved approximately by M. Lagrange. 



The present must be regarded rather in the light of an introduction to a se- 

 ries of Memoirs, than as a complete work in itself. We treat only of motion in a 

 canal of uniform breadth, — nor shall we take even a large portion of that pro- 

 blem. The mode of generating motion, — its effect on the final waves, and on the 

 primary ones, — the variable state of the surface, owing to reflexion from the bot- 

 tom and sides of the canal, — these, and the like questions, will occupy us here- 

 after. We have, however, a sufficiently wide field, without having recourse to 

 these comparatively abstract points. Not to mention the application of the re- 

 sults to the theory of the Tides, an application becoming daily more and more 

 tangible by the labours of Mr Lubbock and Mr Whewell, we have a vast variety 

 of questions to resolve, more obviously belonging to the very threshold of our in- 

 quu-y. What is the correct velocity of a wave in a canal of variable depth ? or 

 in one which is not shallow nor very deep, as compared with the length of a 

 wave ? Will the effect be modified if the canal be a closed one at the commence- 

 ment of motion ? Will the length of the wave depend on the depth, on the quan- 

 tity of fluid flrst put in motion, or on the space over which that motion takes 

 place, or on all these causes ? How will friction modify the form of the wave 

 and the velocity of motion ? And, lastly. What will be the motion in a channel 

 which shallows away at its sides, when the channel is broad, as is the case when 

 reference is had to the motion of the tidal wave ? 



These, and like questions, are of the utmost importance, and demand a care- 

 ful investigation. In my next memoir, I hope to broach at least one, in addition 

 to that at present before us. In the mean time, we proceed to the 



* Transactions of the Cambridge Philosophical Society, vols. iii. and v. 



t Transactions of the Cambridge Philosophical Society. 



X Transactions of the Cambridge Philosophical Society, vol, vi. 



