VII. On tlie Theory of Waves. Part II. By The Rev. P. KELLAND, M.A., 

 F.R.SS.L.$E., F.C.P.S., late Fellow of Queens' 1 College, Cambridge; Pro- 

 fessor of Mathematics, fyc., in the University of Edinburgh. 



(Read 18th January 1841.) 



THE problems on which we were engaged in the preceding Memoir were the 

 following : 1, The complete determination of the velocity of transmission of a 

 wave in a fluid of any depth, provided the depth is uniform throughout ; 2, an 

 application of the hypothesis of parallel sections to the problem in its more com- 

 plicated form, in which the depth is variable in the direction of the breadth of the 

 canal ; 3, The investigation of the motion of a solitary wave. We propose in the 

 following pages to continue the same subjects, and to add the more difficult pro- 

 blem of initial motion. 



The principle on which the determination of the velocity was made to depend 

 is this very simple one, that the variation of pressure in proceeding from point 

 to point in the direction of motion is the same, whether we suppose the pressure 

 a function of x and y, or a function also of z. The former hypothesis refers its 

 determination to the general properties of fluids, the latter to the particular state 

 of the fluid in question at the time. Art. 9. 



This principle is a highly important one, but it requires a slight degree of 

 attention in its application. I deem it, therefore, not superfluous, in discussing a 

 question of so great importance as that now before us, to present other solutions, 

 in which the same principle is viewed in a totally different light. Having done 

 this, I shall give a complete solution of the motion of a wave in a canal of any 

 section, and an approximate one of the motion in a canal of variable depth. 

 Lastly, I shall consider the question of initial motion, not only in the case in 

 which the depth is small, but in the most general case. 



SECTION IV. 



46. We retain the following notation, which corresponds with that used in 

 Art. 20, &c. 



VOL. XV. PART I. E 6 



