PROFESSOR KELLAND ON THE THEORY OF WAVES. 127 



sumes, viz. that it does not lag in the neighbourhood of the shallower part of the 

 canal. 



SECTION VII. 



66. The problem of determining the motion due to a slight disturbance, such 

 as an impact on the surface of the fluid, or the elevation or depression of a por- 

 tion, so as to leave it to regain its original position by disturbing the rest of the 

 fluid ; this problem has occupied the attention of philosophers much, and it 

 would appear that little remains to be done on the subject. We shall, in what 

 follows, adopt the process employed by M. CAUCHY and M. POISSON, viz. that of 

 solving the general equation which results from the hypothesis, that the pressure 

 is a complete differential of the co-ordinates. We shall also adopt their solution 

 in its utmost generality. In so doing we must, however, express a doubt whe- 

 ther it is a complete solution of the problem. That modified form of it which M. 

 CAUCHY has adopted as the ground- work of his Memoir, we have no hesitation in 

 pronouncing far from complete. But to what state of motion the integral applies, 

 if not complete, we can hardly venture to guess. It is probably to a rippling 

 motion or slight, almost vertical, oscillation of the surface, which is very incon- 

 siderable compared with the depth. We make this remark from an examination 

 of the results which M. POISSON has arrived at. Yet though the equation be im- 

 perfect, it will undoubtedly serve as an approximate representation of the form 

 of the function on which the motion depends. A discussion of it, therefore, will 

 probably lead us to some important conclusions relative to the arrangement of 

 the particles at the beginning of the motion, though it fail to give a satisfactory 

 value to the length of the wave. 



We adopt the usual notation, and suppose, as is commonly done, the distur- 

 bance to be small. 



The equations of motion are, 



d d) dd> 

 -~J--u, -f- = v 

 dx ay 



d_u_dv 

 - 



. . ' . . (3) 



,g 1 



where v t , - correspond to the surface of the fluid. 



The integral of equation (1), to which we alluded in the preceding para- 

 graph, is 



