498 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



tigations have been such as apply only to the more abstruse and captivating- 

 branches of the science, whilst the simple extension of its fundamental conditions 

 has been almost, if not altogether, overlooked. From these impressions I have 

 been induced to apply myself to the question. My desire is to simplify as well 

 as to extend ; not that I purpose to remodel the works of those who have gone 

 before me, but, in certain cases, to derive conclusions previously known in sup- 

 port and illustration of my methods of operation. As far, however, as concerns 

 the present memoir, although the results obtained appear to differ but slightly 

 from those of M. Poisson or M. Cauchy, yet, on examination, they will be found 

 very little alike. A most essential element enters into them, which could not, 

 from the nature of the methods employed, appear in either of theirs, viz. a varia- 

 tion dependent on the sphere of excursion of the molecules. Nor is this all. 

 Both these philosophers suppose the oscillations to be complete, — indeed, one of 

 the very first elements on which M. Poisson' s theory is based, amounts only to 

 the expression of this circumstance. Whatever, therefore, may be the value of 

 investigations such as his (and I consider them most valuable), they do not solve 

 the problem which is most wanted, the problem of oscillatory motion, of large or 

 imperfect vibrations. They push to the extreme verge the problem which they 

 actually do contain, but leave the other branches of the subject comparatively 

 unimproved. 



The theory of the tides can derive little assistance from such speculations in 

 the present state of our knowledge, if indeed at all. This, and many other most 

 important divisions of philosophy, if accurately subjected to analysis, require 

 other restrictions than those commonly imposed, and at the same time demand 

 the removal of these. With an ultimate view to problems of this nature, I have 

 drawn up the present memoir, which embraces the motion of regular uniformly 

 sized waves proceeding in one direction, and solves the problem in all its genera- 

 lity. It contains, likewise, an approximation to the motion of a solitary w^ave, 

 such as the tide wave, or, more properly, that examined by Mr Russell, and de- 

 signated by him the Wave of Translation. How far I have succeeded I am not 

 prepared to state. My solution can be regarded only as an approximation, nor 

 does it very accurately agree with observation ; yet the agreement is to my mind 

 very satisfactory, as it shews, as far as I have tested the formula, that the error 

 is due to one, and only one, point : what that point may be, I do not take it on 

 me to conjecture. 



This is the whole substance of the present memoir. Since it is so restricted, 

 I shall not be expected, at the present time, to offer a sketch of the progress of 

 the general problem. It will suffice, that a very brief statement be made of the 

 labours of those who have added to our theoretical knowledge of this branch of 

 Hydrodynamics. 



