PROFESSOR KELLAND ON THE THEORY OF WAVES. ] 15 



It will be seen that the last wave was observed to take a half second, whilst 

 theory makes it only a quarter of a second in proceeding one foot, the slight va- 

 riation of depth in one foot of length producing no appreciable effect. 



SECTION VI. 



58. We proceed to investigate the translation of waves, on the hypothesis 

 that the section of the fluid, perpendicular to the direction of transmission, is not 

 a rectangle. By reference to Art. 28, Part I., it will be seen that we have ob- 

 tained an approximate solution on the hypothesis of parallel sections. The sim- 

 plicity of the formula is such, that we are enabled to perceive at a glance its con- 

 nexion with the hypothesis, and are thus led to suspect that the approximation 

 is an approximation depending on the applicability of the hypothesis. In other 

 words, we conceive that in proportion as this hypothesis approaches nearer to the 

 truth, so does the formula also. Of the applicability of this hypothesis to the 

 waves whose velocity we determined by it, we had great a priori confidence 

 from the circumstance that the fluid was put in motion, in most cases, with an 

 uniform velocity from top to bottom. Were this not the case, our hypothesis 

 would certainly have been violated in the early part of the motion, and it is dif- 

 ficult to see how it could have been satisfied with any degree of accuracy at all. 

 We propose, therefore, to give another solution of this problem on another hy- 

 pothesis. 



Let us now take account of the variation of motion in three dimensions. 



Take x as the axis along which the horizontal motion of transmission takes 

 place, and z vertical. Suppose also, that the origin is at the bottom of the fluid. 

 We have as the equations for determining the pressure, 



dp _ 



and on the hypothesis that p is a complete differential of x, y, and z, we are pre- 

 sented with the following equations, which contain the solution of the general 

 problem, 



dx dy dz 



(2} 



dy\dt dx\dt 



