on the principles of Hydrodynamics, 355 



moreover, an antecedent objection to the supposition of plane- 

 waves. For on making this supposition, a particular and exact 

 integral of the resulting differential equation may be obtained, 

 which, as I have recently shown in this Magazine by argu- 

 ments that need not now be repeated, admits of no interpreta- 

 tion consistent with fluid motion. Such inconsistency must 

 necessarily be significant. The reasoning by which it was 

 arrived at being good, it clearly means that tne general sup- 

 position of plane-waves is not legitimate. 



Again, a general reason for the integrability of m^x + i;^ 

 + 'wdz is given, if it be proved that in every instance of small 

 vibrations the motion is composed of motions in spherical 

 waveSf that is, waves in each of which the motion is directed 

 to or from a fixed centre, and is a function of the distance from 

 the centre. But such composition of the motion does not 

 admit of being proved, because the hypothesis of spherical 

 waves is liable to an antecedent objection. For on making 

 this hypothesis, a result is arrived at inconsistent with the 

 principle of constancy of mass, on which one of the general 

 hydrodynamical equations rests. This I conceive that I have 

 shown in my communication to the Number of the Philoso- 

 phical Magazine for last February. 



It appears, therefore, that vibratory motion is not generally 

 compounded of motion either in plane- waves or spherical waves, 

 and that the integrability of uda^ + vdy + wdz is not accounted 

 for on either of those suppositions. At this stage of the argu- 

 ment it is important to remark, that although inconsistencies 

 have resulted from the general suppositions of plane-waves and 

 spherical waves, it does not thence follow that these are not 

 possible cases of arbitrary disturbance. As such, however, 

 they must plainly be treated by a different process. Before 

 treating these, or any other instances of arbitrary disturbance, 

 it is absolutely necessary to assign a general reason for the 

 integrability o^udx + vdy + isodz. I proceed, therefore, to make 

 another supposition. 



Let the function which we have called rj' be composed of 

 two factors, y and <p, such that /is a function of ^* and y only, 

 and f a function of s: and t only. Then 



_d^_ ^ 

 dx dx 



_d^^ df 

 ^~ dy ~'^ dy 



2 A2 



