192 Prof. Cliallis on the Hydrodynamical Theory of 



analytical result of the solutions wliich does not correspond to 

 a physical reality. This rule is applied in the following infe- 

 rences from the general equation (a). 



5. That equation is plainly equivalent to l-^\ = yrf\ ; whence 



by integration A/r = ^^(/) +C^ Xi[t) being an arbitrary function 

 of the time; and C an arbitraiy quantity independent of t. The 

 argument in art. 17 of the commnnicatiou in the June Number^ 



being conducted so that ;^(/) is included in the function ^^r, 



gives (^\ = 0, and by integration -^^C^ an arbitrary quantity 



not containing t. The present argument shows that the fanc- 

 tion -v/r has in fact this character if ^(^[t) be assumed to be zero^ 

 but that it may also vary in an arbitrary manner with the time. 

 iN'ow^ according to the foregoing rulcj this result must be taken 

 into account as well as the other, both being significant; and 

 the inference to be drawn is that, consistently with the principle 

 of geometrical continuity, there may be tico classes of motions, 

 for one of which the functions -^Ir have constant values and the 

 surfaces of displacement have fixed positions in space, and for 

 the other these functions vary with the tniie^ and the surfaces of 

 displacement are continually shifting their positions. Genei'ally, 

 the former is the class of steady motions, and the other that of 

 unsteady or vibratory motions. It should be observed that the 

 actuality of such motions cannot be demonstrated without taking 

 account of the other general equations, and that the foregoing 

 reasoning only shows that then- existence would be compatible 

 with the principle of geometrical continuity. 



6. To proceed in logical order, it would next be required to 

 investigate mathematically the general differential equation 



|^^_p«^^- ./^-^ .... (b) 



dt dx dy dz 



\\-hich is derived from the principle of constancy of mass, and 

 the three dynamical equations, 



dp __ fdii\ dp _ fdr \ iP__j_ A^? A / •. 



pdx-^'KdiP pdy-^ Kdtr pdz-^ \dtr '' 



which are given by D'A^lembert's Principle; but these investi- 

 gations are so well known that they need not be introdireed here. 

 X.B. The expression udx + vdy -ricdz occurs so frequently 

 that in future it will be designated by \jLidx']. 



7. The reasoning thus far would be the same whether we sup- 

 posed [wf/.r] to be an exact differential, or to become such by 



