a new General Equation in Hydrodynamics. 159 

 must be integrable by a factor. But since the criterion 



(dv dw\ (dw du\ (du dv\ _ ,, , 



Iz-d^)+'\-d^-Tj+''\Aj-d^)=^ ■ (*•) 



is not satisfied, this is not the case ; and therefore, according to 

 the axiom, the motion expressed by {a.) is such as could only 

 belong to a set of indefinitely small discrete atoms. Yet it is 

 simply a motion of rotation compounded with a motion of trans- 

 lation ; a kind of motion which even a solid body may possess, 

 and which is very nearly exhibited in the case of water revolving 

 in a unifoi-mly descending bucket, and might almost exactly be 

 exhibited were it not for the friction against the sides and bot- 

 tom of the bucket. 



Second argument. It is plain that Professor Challis admits 

 the possibility of the existence of motions in a fluid such that 

 udx + vdy + ivdz is integrable by a factor, though not by itself, 

 so that on this point we are agreed. Let u, v, w refer to such a 

 motion, which therefore must satisfy equation [b.). Compound 

 this motion with a motion of translation for which the constants 

 a, b, c are the components of the velocity. If the resulting 

 motion be such that the directions of motion may eveiywhere be 

 cut at right angles by continuous sm'faces, 



(m + a)dx + {v + b)dy+ {w + c)dz 



must be integi-able by a factor. Substituting in the equation 

 which expresses the criterion of integrability, and taking account 

 of {b.), we see that 



(dv dw\ , /dw du\ I du dv\ „ , , 



and since a, b, c are arbitraiy, an infinite number of systems of 

 values of a, b, c may be assigned for which (c.) shall be violated, 

 unless 



dv drv _„ dw du du ^^ _o 



dz dy ' dx dz ' dy dv ' 



which is contrary to hypothesis, since udx + vdy + wdz is supposed 

 not to be an exact differential. Hence, if the second axiom be 

 granted, we are obliged to suppose that a motion which may 

 belong to a fluid changes its character so completely by mere 

 composition with a uniform motion of translation, that the re- 

 sulting motion can only belong to a set of discrete atoms. At 

 this rate we could not tell whether a proposed motion, defined, 

 we will su|)po.se, relatively to the walls of a room, could or could 

 not belong to a fluid, without knowing the motion of the solar 

 system in space. 



Since a mutiou for which udx-\- &;c. is integrable by a factor, 



