ON RECENT RESEARCHES IN HYDRODYNAMICS. 8 
first eliminates the pressure by differentiation from the three partial differential 
equations of motion. He then changes the independent variables in the 
three resulting equations from 2, y, 2, é to a, b, ec, t, where a, b, ¢ are the 
initial co-ordinates of the particle whose co-ordinates at the time ¢ are 2, y, 2. 
The three transformed equations admit each of being once integrated with 
respect to ¢, and the arbitrary functions of a, 6, ¢ introduced by integration 
are determined by the initial motion, which is supposed to be given. The 
theorem in question is deduced without difficulty from the integrals thus 
obtained. It is easily proved that if the velocity is suddenly altered by 
means of impulsive forces applied at the surface of the fluid, the alteration is 
such as to leave (A.) an exact differential if it were such before impact. 
M. Cauchy’s proof shows moreover that if (A.) be an exact differential for 
one portion of the fluid, although not for the whole, it will always remain so 
for that portion. It should be observed, that although M. Cauchy has proved 
the theorem for an incompressible fluid only, the same method of proof 
applies to the more general case in which the density is a function of the 
pressure. 
In a paper read last year before the Cambridge Philosophical Society, I 
have given a new proof of the same theorem*. This proof is rather simpler 
than M. Cauchy’s, inasmuch as it does not require any integration. 
In a paper published in the Philosophical Magazine+, Prof. Challis has 
raised an objection to the application of the theorem to the case in which 
the motion of the fluid begins from rest. According to the views contained 
in this paper, we are not in general at liberty to suppose (A.) to be an exact 
differential when w, » and w vanish: this supposition can only be made when 
the limiting value of ‘—2 (wda+vdy+wdz) is an exact differential, where 
@ is so taken as that one at least of the terms in this expression does not 
vanish when ¢ vanishes. 
It is maintained by Prof. Challis that the received equations of hydro- 
dynamics are not complete, as regards the analytical principles of the science, 
and he has given a new fundamental equation, in addition to those received, 
which he calls the equation of continuity of the motion}. On this equation 
Prof. Challis rests a result at which he has arrived, and which all must allow 
to be most important, supposing it correct, namely that whenever (A.) is an 
exact differential the motion of the fluid is necessarily rectilinear, one peculiar 
case of circular motion being excepted. As I have the misfortune to differ 
from Professor Challis on the points mentioned in this and the preceding 
paragraph, for reasons which cannot be stated here, it may be well to apprise 
the reader that many of the results which will be mentioned further on as 
satisfactory lie open to Prof. Challis’s objections. 
___ By virtue of the equation of continuity of a homogeneous incompressible 
- fluid, the expression wdy—vd-z will always be the exact differential of a 
function of x and y. In the Cambridge Philosophical Transactions§ there 
will be found some applications of this function, and of an analogous function 
_ for the case of motion which is symmetrical about an axis, and takes place 
: in planes passing through the axis. The former of these functions had been 
_ previously employed by Mr. Earnshaw. 
i II. In the investigations which come under this head, it is to be-understood 
that the motion is supposed to be very small, so that first powers only of 
i small quantities are retained, unless the contrary is stated. 
* Transactions of the Cambridge Philosophical Society, vol. viii. p. 307. 
+ Vol. xxiv. New Series, p. 94. 
__ } Transactions of the Cambridge Philosophical Society, vol. viii. p. 31; and Philosophical 
_ Magazine, vol. xxvi. New Series, Ps 425, § Vol. vii. p. 489, 
By B2 
