g ' REPORT—1846. 
III. The discharge of gases through small orifices. 
IV. Theory of sound. 
V. Simultaneous oscillations of fluids and solids. 
VI. Formation of the equations of motion when the pressure is not sup- 
posed equal in all directions. 
{. Although the common equations of hydrodynamics have been so long 
known, their complexity is so great that little has been done with them 
except in the case in which the expression usually denoted by 
edepodyptwde ssi. Se ee FAD 
is the exact differential of a function of the independent variables 2, y, 2*. 
It becomes then of the utmost importance to inquire in what cases this sup- 
position may be made. Now Lagrange enunciated two theorems, by virtue 
of which, supposing them true, the supposition may be made in a great 
number of important cases, in fact, in nearly all those cases which it is most 
interesting to investigate. It must be premised that in these theorems the 
accelerating forces X, Y, Z are supposed to be such that Xdx+ Ydy+ Zdz is 
an exact differential, supposing the time constant, and the density of the fluid is 
supposed to be either constant, or afunction of the pressure. Thetheoremsare— 
First, that (A.) is approximately an exact differential when the motion is 
so small that squares and products of w, v, wand their differential coefficients 
may be neglected. By calling (A.) approximately an exact differential, it is 
meaut that there exists an expression udx +4+v,dy+wdz, which is accurately 
an exact differential, and which is such that uw, v, w, differ from wu, v, w 
respectively by quantities of the second order only. 
Secondly, that (A.) is accurately an exact differential at all times when it 
is so at one instant, and in particular when the motion begins from rest. 
It has been pointed out by Poisson that the first of these theorems is not 
true+. In fact, the initial motion, being arbitrary, need not be such as to 
render (A.) an exact differential. Thus those cases coming under the first 
theorem in which the assertion is true are merged in those which come under 
the second, at least if we except the case of small motions kept up by dis- 
turbing causes, a case in which we have no occasion to consider initial motion 
at all. This case it is true is very important. ; 
The validity of Lagrange’s proof of the second theorem depends on the 
legitimacy of supposing w, v and w capable of expansion according to posi- 
tive, integral powers of the time ¢, for a sufficiently small value of that varia- 
ble. This proof lies open to objection; for there are functions of ¢ the 
expansions of which contain fractional powers, and there are others which 
cannot be expanded according to ascending powers of ¢, integral or fractional, 
even though they may vanish when ¢=0. It has been shown by Mr. Power 
that Lagrange’s proof is still applicable if «, 7 and w admit of expansion 
according to ascending powers of ¢ of any kind{. The second objection 
however still remains: nor does the proof which Poisson has substituted for 
Lagrange’s in his ‘Traité de Mécanique’ appear at all more satisfactory. 
Besides, it does not appear from these proofs what becomes of the theorem if 
it is only for a certain portion of the fluid that (A.) is at one instant an exact 
differential. 
M. Cauchy has however given a proof of the theorem§, which is totally — 
different from either of the former, and perfectly satisfactory. M. Cauchy 
* In nearly all the investigations of Mr. Airy it will be found that (A.) is an exact differen- 
tial, although he does not start with assuming it to be so. 
+ Mémoires de l’Académie des Sciences, tom. x. p. 554. 
+ Transactions of the Cambridge Philosophical Society, vol. vii. p, 455, 
§ Mémoires des Savans Etrangers, tom. i. p. 40, 
