IN HYDRODYNAMICS. 



35 



First, let the fluid be incompressible. It has been shewn in Art. 5 that when udai + vdy + wdz 



is an exact differential, the dynamical equation may be integrated from any one to any other 



point of the fluid. But the result obtained by integrating that equation in this manner, 



does not give a possible motion unless the equation (8) be similarly integrable. Let this be 



the case. Then the first condition that must be satisfied is, that each surface of displacement 



dV 

 be a surface of equal velocity. For on no other supposition can the differential coefficient ■— 



remain the same, in passing from a given point to another indefinitely near in an unlimited 

 number of directions. In the annexed figure P and Q are any 

 two points of the fluid ; QR is an orthogonal trajectory to the 

 surfaces of displacement situated at a given instant between 

 P and Q ; PR is a line drawn on the surface of displacement 

 which passes through P, and intersecting QR in R. Now by 

 hypothesis the integral of equation (8) may be taken between 

 arbitrary limits. Therefore the integral from P to Q along PQ 

 is the same as the integral along PR and RQ. But the integral 

 along PR is nothing, because PR is on a surface of equal velocity. 

 Therefore the integral from P to Q is the same as the integral 



from R to Q. Supposing therefore the surface of displacement through P and the velocity in 

 this surface to be given at a given instant, the velocity at any point Q is a function of the line 

 QR. Let QR = s. Then Yds is a differential of a function of s and the time. Since, there- 

 fore, d(p = Yds, (j) is also a function of s and the time. But the equation ^ = is the equation 

 of a surface of displacement. Hence for a given surface of displacement s is constant. This 

 proves that the surfaces of displacement are parallel to each other, the orthogonal trajectories 

 are straight lines, and the motion is rectilinear. 



Again, let da be the increment of any line drawn arbitrarily on any surface of displacement. 

 Then since the direction of the variation of co-ordinates in the equation (7) may be any whatever, 

 we shall have, 



dX rl'^r 



d'(h 



1 '— 



d(j dadt 



aa 



d\ 



But since — is the effective accelerative force perpendicular to the direction of motion, and 

 da 



, d\ A, <^^ « 



smce, as we have seen, the motion is rectilinear, it follows that -j- = 0- Also 



Consequently 



d?(f) 



da 

 du 



= 0. 



This proves that the equations udie + vdy + wdz = 0, and — 



dx 



dadt 



+ — dy + dz = are true at the same time. The latter equation is the former differentiated 



dt dt 



with respect to t, on the supposition that d.r, dy, dx do not vary with the time. It follows 



with respect to the first is liable to this objection : — he concludes 



that 



fhi 



dw 



, from approximate equa- 



_ dv du _ dvj dv 

 Jy~ dx' dz~ dx ' dz ~ dy 

 tions, whence it follows that those equalities are approximate; 

 whilst the inference that udx + vdij 4- wdz is a complete differ- 

 ential, requires that they should be exact. No reason is assigned 

 by Lagrange for the other Theorem. The following argument 

 shews it to be without foundation. Jf'each of the quantities w, 

 t), w vanishes for a certain value ti of t^ they must each con- 



tain t-h as a factor. We may therefore assume that udx+vdij 

 + wdz = {t - h)'^{Udx + VdyJr Wdz), one at least of the 

 quantities f. (', [r not vanishing when l-h. Since ( — A is 

 unaffected by the sign of differentiation, if the left-hand side of 

 the equality be an exact differential, Udx -(■ Vdy + Wdz must 

 be an exact differential also. But the latter quantity is not 

 necessarily an exact differential when t = h; therefore neither 

 is tlie other. 



E2 



