31-32] RELATIVE MOTION IN A FLUID ELEMENT. 35 



It is easily seen that the above resolution of the motion is 

 unique. If we assume that the motion relative to the point 

 (x, y, z) can be made up of a strain and a rotation in which the 

 axes and coefficients of the strain and the axis and angular 

 velocity of the rotation are arbitrary, then calculating the relative 

 velocities u, v, w, we get expressions similar to those on the right- 

 hand sides of (2), but with arbitrary values of a, b, c,f, g, h, f, 77, f. 

 Equating coefficients of x, y, z, however, we find that a, b, c, &c. 

 must have respectively the same values as before. Hence the direc 

 tions of the axes of the strain, the rates of extension or contraction 

 along them, and the axis and the angular velocity of rotation, at 

 any point of the fluid, depend only on the state of relative motion 

 at that point, and not on the position of the axes of reference. 



When throughout a finite portion of a fluid mass we have 

 f , 77, f all zero, the relative motion of any element of that portion 

 consists of a pure strain only, and is called irrotational/ 



32. The value of the integral 



f(udx + vdy + wdz), 



[f dx dy dz\ 7 



or I (u -j- + v -/- + w -y- }ds, 



J\ds ds ds J 



taken along any line ABCD, is called* the flow of the fluid 

 from A to D along that line. We shall denote it for shortness by 

 I (ABCD). 



If A and D coincide, so that the line forms a closed curve, or 

 circuit, the value of the integral is called the circulation in that 

 circuit. We denote it by I (ABC A). If in either case the inte 

 gration be taken in the opposite direction, the signs of dx/ds, 

 dy/ds, dzjds will be reversed, so that we have 



I(AD) = -I(DA), and I (ABC A) = - I (ACBA). 

 It is also plain that 



/ (ABCD) = I (AB) + / (BC) + / (CD). 



Let us calculate the circulation in an infinitely small circuit 

 surrounding the point (x, y, z). If (x + x, y + y, z + z) be a 

 point on the circuit, we have, by Art. 31 (2), 

 uox + vdy + wdz = Jd (ax 2 + 6y 2 + cz 2 + 2/yz + 2#zx + 2/zxy) 

 + ? (ydz - zdy) + 77 (zdx - xdz) + f (xdy - ydx). . .(I). 



* Sir W. Thomson, &quot;On Vortex Motion.&quot; Edin. Trans., t. xxv. (1869). 



32 



