34 IRROTATIONAL MOTION. [CHAP. III. 



tion, 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, all zero, the motion of any element of that portion consists of 

 a translation and a distortion only. We follow Thomson in calling 

 the motion in such cases irrotational, and that in all other 

 cases rotational. 



39. The value of the integral f(udx 4- vdy -f wdz), or, other 

 wise, \\u~J~ + v j + w j-)ds, 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(ABCA). If in either case the inte- 



dx 

 gration be taken in the opposite direction, the signs of -y-, &c., 



will be reversed, so that we have 



I(AD) = - 1 (DA), and I (ABC A) = - I (ACS A). 



It is also plain that 



I(ABCD] = I(AB) + I(BC) + I (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 (2), 



UdX + VdY+ WdZ=d(UX+ FF+ WZ) 



+ | d(a X 2 + bY* + cZ* + 2/YZ + 2gZX + 2hXY) 



+ ( YdZ- ZdY)+r) (ZdX- XdZ) + (Xd Y- YdX}. 



The first two lines of this expression, being exact differentials of 

 single-valued functions, disappear when integrated round the cir- 



* Thomson, On Vortex Motion. Edin. Trans. Vol. xxv., 1869. 



