GENERAL PRINCIPLES OF HYDRAULICS. 443 
horizontal stream between the discs above datum, then by 
i’s theorem 
a Pv P v2 2 
Bat 99 = Ha constant. Hence H — h — 7 "99 8gm2aPr2” and 
P Q? 
(Hr h-2)= Sgmigt = 9 constant. 
_ ‘If rand : be plotted in a plane Sintainih the axis of the disce, a 
curve wits is Barlow’s curve is obtained. 
The foregoing discussion will obviously also apply to the case where 
the current is reversed, flowing inwards instead of outwards, 
377. Vortices.—A mass of rotating fluid is called a vortex. When the 
motion is produced by the action of forces of weight and fluid pressure only, 
the vortex is called a free vortex. When the law of motion in a vortex 
is different from that of a free vortex, it is called a forced vortex. The 
- simplest form of forced vortex is that in which all the particles have the 
same angular velocity ; this form of forced vortex is considered in Art. 
380, under the heading of “ whirling liquids.” 
_ $78. Free Circular Vortex.—If instead of having simple radial 
motion the water between the discs in Fig. 720 moves in circular currents, 
i and at the same time moves slowly in a radial direction from one circular 
_ current to another, assuming freely the velocities proper to the currents 
_ which it enters, a free circular vortex is produced. 
| Consider a portion ABCD (Fig. 721) of a ring of the water in a free 
) circular vortex. Let r be the internal radius of 5 yap oP 
_ this ring, and dr its radial thickness. Let the ‘ 
length and depth of ABCD be such that the area 
of the vertical face AB is unity. The faces AD 
and BC are radial, and since the thickness dr of the 
ring “is very small, the area of the face CD may 
also be taken as unity. Hence the volume of ABCD ‘ 
_ is dr, and its weight wdr. Let Pand P+dP be the Fia. 721. 
fluid pressures, and v and v+dv be the velocities at the inner and outer 
_ faces of the ring respectively. 
The centrifugal foree of ABCD is , and this must be balanced 
_ by the difference between the fluid ‘eaahires on the outside and inside, 
wedr 
wedr 
namely, dP. Hence, dP= Again, by Bernoulli's theorem, 
gr 
. dP d vw dP vd 
| RES eee CJ ar Sipe » which gives the result —+"~=0, 
w 29 29 g 
| ra 
_ Substituting for dP its value ———, the result vdr +rdv =0 is obtained. 
Hence vr = constant, and v varies inversely as 7 as in the radiating current. 
_ It follows that the law of variation of hie a will also be the same as in 
the radiating current. 
379. Free Spiral Vortex. a superposing on the fluid particles the 
motions of a radiating current and of a free circular vortex, a free spiral 
