If w vanishes throughout the fluid at any instant, equation (I I) shows that it will always be zero, and the fluid motion is then called **irrotational**; and a function 4) exists, called the velocity function, such that udx+vdy-{-wdz =-d, (13) and then the velocity in any direction is the space-decrease or downward gradient of cp.

D - K dK dK _ dK dK dK ?dx n dyd °, udx dz - ° and K=fdp/o+V+2q 2 =H (3) is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is **irrotational** H is a constant throughout the fluid.

Y If the motion is **irrotational**, u=-x-- dy' 2' d y = dx' y y so that :(, and 4' are conjugate functions of x and y, 0+4,i = f(x + y i), v 2 4 =o, v 2 0 =o; or putting 0+0=w, +yi=z, w=f(z).

When the motion is **irrotational**, dq_ _I d deId> G =o, a=-dxy dy, v dy ydx' v 21, ' = o, or dx + dy -y chi, '1/4724, 4 1 1+1 Rx2 = $Rc 2 (ch 2 a1 +I), +h+I Ry2 = 8Rc 2 (ch 2a 1 - I), (6) (7) b2)2/(a2 + b2).

**Irrotational** Motion in General.-Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to a single-valued potential function; any motion set up in the liquid must be due to a movement of the boundary, and the motion will be **irrotational**; for any small spherical element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation.

= -dQ+1dg2, and integrating round a closed curve (udx+vdy+wdz) =0, and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential **irrotational** and due to a velocity function, the circulation is zero round all reconcilable paths.

If this relation is true along all paths, the velocity of the aether must be of **irrotational** type, like that of frictionless fluid.

Moreover, this is precisely the condition for the absence of interference between the component of a split beam; because, the time of passage being to the first order fds/V f(udx+vdy+wdz)V2, the second term will then be independent of the path (43 being a single valued function) and therefore the same for the paths of both the interfering beams. If therefore the aether can be put into motion, we conclude (with Stokes) that such motion, in free space, must be of strictly **irrotational** type.