and when q is stationary, the acceleration is normal to the surface H = constant, and the stream line is a geodesic. Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary. Equation (3) is called Bernoulli's equation, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow. If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli's equation may be written H = p/[rho] + z + q²/2g = P/[rho] + h, (8) where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward. In particular, for a jet issuing into the atmosphere, where p = P, q²/2g = h - z, (9) or the velocity of the jet is due to the head k - z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics. 27. _Uniplanar Motion._--In the uniplanar motion of a homogeneous liquid the equation of continuity reduces to du dv -- + -- = 0, (1) dx dy so that we can put u = -d[psi]/dy, v = d[psi]/dx, (2) where [psi] is a function of x, y, called the stream- or current-function; interpreted physically, [psi] - [psi]0, the difference of the value of [psi] at a fixed point A and a variable point P is the flow, in ft.³/second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity. Thus if d[psi] is the increase of [psi] due to a displacement from P to P´, and k is the component of velocity normal to PP´, the flow across PP´ is d[psi] = k·PP´; and taking PP´ parallel to Ox, d[psi] = vdx; and similarly d[psi]= -udy with PP´ parallel to Oy; and generally d[psi]/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock. In the equations of uniplanar motion dv du d²[psi] d²[psi] 2[zeta] = -- - -- = ------ + ------ = -[Nabla]²[psi], suppose, (3) dx dy dx² dy² so that in steady motion dH d[psi] dH d[psi] dH -- + [Nabla]²[psi]------ = 0, -- + [Nabla]²[psi]------ = 0, ------ + [Nabla]²[psi] = 0, (4) dx dx dy dy d[psi] and [Nabla]²[psi] must be a function of [psi]. If the motion ia irrotational, d[phi] d[psi] d[phi] d[psi] u = - ------ = - ------, v = - ----- = ------, (5) dx dy dy dx´ so that [psi] and [phi] are conjugate functions of x and y,