Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/465

449 HYDRODYNAMICS.] HYDROMECHANICS Cauchy s integrals reduce to the single equation 449 or where and therefore Now therefore i and therefore _ _ &quot; da db d(a, b) d(v ,b) _ d(u n ,a) 1- - -&amp;gt;T 2 sinh aud the motion cannot therefore have been generated from rest by natural forces ; the fluid must have been created with the proper amount of spin at every point. c / ,. r, dll ~ c We have -^ - core c sm dt ,7 ft / dv 9 T / /, a. __ =-. -, core cos I cat -&amp;lt; I , JK therefore the dynamical equations (1) and (2) become - cu-e c sin ( at - I =, da. c , -7 , &amp;gt; C c COS a&amp;gt; H dp ~  c and therefore the integral of these two equations is Q f c 2 w 2 c c cos ( ut + j -  c 2 a&amp;gt; 2 c &quot; c = H , a constant. v r /&quot;dp -.r P Now y = / + V, - Q ?/ , and therefore -a/8 - ace c cos P
 * 2 - a)c c COS ( cat + J - i C 2 w 2

2 =H. free surface is possible if a; id then P and the pressure at a particle is constant. The wave length A = 2-irc ; and the velocity of propagation c&amp;lt;a = Vgc -- The surfaces of equal pressure are trochoids, obtained by rolling a circle of radius c on the under side of a line at a depth - c, the distance of thp carried point from the centre being cc c. Irrotational Motion. If liijuid originally at rest be contained in a singly- connected space, then forces due to a singly-valued function V are not capable of setting up any motion in tin: liquid, and any motion must be due to the motion of the bounding surface. For, &amp;lt;(&amp;gt; denoting the velocity function, by Green s theorem the kinetic energy and therefore, if -^- = 0, then T = 0, and therefore dn d-, ,V0V -,-,- dx dij d; If we suppose the actual motion at any instant to have been in stantaneously generated from rest by the application of proper im pulses at the bounding surface, then, since no natural forces can act impulsively throughout the liquid, the equations of impulse are 1 d-a _ 1 d-ar 1 d-ar T&quot;~ ^^ 1 ~1 :== ^ 1 &quot;I ^^ Zi&quot; i p dx p dy p dz a- denoting the impulsive pressure at any point of the liquid ; and therefore, if (/&amp;gt; denote the velocity function, we can put Since the work done by an impulse is the product ot the impulse into half the sum of the initial and final velocities, we see how it is that the kinetic energy of the liquid Also the kinetic energy acquired thus due to the velocity function &amp;lt;p will be less than the kinetic energy of any other motion, wholly or partially rotational, but satisfying the equation of continuity, and the condition at the boundary that the normal velocity of the liquid is the normal velocity of the boundary. For, if MJ, v lt ?! be the velocities at any point of this new motion, and T! the whole kinetic energy, - 2 dxdydz. But fff w(j - M) + K^i - v) + w ( u i - v } dxdydz fff dx &quot; ff&amp;lt;t&amp;gt; i l(&quot;i - w ) + &quot;( r i - r ) + &quot;( u i ~ w ) 1 ds JJ V ( = 0. Then T l - T = i p fff j a positive quantity ; and therefore Tj is always greater than T, a theorem due to Sir W. Thomson. If, however, &amp;lt;p be multiply- valued, and the space occupied by the liquid multiply-connected, we can have circulation existingin the different circuits of the space even when the bounding surface is at rest, and the motion may still be differentially irrotational, and any motion of the bounding sur face will not affect these circulations. For instance, we may have &amp;lt;p = tan &quot;&quot; ^-, and the liquid circulating in any ring-shaped surface, whose axis of figure is the axis of -. To find the kinetic energy of a liquid in a multiply-connected space, the motion being differentially irrotational, but circulations existing in the circuits, the space occupied by the liquid must be rendered acyclic by barriers, which may be supposed to be membranes, moving with the velocity of the liquid ; and then, if k be the cyclic constant of the value of &amp;lt;f&amp;gt; in any circuit, we must suppose the value of on one side of the membrane to exceed tin- value of d&amp;gt; on the other side bv k, so that the integral //d, --? &amp;lt;/S JJ dn over the membrane must be replaced by k // -? e/R ; JJ an the term .^ over dn the outside surface must be added number of terms of the form A pk // -?&amp;lt;/S, to express the due to the circulation in the circu continuity shows that /Y- o?S over which render a circuit acyclic is independent of th membrane. XII and the condition of one of these membrane form of the 57