448 HYDROMECHANICS [HYDRODYNAMICS. These principles assumed enable us to give a general explanation of the working of Giffard s injector. For, if the steam jet and water jet be directed at each other, with a small interval between, the superior energy and equal momentum of the steam jet will overcome the water jet, and the steam will now back into the boiler. But the steam jet, without losing its momentum, is capable of being mixed with water to such an extent as to become a condensed water jet, moving with the velocity of the water jet, and still entering the boiler, a valve preventing the reversal of the motion. Conse quently, the amount of water carried into the boiler per unit of time will theoretically be at most the difference between the quantities which would escape by the water and the steam jets, and therefore and the efficiency of the injector, that is, the ratio of the water pumped in to the quantity of steam used, will be -1, (T the efficiency of a pump being -- . a "With C.G.S. units, and a pressure of 8 atmospheres, for instance, p - P = 7 x 10 6 very nearly, . / -- = 1 5, and p = 1 . Therefore, if the diameter of the nozzles of the injector be d centi metres, the delivery in grammes per second and since 1 gallon is 4541 cubic centimetres, the delivery in gallons per minute 4541 = 2878d 2 nearly. The Lagrangian Form of the Equations. Here the independent variables which define a particle are the time t, and a, b, c, the initial values of the coordinates x, y, z of & particle of fluid (or else functions of the initial coordinates, but it is best to consider a, b, c as the initial coordinates themselves). Here x, y, z do not refer to a fixed point in space, but are the variable coordinates of a fluid particle, and are functions of a, b, c, t, the independent variables ; and consequently _ dx _ dy _ dz . ~~di ~~dT ^ ~~dt and the component accelerations of the fluid particle are du dv dw dt ~dt Tt Consequently the equations of motion, assuming the existence of the potential V, and putting P=y , and P + V = Q , ar du dQ . dv dQ dw | = dx dt dQ dv dy dt dz dt or multiplying by , V- , da da da dQ, du da dt dx dv da dt and adding, dy dw da dt da (i); with two similar equations dQ du dx db dt db dQ du dc dt (2), . . (3). Since the elementary parallelepiped whose edges were initially da, db, dc, becomes strained into a parallepiped of volume d(a,b,c) therefore the equation of continuity is d(x,y,z) P ^M) = p0 or, if the fluid be a homogeneous liquid, d(a,b,c) When a, b, c are not the coordinates of a point actually occupiec by the fluid particle, this equation of continuity must be replaced by d_ ( d(x,y,z) dt Cauchy s Integrals of Lagrange s Equations. Eliminating Q by differentiation between (2) and (3) dx d"u dx . d*v dy d 2 v dy . dho dz d 2 w dz _ _ ~ ~ d~u dx d 2 v ~ a .. . ~a ,- dtdb dc dtdc db dtdb dc dtdc db dtdb dc dtdc db ind integrating with respect to t, du dx du dx ^dv dy _ dv dy db dc du dx dv dy dc db db dc dw dz dc db db dc dw dz dw Q dv dc db db dc u , v , W Q being the initial values of u, v, w, and a, b, c the initial values of x, y, z. Now du _ du dx du dy du dz da dx da dy da dz da and therefore dw dv d(y,z) /du dwd(z,x) /dv dud(x,y) _< dy dz Jd(b,c) or putting _ dxjd(b,c) dx dyjd(b,c) db dc dy dz * du _ ~dz~ d(z,x) with two similar equations f d(y,z] S~T, s f t d(y,z) d(z,x) Therefore dx dx ] ~db dy_ dz dx ~ where d(a,b,c) or, since J = -^2-, therefore P da da 2h Po p db lib dz_ ~db dx dc ~dc dz Po (4), (5), (6). Consequently if ,77, are ever zero they are always zero, and then dd> dd> dd> u = ~ , v = -j-, 10 = , ax ay dz and a velocity function $ exists. For instance, if motion be generated from rest in a non-viscous fluid under forces due to a potential, a velocity function always exists, and the discovery of this velocity function for different cases is one of the chief problems to be solved in hydrodynamics. A good example of the use of the Lagrangian equations of motion is given by the state of wave motion in deep water invented by Rankine ; he puts
- = a + ce c sin ( cat +
c cos at + and therefore the coordinates of a particle are given in terms of t and a and /3. But a and /3 are not the initial coordinates of a particle ; for putting t = 0, then the coordinates are Therefore and = 1 ,, ,. therefore d(x.y) , -rr-^ > = 1 ! d(a,b)
and the equation of continuity is satisfied.
