The London, Edinburgh and Dublin philosophical magazine and journal of science . )above in the following manner. First integrate along thepath of which PP is an element from an initial point A to afinal point B. We obtain from (6) XA~ —% 1 cogsq sin 6 ds — 1 w XB-XA= -* \ cogs/qsm6ds- \ ^ds. . (8) AB AB Now as q is the resultant velocity of the fluid at any pointP on the path AB, q ( = qcos0) is the component at Palong the tangent drawn there to AB, while q sin 6 is thevelocity with which each particle P on the path AB is beingcarried by the motion towards the right (see fig. 2) in theplane of
The London, Edinburgh and Dublin philosophical magazine and journal of science . )above in the following manner. First integrate along thepath of which PP is an element from an initial point A to afinal point B. We obtain from (6) XA~ —% 1 cogsq sin 6 ds — 1 w XB-XA= -* \ cogs/qsm6ds- \ ^ds. . (8) AB AB Now as q is the resultant velocity of the fluid at any pointP on the path AB, q ( = qcos0) is the component at Palong the tangent drawn there to AB, while q sin 6 is thevelocity with which each particle P on the path AB is beingcarried by the motion towards the right (see fig. 2) in theplane of the diagram. The product q sin 0 dsf is thereforethe rate at which an area of which ds is an element ofboundary, and which is situated to the left of AB, is in-creasing (or if the area is situated to the right of AB, isdiminishing) in consequence of the motion of ds as a whole, atright angles to itself, in the plane of the diagram. We see,therefore, that the first term on the left is twice the rate atwhich the surface integral of elemental rotation, taken over Fior. Prof. A. Gray : Notes on Hydrodynamics. 5 any area of which AB is part of the boundary, is changingin consequence of the fact that each element ds of AB isbeing carried towards the right by the motion of the fluid. Fig. 3 shows the effect of thismotion for a closed path of inte-gration. The area between twostream-line elements ds, and thetwo positions of the connecting ds,is evidently dsds sin 6. The second term on the right isthe rate at which as time passesthe flow along ^B~4schangingapart from the mo£ion>Qf theelements ds. , If the path- AB beleft-hand side of (8) vanishes and we hAte^ where (j) denotes integration round the 7. Now since the line integral xqds taken rouncclosed path is twice the surface-integral of elemental rotationabout normals drawn to the elements into which any surfacebounded by the path may be divided, if we calculate thewhole change of rate of flow along AB due to the variousc
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