1915] NOTE ON THE FORMULA. FOR THE GRADIENT WIND 277 and dv dv dv „ ,, _ • +u- -- f- fl _ -|- 2 6)16 — CO-?/*. at dx dy J Since the relative motion is supposed to be steady, du/dt, dv/dt disappear, and the dynamical equations are 1 dp 2l0 du du n. — f- = &ro3 + 2wv —u-^ -- v -T~ . ..................... (1) p dx dx dy I dp dv dv /0. -- r- = ft)2?/ — 2d)U — U^j -- V -r- ...................... (2) p dy J dx dy N The velocities u, v may be expressed by means of the relative stream-function -\/r : u = dtyjdy, v — Equations (1), (2) then become 1 4 * _i*J(**)V*m+vv.^, ...... W 2 cfcc (V<&»/ \dyJ } r dx -+ + V^.; ...... (4) p dy dy 2 dy \dxj \dy) f Y rfy and on integration, if we leave out the part of p independent of the relative motion, in which by a known theorem V2-v|r is a function of ^ only. If <a be omitted, (5) coincides with the equation given long ago by Stokes f. It expresses p in terms of •x/r ; but it does not directly allow of the expression of -\|r in terms of p, as is required if the data relate to a barometric chart. We may revert to the more usual form, if in (1) or (3) we bake the axis of x perpendicular to the direction of (relative) motion at any point. Then u = 0, and 1 dp dty dty /ftN - -/- = 2wv + -3- —^ ......................... (6) p dx dx dy* But since dty/dy = 0, the curvature at this place of the stream-line (^ = const.) s ^ . ~ r " dyz ' dx' and thus — f = 2cov + — , .............................. (7) p dx ~ r * Lamb's Hydrodynamics, § 206i f Carnb. Phil. Trans. Vol. vn. 1812 ; Math, and, Phys. Papers, Vol. i. p. 9.t yet been proved theoretically that the stability can fail for finite disturbances on the supposition of perfectly smooth walls ; but such failure seems probable. We know from the work of Keynolds, Lorentz, and Orr that no failure of stability can occur unless @D*jv > 177, where D is the distance between, the walls, so that j3D represents their relative motion.