When the motion is such that d[phi] d[psi] d[phi] d[psi] d[phi] d[psi] u = - ------ - m------, v = - ------ - m------, w = - ------ - m------, (6) dx dx dy dy dz dz as in §25 (1), a first integral of the equations in (5) may be written _ / dp d[phi] d[psi] / d[phi] d[psi] \ | ----- + V + ½q² - ------ - m------ + (u - u´) ( ------ + m------ ) _/ [rho] dt dt \ dx dx / / d[phi] d[psi] \ / d[phi] d[psi] \ + (v - v´)( ------ + m------ ) + (w - w´)( ------ + m------ ) = F(t), (7) \ dy dy / \ dz dz / in which d[phi] d[phi] d[phi] d[phi] ------ - (u - u´)------ - (v -v´)------ - (w - w´)------ dt dx dy dz d[phi] d[phi] d[phi] d[phi] = ------ - (U - yR + zQ)------ - (V - zP + xR)------ - (W - xQ + yP)------ (8) dt dx dy dz is the time-rate of change of [phi] at a point fixed in space, which is left behind with velocity components u - u´, v - v´, w - w´. In the case of a steady motion of homogeneous liquid symmetrical about Ox, where O is advancing with velocity U, the equation (5) of § 34 p/[rho] + V + ½q´² - [f]([psi]´) = constant (9) becomes transformed into p U d[psi] ----- + V + ½q² - --- ------ + ½U² - [f]([psi] + ½Uy²) = constant, (10) [rho] y dy [psi]´ = [psi] + ¼U², (11) subject to the condition, from (4) §34, y^(-2)[nabla]²[psi]´ = -[f]´([psi]´), y^(-2)[nabla]²[psi] = -[f]´([psi] + ½Uy²). (12) Thus, for example, with [psi]´ = ¾Uy²(r²a^(-2) - 1), r² = x² + y², (13) for the space inside the sphere r = a, compared with the value of [psi]´ in §34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface. Inside the sphere d / 1 d[psi]´\ d / 1 d[psi]´\ 15 y 2[zeta] = --- ( --- ------- ) + --- ( --- ------- ) = ---U ---, (14) dx \ y dx / dy \ y dy / 2 a² so that §34 (4) is satisfied, with 15 15 [f]´([psi]´)= ---Ua^(-2), [f]([psi]´) = ---U[psi]´a^(-2); (15) 2 2 and (10) reduces to _ _ p 9 | / x² \² / y² \² | ----- + V - ---U | ( --- -1 ) - ( --- - ½ ) | = constant; (16) [rho] 8 |_ \ a² / \ a² / _| this gives the state of motion in M. J. M. Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity.