SIR WILLIAM THOMSON ON VORTEX MOTION. 257 



fluid, Let <p q> be the velocity potentials of two motions fulfilling the prescribed 

 conditions, and let 



At every point of the boundary (the barriers not included) the prescribed con- 

 ditions require that &p = U<p', and therefore tn// = 0. Again, the cyclic constants 

 for q>' are equal to those for <p ; those for xjj, being their differences, must there- 

 fore vanish. Hence, if the <p and <p' of § 57 (7) be made equal to one another and 

 to avoid confusion with our present notation we substitute xp for each, the second 

 members of that double equation vanish, and it becomes simply 



which, as before (§ 61), proves that i// = 0, and therefore <p' = p; and so establishes 

 our present proposition. 



Example (1). The solution <p = tan - considered in § 56, fulfils Laplace's equa- 

 tion, v 2 P = 0; and obviously satisfies the surface condition, not merely for the 

 annular space with rectangular meridional section there considered, but for 

 the hollow space bounded by the figure of revolution obtained by carrying a 

 closed curve of any shape round any axis (OZ) not cutting the curve ; which, for 

 brevity, we shall in future call a hollow circular ring. Hence the irrotational 

 motion possible within a fixed hollow circular ring is such that the velocity poten- 

 tial is proportional to the angle between the meridian plane through any point, 

 and a fixed meridian. 



Example (2). The solid angle, a, subtended at any point {x, y, z), by an 

 infinitesimal plane area, A, in any fixed position, fulfils Laplace's equation y 2 a= 0. 

 This well-known proposition may be proved by taking A at the origin, and per- 

 pendicular to OX, when we have 



Ax -A d =i (5) 



{x 2 +y 2 + « 2 )# _ ch (x 2 + y 2 + z 2 )\ 



for which V 2 <* = is verified. 



The solid angle subtended at (cc, p, z) by any single closed circuit is the sum 

 of those subtended at the same point by all parts into which we may divide any 

 limited surface having this curve for its bounding edge. [Consider particularly 

 curves such as those represented by the first three diagrams of § 58.] Hence 

 if we call <f> the solid angle subtended at (#, y, z) by this surface, Laplace's equa- 

 V 2 <£ is fulfilled. Hence (f> represents the velocity potential of the irrotational 

 motion possible for a liquid contained in an infinite fixed closed vessel, within 

 which is fixed, at an infinite distance from the outer bounding surface, an in- 

 finitely thin wire bent into the form of the closed curve in question. 



VOL. XXV. PART II. 3 U 



