SIR WILLIAM THOMSON ON VORTEX MOTION. 237 



the equality to zero asserted in this proposition is proved in § 48 to be approxi- 

 mated to when the planes are extended to distances all round, which, though infi- 

 nitely short of the distances to the containing vessel, are very great in comparison 

 with their perpendicular distances from the most distant parts of the field of 

 motion. 



50. The convergencies concerned in § 46, 1., III. may be analysed thus. Per- 

 pendicular to the resultant impulse draw any two planes on the two sides of the 

 field of motion, with all the moving solids and vortices between them, and divide 

 a portion of the space between them into finite prismatic portions by cylindrical 

 (or plane) surfaces perpendicular to them. Suppose now one of these prismatic 

 portions to include all the moving solids and vortices, and without altering the 

 prismatic boundary, let the parallel planes be removed in opposite directions to 

 distances each infinite (or very great) in comparison with distance of the most 

 distant of the moving solids or vortices. By § 46, 1., the momentum of the motion 

 within this prismatic space is (approximately) equal to the force-resultant, I, of 

 the impulse, and that of the motion within any one of the others is (approximately) 

 zero. 



But the sum of these (approximately) zero values must, on account of § 46, 

 III., be equal to — I, if the portions of the planes containing the ends of the 

 prismatic spaces be extended to distances very great in comparison with the dis- 

 tance between the planes. To understand this, we have only to remark that if </> 

 denotes the velocity potential at a point distant D from the middle of the field, 

 and x from a plane through the middle perpendicular to the impulse, we have 

 (§ 53) approximately, 



(15), 



4*-D 3 



provided D be great in comparison with the radius of the smallest sphere enclos- 

 ing all the moving solids or vortices. Hence, putting x = ± a for the two planes 

 under consideration, denoting by A the area of either end of one of the prismatic 

 portions, and calling D the proper mean distance for this area, we have (§ 46) for 

 the momentum of the fluid motion within this prismatic space, provided it con- 

 tains no moving solids or vortices, 



- 2 4VT? A ( 16 > 



This vanishes when ^ is an infinitely small fraction (as j. is at most unity) ; but 



A a 



it is finite if ™ is finite, provided y. be not infinitely small. And its integral 



value (compare § 48, footnote) converges to — I, when the portion of area in- 

 cluded in the integration is extended till tt is infinitely small for all points of its 



boundary. 



vol xxv. part i. 3 P 



