224 SIR W. THOMSON ON VORTEX MOTION. 



If 



7Q 



S t - ~ sin 6 (16 cty = , 



except when i' = i. But this is true also when i' = i because 



o d°i 17 (^1 ) 



1 d-4, - * a cty ' 



and therefore, as in § 14, the integration for \^, from \^ = Oto^ = 2-7r gives zero. 

 Hence (11) gives 



^ = 



(It ' 

 This and § 11 establish § 10. 



16. Lemma (1) of § 11, and § 10 now proved, show that in any motion whatever 

 of an incompressible liquid, whether with solids immersed in it or not, udx + 

 rdy + wdz is always a complete differential through any portion of the fluid, for 

 which it is a complete differential at any instant, to whatever shape and position 

 of space this portion may be brought in the course of the motion. This is the 

 ordinary statement of the fundamental proposition of fluid motion referred to in 

 § 9, which was first discovered by Lagrange. (For another proof see § 60.) I have 

 given the preceding demonstration, not so much because it is useful to look at 

 mathematical structures from many different points of view, but (§ 19) because the 

 dynamical considerations and the formulae I have used are immediately available 

 for establishing the theory of the impulse (§§ 3 . . . 8), of which a fundamental pro- 

 position was stated above (§ 5). To prove this proposition (in § 19) I now proceed. 



17. Imagine any spherical surfaces to be described round a moveable solid or 

 solids immersed in a liquid. The surrounding fluid can only press (§ 1) perpen- 

 dicularly; and therefore when any motion is (§ 3) generated by impulsive forces 

 applied to the solids, the moment round any diameter of the momentum of the 

 matter within the spherical surface at the first instant, must be exactly equal to 

 the moment of those impulsive forces round this line. And the moment round 

 this line, of the momentum of the matter in the space between any two concentric 

 spherical surfaces is zero, provided neither cuts any solid, and provided that, if 

 there are any solids in this space, no impulse acts on them. 



18. Hence, considering what we have defined as "the impulse of the motion," 

 (§ 6), we see that its moment round any line is equal to the moment of momen- 

 tum round the same line, of all the motion within any spherical surface having its 

 centre in this line, and enclosing all the matter to which any constituent of the 

 impulse is applied. This will still hold, though there are other solids not in the 

 neighbourhood, and impulses are applied to them : provided the moments of mo- 

 mentum of those only which are within S are taken into account, and provided 

 none of them is cut by S. 



19. The statements of §11, regarding fluid occupying at any instant a fixed 

 spherical surface, are applicable without change to the fluids and solids occupying 



