460 Sir William Thomson on the Stability of 



fluid will sometimes be considered as absolutely free in space 

 undisturbed by gravity or other force ; and sometimes we 

 shall suppose it to be held absolutely fixed. Bat more fre- 

 quently we may suppose it to be held by solid supports of 

 real, and therefore viscously elastic, material ; so that it will 

 be fixed only in the same sense as a real three-legged table 

 resting on the ground is fixed. The fundamental philoso- 

 phic question, What is fixity ? is of paramount importance 

 in our present subject. Directional fixedness is explained in 

 Thomson and Tait's * Natural Philosophy,' 2nd edition, Part I. 

 § 249, and more fully discussed by Prof. James Thomson in 

 a paper " On the Law of Inertia, the Principle of Chrono- 

 metry, and the Principle of Absolute Clinural Best and of 

 Absolute Rotation." For our present purpose we shall cut 

 the matter short by assuming our platform, the earth or the 

 floor of our room, to be absolutely fixed in space. 



3. The object of the present communication, so far as it 

 relates to inviscid fluid, is to prove and to illustrate the proof 

 of the three following propositions regarding a mass of fluid 

 given with any rotation in any part of it : — 



(I.) The energy of the whole motion may be infinitely in- 

 creased by doing work in a certain systematic manner on the 

 containing-vessel and bringing it ultimately to rest. 



(II.) If the containing-vessel be simply continuous and be 

 of natural viscously elastic material, the fluid given moving 

 within it will come of itself to rest. 



(III.) If the containing-vessel be complexly continuous and 

 be of natural viscously elastic material, the fluid will lose 

 energy ; not to zero, however, but to a determinate condition 

 of irrotational circulation with a determinate cyclic constant 

 for each circuit through it. 



4. To prove 3 (I.) remark, first, that mere distortion of the 

 fluid, by changing the shape of the boundary, can increase 

 the kinetic energy indefinitely. For simplicity, suppose a 

 finite or an infinitely great change of shape of the containing- 

 vessel to be made in an infinitely short time ; this will distort 

 the internal fluid precisely as it would have done if the fluid 

 had been given at rest, and thus, by Helmholtz's laws of vor- 

 tex motion, we can calculate, from the initial state of motion 

 supposed known, the molecular rotation of every part of the 

 fluid, after the change. For example, let the shape of the 

 containing-vessel be altered by homogeneous strain ; that is 

 to say, dilated uniformly in one, or in each of two, directions, 

 and contracted uniformly in the other direction or directions, 

 of three at right angles to one another. The liquid will be 

 homogeneously deformed throughout ; the axis of molecular 



