in Moving Viscous Liquids. 471 



suppose that the liquid under consideration is incompressible; 

 we shall further neglect the effect due to external forces, such 

 as gravitation. After all, the part played by these auxiliary 

 hypotheses in our reasoning will be a small one. 



§ 2. The component of the velocity of M parallel to the 

 axis of z is zero. The two other components are 



u ~ ±2 sin® ; u=+</cos© (1) 



The double sign ± in these equations has the same mean- 

 ing as in equations (2) and (3) in the preceding paragraph ; 

 it must, of course, be taken in the same consecutive order 

 wherever it occurs. 



Denoting by F any function of the variables r 3 ©, we have 



BF_^BF ydF 



■bx ~r -dr ~V3@' ....</«; 



by ~ t ~dr + r»3"e K '' 



The equations (2), along with (1), enable the components of 

 the velocity of apparent f deformation to be calculated. 

 Among these components, those which we shall have to con- 

 sider are the following : — 



, = |!! = + (^_i\ sin @ cos@) . . . (3 ) 



dx —\dr r J 



/= f- = +('''/ _2\ gi „eco3 a ... (4) 



J 3« \dr r ) 



C ~%.^ 



§ 3. In a previous paper (to which reference has been 

 made) we gave the definition of what we term the true de- 

 formation. Let 



6*, </>*, f *, **, /3*, 7* .... (1) 



be the components, referred to the axes Ox, Oy, Oz, of this 

 deformation. We shall refer the same deformation to three 

 new axes Ox v 0y v 0z x chosen as follows : — the direction of the 

 axis O.r, is at every instant the same as that of the velocity g; 

 the axis 0//i is coincident with r ; and Oz i coincides with the 

 axis of rotation, i. e. with the axis Oz. Referred to these 

 axes, the actual deformation has for its components 



«t%#^ *,*,«£, 04* 7r ( 2 ) 



t For the justification of this term the reader is referred to our pre- 

 ceding paper, '" On the Laws of Viscosity," supra, p. 342. 



212 



