290 



JlK. STOKES, ON THE FRICTION OF FLUIDS IN MOTION, 



due to any motion of rotation may be eliminated without affecting the differences of the pressures 



above mentioned. 



Let us see how far this principle will lead us when it is carried out. 



2. It will be necessary now to examine the nature of the most general instantaneous motion 

 of an element of a fluid. The proposition in this article is however purely geometrical, and 



may be thus enunciated: "Supposing space, or any portion of space, to be filled with an 



infinite number of points which move in any continuous manner, retaining their identity, to 

 examine the nature of the instantaneous motion of any elementary portion of these points." 



Let u V w he the resolved parts, parallel to the rectangular axes Oai, Oy, Oz, of the 

 velocity of the point P, whose co-ordinates at the instant considered are .v, y, x. Then the 

 relative velocities at the point P, whose co-ordinates are a) + ic' , y + y', z + x', will be 



du 



d zi , du , 

 -— ai + — 2/ 

 dx dy 



H » parallel to so, 



dx 



dv dv , du , 



— X + 3-2/ + ^-i». 

 dx dy 



dss 



dw , dw , dw , 

 -— X + -—y + ^- X . 

 dx dy ^ ■^~ 



dz 



neglecting squares and products of x', y', z. Let these velocities be compounded of those due 

 to the angular velocities w', w", to'" about the axes of x, y, z, and of the velocities U, V, W 

 parallel to x, y, z. The linear velocities due to the angular velocities being w"z' - w'y , 

 J"x' - w'z, w y - u)"x' parallel to the axes of x, y, z, we shall therefore have 



du , 



U = -r "> + 

 dx 



du 



dy^"' J^ + U-"')"' 

 idn „,\ , d\> , Idv ,\ , 



Idw „\ , (dw ,\ , dw , 



Since w, u>", u>"' are arbitrary, let them be so assumed that 

 dUdV dV _dW dW _ dU 

 dy dic" dz' dy ^ dx' dz' 



which gives 



/dw dv\ „ /du du 



lu dw\ , 

 ^ \dy ^ dxj^ ^ ^ \dz dx I 



dv 

 dx 



du\ 

 Tyh 



....(1) 



dti , , /du dv\ , 

 U=^x +1(^^+ —I y + 



dx 



, idv du\ , dv , , /dv 



W ■ 



dyl 



/dw dw 

 ^\dx dz 



*' + i 



/dw dv\ , 

 Kdy'^Jz)'^ 



dw\ , 

 dy)^^ 



dw , 

 + -— «. 

 oar 



(2) 



The quantities w, id", w" are what I shall call the angular velocities uf the Jtuid at the 

 point considered. This is evidently an allowable definition, since, in the particular case in which 



