► 



ON SOME CASES OF FLUID MOTION. 411 



culate is that which arises from a motion of rotation of the box about an axis parallel to that of z 

 and passing through the centre of the parallelepiped. Consequently in applying (2) we must for a 

 moment conceive the axis of z to pass through the centre of the parallelepiped, and then transfer the 



origin to the corner, and we must therefore write w l.i; - -J ly ] for wxy. In the present case 



the cylindrical surface consists of the four faces which are parallel to the axis of a', and the remain- 

 ing faces form the plane ends. The motion of the face wy and the opposite face has evidently no 

 effect on the fluid, so that ^_, will be the part of (p due to the motion of the face xz and the opposite 

 face. The value of this quantity is given near the top of page 133 in my former paper. We have 

 then by the second of the formulae (2) 



njrb _ "jrw nirf' nny 



e " — e " 



the sign 2o denoting the sum corresponding to all odd integral values of n from 1 to co . This 

 value of <p expresses completely the motion of the fluid due to a motion of rotation of the box about 

 an axis parallel to that of z, and passing through the centre of its interior. 



Suppose now the motion to be very small, so that the square of the velocity may be neglected. 



It 



in finding — we may suppose the axes to be fixed in space, since by takimr account of their 



at J a 



motion we should only introduce terms depending on the square of the velocity. In fact, if for 

 the sake of distinction we denote the co-ordinates of a fluid particle referred to the moveable axes by 

 x, y\ while ,r', y denote its co-ordinates referred to axes fixed in space, which after differentiation with 

 respect to t we may suppose to coincide with the moveable axes at the instant considered, and if we 



denote the differential coefficient of <p with respect to / by I — J when go, y, t are the independent 



variables, and by — -^ when .r', y ■, t are the independent variables, we shall have 

 ■ dt 



ld<p\ d<p d(p dx d(p dy' d(h dai dy' ''^ 



\dfl^'di"*'d^''dJ'^d^'7~t'^dt'*'^'dt'^^'di '" 



dd> d(t> , , , , deb dd) , 



'or — , , —L mean absolutely tlie same as — ^ , — c , and are therefore equal to u, v respectively. 



dx dy' 

 Now — , — - , depending on the motion of the axes, are small quantities of the order w ; their 



values are in fact uiy, - wx; so that, omitting small quantities of the order w', we have 



Then, p denoting the part of the pressure due to the motion, we shall have p = - n -i Also 



(S) = 



(d(j)\ d(f) 

 ~ If ' 



We shall therefore find the value of j) from that of (h by merely writing - p — for w. In order 



* Ft may be very easily proved by means of this equation, i eW'ccl on tiic motion of llic box as the solid of wliitli tin.' moment 

 combined with tbe general cijualion which determines ;;, that of inertia is determined in this paper on the supposition that the 

 whether the velocity be great or small the Hviid will have the same nmtion is small. 



.3 2 



