134 Mr. stokes, on SOME CASES OF FLUID MOTION. 



- p-^y\^(p, a,y, b) - yj^Cy, b, 0, a)}dydz. 

 ' dt 



Substituting the values of the functions, integrating from y = to y = 6, and from ^ = to « = c, 



1 IT* 



renlacine- 2 bv its value — , and reducing the other terms, it will be found that the couple 



due to the pressure on the plane yz \i 



pa'bc doT _ Spa'c dm" J_ 1 - e'^ _ Spb'c dw'" ^ ^ IjU^ 

 24 dt ir^ dt n ^^^-— -k dt n ^ ^ ^—r 



AVe shall get the couple due to the pressure on the plane xx by interchanging a and 6, 



changing the sign of ui", and measuring the couple in the opposite direction, or, which is the same, 



by merely interchanging a and 6. Adding together these two couples and doubling their sum 



, duJ" . _ dw" . 



we shall find that the couple due to -— - is - C -^ , where 



1 + e " 1 + £ * 



will be — .4 — — 



dt dt 



it ^ t 



Similarly, the couple due to - — will be -A——, tending to turn the box from y to x, and 

 •" ' dt dt 



fl " d " 



that due to — ^ will be - B , tending to turn the box from z to w, where // and B are 



dt dt 



derived from C by interchanging the requisite quantities. Hence, considering the motions both 

 of translation and rotation of the box, we see that the small motions of the box will take place 

 as if the fluid were replaced by a solid having the same mass, centre of gravity, and principal 

 axes, and having A, B and C for its principal moments. This will be true whether forces act 

 on the fluid or not, provided that if there are any they are of the kind mentioned in Art. 1. 



Putting A , B , C^, for the principal moments of inertia of the solidified fluid we have 



o(tbc 

 12 ^ ' 



Taking the ratio of C to C , replacing each term such as 



\-e~~ , 2e~ - . „ 384. 1 

 ^ by 1 j^, putting for -^^0-, 



1 + e~ n 1 + e" » 



, „ 384 . . , J , . , 



its approximate value f260497, and for — - its approximate value 1 ■254821, and employing sub- 

 sidiary angles, we have 



C ^ a' + b 



— = 1-260497 -TT-, .,, 



C ab^a' + b^) 



1-260497 ", 7 "^ .^ - 1*254821 1--^ — — S„ - versin 20„ 



+ -7-1 — F^ ^o - versin 2 &„ \-\, 

 a {a' + b') n J 



nnb n T « 



where tan 0„ = e" ^n , tan 6'„ = e~ ■'i' , so that 



L tan 9„ = 10 - k — , L tan ff„ = 10 - A; — , where k = -6821882. 

 a b 



