412 Mr. STOKES'S SUPPLEMENT TO A MEMOIR 



to determine the motion of the box it will be necessary to find the resultant of the fluid pressures on 

 its several faces. As shown in my former paper, these pressures will have no resultant force, but 

 only a resultant couple, of which the axis will evidently be parallel to that of x. In calculating 

 this couple, it is immaterial whether we take the moments about the axis of x, or about a line 

 parallel to it passing through the centre of the parallelepiped : suppose that we adopt the latter 

 plan. If we reckon the couple positive when it tends to turn the box from the axis of x to that 



of y we shall evidently have - Jo fo P^^o (^ ]d,vdx for the part arising from the pressure on 



the face .vx, and f^ /„' p^^„ (y ) dydz for the part arising from the pressure on the face yz. 



It is easily seen from (4) that p^^^ = - p^,,, and p^^j = - Py,„, so that the couples due to the pres- 

 sures on the faces xx, y a are equal to the couples due to the pressures on the opposite faces 

 respectively. In order, therefore, to find the whole couple we have only got to double the part 

 already found. As the integrations do not present the slightest difficulty, it will be sufficient to 



write down the result. It will be found that the whole couple is equal to —C— , where 



_mrfc 

 1 -t- £ " 



This expression has been simplified after integration by putting for 2„ — its value — ; . 



It appears then that the effect of the inertia of the fluid is to increase the moment of inertia 

 of the box about an axis passing through its centre and parallel to the edge c by the quantity C. 

 In equation (40) of my former paper, there is given an expression for C which is apparently very 

 different from that given by (5), but the numerical values of the two expressions are necessarily 

 the same. If we denote the moment of inertia of the fluid supposed to be solidified by C , we 



shall have C = {a^ + h') ; and if we put 



12 



6='-' C=-^^''^' 



and treat (5) as equation (40) of my former paper was treated, we shall find 



/(r) = (1 + r-)-' \l -Sr" + 2r'(1.260497 - 1.254821 2o — versin 26,,)} (6), 



n 

 wliere, tab. log tan 9„= 10 - .6821882 - . 



r 



The equation (6) is true, (except as regards the decimals omitted,) whatever be the value of r; 

 but for convenience of calculation it will be proper to take r less than 1, that is, to choose for a 

 the smaller of the two a, b. The value of /(r) given by (6) is apparently very different from 

 that given at the bottom of page 134 of my former paper, but any one may easily satisfy himself 

 as to the equivalence of the two expressions by assigning to r a value at random, and calculating 

 the value of f{r) from the two expressions separately. The expression (6) is however preferable to 

 the other, especially when we have to calculate the value of f(r) for small values of r. The 

 infinite series contained in (6) converges with such rapidity that in the most unfavourable case, that 

 is, when r = 1 nearly, the omission of all terms after the first would only introduce an error 

 of about .000003 in the value of /(»•). 



