568 Mr. stokes, on THE CRITICAL VALUES OF 



with the particular conditions 



— - = w(a; — ia), when y = or = b (76), 



dy 



d(p 

 dm 



= - (o(j/ - ^6), when a; = or = a (77). 



This is the problem in pure analysis to which we are led in seeking to determine the motion 

 of a liquid within a closed rectangular box which is made to oscillate. 



For a given value of y, the value of (p can be expanded in a convergent series of cosines of 



— and its multiples; for another value of y, <p can be expanded in a similar series with different 

 a 



coefficients, and so on. Hence, in general, (p can be expanded in a convergent series of the form 



„ mra: 



SF„cos (78), 



a 



where Y„ is a certain function of y, which has to be determined. 



In the first place the value of (p given by (78) must satisfy (7.5). Now the direct developement 

 of -— ^ in a series of cosines will be obtained from (78) by differentiating under the sign of sum- 

 mation ; the direct developement of ---^ will be given by the formula (D). We thus get 



^frf-'F,, w'^ir- „ 2a), , ^ , ^ ,,1 nvx 



-\df- a- h.^~{l-i-m(y-^b)^cos^=0; 



and the left-hand member of this equation being the result of directly developing the right-hand 

 member in a series of cosines, we have 



dy- a' a '' 



according as ti is odd or even. This equation is easily integrated, and the integral contains two 

 arbitrary constants, C,„ D„ suppose. It only remains to satisfy (76). Now the direct developement 



dV, . 

 of —— will be obtained by differentiating under the sign of summation, and the direct develope- 

 ment of w(x - ^a) is easily found to be - 2o — ;— j cos , the sign 2o denoting that odd values 



only of n are to be taken. We have then, both for y = and for y = h, 



dV„ 4(0 o 



dy IT w 



according as n is odd or even, which determines C„ and D„. 



It is unnecessary to write down the result, because I have already given it in a former paper*, 

 where it is obtained by considerations applicable to this particular problem. The result is con- 

 tained in equation (4) of that paper. The only step of the process which I have just indicated 

 which requires notice is, that the term containing {x - lo) (y - ^6) at first appears as an infinite 



' Supplement to a Memoir 'On some Cases of Fluid .Motion,' p. 401* of the present Volutue. 



