a Rectangular Sheet of Rotating Liquid. 299 



or, since the A's are odd functions of y, 



A 2 = B 2 smh (\/3 .?/), A 4 = B 4 sinh (n/15 .y), &c. 

 Also A =B sin?/. 



In these equations the B's are absolute constants. 



The boundary conditions at ?/ = +yi now take the form 



= — cos x — 15 cos y x 



(7 



5- B 2 cosh (>v/3 . y x ) . cos 2x 



z 



— B 4 cosh (\/15. yi) .cos 4=x — , . (11) 



which can be satisfied if cos x be expressed between the 

 limits of x in the series 



cos a?=C + 2 cos 2a? + C 4 cos 4# + (12) 



By Fourier's theorem we find that (12) holds between x= — \ir 

 and x— + \iT) if 



2 4 4 4 



0.= -, c 2 =^, 0*=-^, c 2m =-(-i)'» (4m2 _ 1)7r 



. . . (13) 

 The B's are thus determined by (11), and we get 



. _ 2m' 2 sin?/ . _ 2m 2 4 sinh (\/3 .y ) 



0— <r TTCOSyi . 2— cr ^oTrcosh (v^3 .^ 1 )^ 



_2m 2?rc 4( — l) m + 1 sinh(y\/4m 2 -l) 

 2ni ~ : V V (4m 2 -l) (4m 2 — 1)tt cosh^ v /4m 2 -l)- 



Hence, finally, for the complete values of ij and v to this 

 order of approximation 



2m J 8 sin 2a? sinh(i/3.y) 1 m , 



A + o- \3V3.7T cosh (4/3.^) + * "J ' * ' ^ } 



2m f 2 cos ?/ 4 cosh ( V'S . y) 1 



v— *> cos a? * g-cos2a; — u / /o \ + • • • r- 



CT |^ 7TCOS?/! 07T COSh ( \/ 3 . ?/i) J 



. . . (15) 



The limiting values of x have been supposed, for the sake 

 of brevity, to be +J 71 "* If we denote them by +# 1} we are 

 to replace x, y, y x in (14), (15) by ^ir x/x u Jw y/x h hiry^v^ 

 At the same time (4) becomes 



^w (lt>) 



w = cos 



