FLUIDS AND THE DETERMINATION OF THE CRITERION. 161 
Having regard to the boundary conditions, &c., and omitting all possible terms 
which increase the numerator without affecting the denominator, the most general 
form appears to be 
@ = Y" (Qe. sin (25 + 1) p], ) 
B = 3," [by sin (2ép)}, ea rat ieeene ene anes 9" (08) 
where | 
P = ty/2by J 
To satisfy the boundary conditions 
s = 2r, when s is even, s = 2r + 1, when s is odd. 
t= 2r + 1, when ¢ is odd, t = 2(r + 1), when ¢ is even. 
Since a = 0, when p= + 37, >) 
| 
| 
yo (Gs 41 s Csr43) = 0, 
| GD: 
and since dB/dy = 0, when p = + $7, : | 
ay (4r or 2) Dir + 4 (r+ DV Oiyat — 04 
From the form of K, it is clear that every term in the series for a and B increases 
_ the value of K, and to an extent depending on the value of rv. K, will therefore be 
minimum, when 
(70), 
a= ad, sin p + a, sin 3p 
B= b, sin 2p +b, sin 4p | 
which satisfy the boundary conditions if 
dies Bagg Nati us Sus aU ati muse Des tee (Yet 
by = a) ay) 
Therefore we have, as the values of « and £, which render K, a minimum for 
any value of l 


a/a,=sinp-+sin 3p, B/b,= sin 2p + ¥sin 4p. 5) 
And | 
2by da 2b), dB _ a3 
Rag = CEP + 3 cos 3p, BR Fc 2 cos 2p + 2 cos 4p (72) 
2b, [adB Bdz 1 ; ; : ; | 
—_— —_— g — —— y, 2 
rap, ( dy dy ) aaa 3 sin p — 3 sin 3p + sin 5p + sin 7ps | 
MDCCCXCV.—A. Y 
