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```1916]                               HORIZONTAL LAYER  OF  FLUID                                   445
If for the moment we omit the terms of the second order, we have AC, J0' + M0 J0" {oin cos n& + j3n sin nQ] + Jn' [An cos n6 -f Bn sin n6] = 0; (62) so that                                           Jo (z) = 0, and            M0 Jo" . 0^ + Jn . 4n = 0,        MO Jo" • £n + Jn- -Bn = 0.......(63)
To this order of approximation z, — ka, has the same value as when p = 0; that is to say, the equivalent radius is equal to the mean radius, or (as we may also express it) k may be regarded as dependent upon the area only. Equations (63) determine An, Bn in terms of the known quantities an, @n.
Since Jo' is a small quantity, cos \$ in (61) may now be omitted. To obtain a corrected evaluation of z, it suffices to take the mean of (61) for all values of 6. Thus
A0 {2J-0' + P'JV"(an2 + £„»)] + [kJn"-n*Jn/<u} K4B +£nJBBJ = 0, or on substitution of the approximate values of An, Bn from (63),
Jo  = P2 («w2 + fin) ] -ft (Jn'------—----n~\.............(64)
(Jn   \            &   )       2   J                       '
This expression may, however, be much simplified.    In virtue of the general
equation for Jn,
rt1                J '
T  ft                   T                       W>            T
" n           £ ™ n =         ~J n >
and since here J0' = 0 approximately,
r //__ __   T           r /// _ __ „—i   r it__   —j   r
Thus                         J0'(z) = P2J0. 2 («,i2 + /9W2) I-?1/ + 5-r,...............(65)
(yn        ^J
the sign of summation with respect to n being introduced.
Let us now suppose that a + da is the equivalent radius, so that J0'(/ca+ kda) = Q, that is the radius of the exact circle which corresponds to the value of k appropriate to the approximate circle. Then
and                       da=-7r4r/ = A?S(anl + £na) J5y7 + T-[.............(66)
Again, if a + da' be the radius of the true circle which has the same area as the approximate circle
] da' = j1 2 («n2 + fin2)>........................(67)
and                               do! — da = — ^  n 9        ~r^T \ > ..................(68)
2a      Jn {z)
where z is the first root (after zero) of J0' (z) = 0, viz. 3'832.dition w = 0, we may convince ourselves that the value of Jc* for the hexagon cannot differ much from that appropriate to a circle of the same area. Thus if a be the radius of this circle, k is given by J0' (ka) = 0,
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