1916]	HORIZONTAL LAYER  OF  FLUID	445
If for the moment we omit the terms of the second order, we have
ac, J_0' + M_0 J_0" {oi_n cos n& + j3_n sin nQ] + J_n' [A_n cos n6 -f B_n sin n6] = 0; (62)
so that                                           Jo (z) = 0,
and	M_0 Jo" . 0^ + Jn . 4_n = 0,        mo Jo" • £n + Jn- -B_n = 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 A_n, B_n in terms of the known quantities a_n, @_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
A_0 {2J-_0' + P'JV"(a_n^2 + £„»)] + [kJ_n"-n*J_n/<u} K4_B +£_nJB_BJ = 0, or on substitution of the approximate values of A_n, B_n from (63),
Jo  = P^2 («w^2 + fin) ] -ft (Jn'	—	n~\	(64)
(J_n   \	&   )	2   J	'
This expression may, however, be much simplified.    In virtue of the general
equation for J_n,
rt^1	J '
T  ft	T	W>	T
" n	£ ™ n ^=         ~J n >
and since here J_0' = 0 approximately,
r //	 	   t	r /// _ 	 „—i   r it	   —j   r
Thus	J_0'(z) = P^2J_0. 2 («,i^2 + /9_W^2) 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 J_0'(/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=-7_r4r_/ = A?S(a_n^l + £n^a) J5y7 + T-[	(66)
Again, if a + da' be the radius of the true circle which has the same area as the approximate circle
]
da' = j^1 2 («_n^2 + fin^2)>	(67)
and	do! — da = — ^  ^n _9        ~r^T \ > 	(68)
2a      J_n {z)
where z is the first root (after zero) of J_0' (z) = 0, viz. 3'832.