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,