so that if H。= 21 ft., A/a = 17,000. In the case of the first Ho set of Prof. Gibson's curves, where the period of the tide was taken as 13 hours, the corresponding value of A/a would be 8.15.101 H2-1/2, so that if Ho=8, the best value of A/a is 28,800. By inference from his graphical results, Prof. Gibson estimated these two ratios of A/a as 16,000 and 30,000 respectively. $5. It remains to consider how far the neglect of the higher harmonic terms in h is likely to affect the estimate of the available power. The coefficients of the neglected third and fifth harmonics in 12 in (19) are 1/7 and 5/77 times the coefficient of the leading term in 2, and it would seem as if the odd harmonics in the true value of h bear smaller ratios than these to the leading term, while the even harmonics are small, if not zero. The coefficients of the second harmonics in H' or h' corresponding to the four values of A/a in Prof. Gibson's first set of curves are 0.03, 0·04, 0·00, 0·06, while the corresponding coefficients of the third harmonics are 0.39, 0.28, 0.57, 047; part of these may be due to slight errors in measuring the curves. Now the utmost extent to which the mean value of h can be affected by a harmonic of frequency n and amplitude a, is easily seen. to be a/n. In the cases cited this is quite negligible. Even for the spring-tide curves discussed in §3, where H was itself not a pure sine-function (ef. 32) the second and third harmonic terms in h, as derived graphically, were only 147 and 0.70 (A/a=12,000) and 0·70, 0·63 (A/a = 16,000). The second harmonics would in these cases affect the mean value of h very little, because the periods of inflow and outflow for the reservoir are very nearly equal. Thus, though the analysis of this paper could be extended, if necessary, so as to afford an approximation to at least the next most important harmonic component in h after the first, there appears to be no practical necessity for so doing. Summary. §6. The difference of height, h, between (i.) a sea or estuary undergoing a tidal change of water-level, in half a lunar day, represented by Ho sin at, and (ii.) a tidal basin of area A connected with the sea by sluices of area a, is represented by the approximate formula h=h, sin (at + €), where ho and e are given by h = (H ̧2+h2)1/2 — k, tan e=(2k/h)1/2 0 in terms of the number k, where k=2·018. 10° (a/A)2. The ratio of A to a which corresponds to the maximum power-development from this difference of head is given by A/a 7.79.10 H-12, where Ho is supposed measured in feet. VIII. Note on the Magnetic Field produced by Circular Currents. By T. J. I'A. BROMWICH, Sc. D., F.R.S., Fellow and Prelector in Mathematical Science, St. John's College, Cambridge. PROE ROF. H. NAGAOKA has investigated these magnetic fields by means of q-series in a recent number of this journal; but it does not seem to have been noticed that his results can be found in a very elementary way by using Legendre's theorem that the potential of a symmetrical system can be determined when the form of the potential is known along the axis of symmetry. 1. The field of a single circle (of radius a).-At points on the axis of the magnetic force due to unit current in the circular wire is equal to provided that is numerically less than a. • Communicated by the Author. -...), Now the component of magnetic force in any fixed direction satisfies Laplace's equation; and in particular this is true of y, the component parallel to the axis of symmetry. Thus at points not on the axis the magnetic force y is given by the series The formula (1) agrees with Prof. Nagaoka's (18), (19) ;. to obtain (20) we must include the following term in the bracket, which is It will be observed that the coefficient of p is apparently erroneous in Prof. Nagaoka's formula. The value of the magnetic potential V can now be found, by integrating the equation av 2 This gives, at points on the axis, V。- V=2π 0 ( 2 1 23 1.3 25 1.3.5 + a 2 a1 2.4 a 2.4.6 a +...), where V, is a constant; and so, at any point, we have theformula When the last formula is written out in full we obtain the Thus on differentiation with respect to p we find the transverse component of the magnetic field The formula (4) agrees precisely with (15), (16), and (17) of Prof. Nagaoka's. 2. Field near the centre of Helmholtz's galvanometer (double coils). Let the section be represented by the adjoining diagram (fig. 2) in which a=2c, and 12=a2+c2=5c2. Since the field is symmetrical about the line of centres AA' (which we take as Oz), it will suffice to calculate the field at a point P on the axis (and near to 0). The magnetic force at the point P due to unit current in the coil A is known to be Now we have 2 + C)2 + (4) * . a2 + (e + :)2 = l2 + 2c; + ÷ 2= (1 + ©) * Thus, since a/12 is small, we can write To obtain the same degree of accuracy as would be given by Prof. Nagaoka's formulæ, we must expand as far as terms in. Then the result is The magnetic force y2, due to the same current in the second coil, is given by changing the sign of in y1; and so the complete field at P is こ correct as far as terms in . So far we have not used the special feature of the Helmholtz arrangement: namely, the relation a=2c, which provides for the disappearance of the second term in y. Inserting this value, we find that on the axis the magnetic field is given by the approximate formula To obtain the field at other points we must calculate first the magnetic potential at P; this is given by Accordingly the value of V at any other point near O is given by the approximate formula |