a higher order of greatness at one of the limits of integration, while n'n at another of the limits. These considerations reduce the important terms in the equivalent of the square bracket in formula (26) to 00 4π 2kg sinh/47n \ cos($ +yz) X exp{i(-) λ λ which reduces, after omission of some negligible elements, to a form whose limit, for to, on substitution in (26) yields the formula 11. The difference motion. If formula (27) be compared with formula (13) and (14) it is seen that v2+iu2−(v1 + iu1) >0. Hence if (u, v) specify the difference motion, so that Thus the difference motion is an irrotational motion, vanishing at infinity and having at the boundary a normal velocity corresponding to rotation about the point x exp (iy). It is free from circulation, as a circulation would involve, for 7 infinite, a definite limit value of u different from zero. It therefore constitutes the solution of the problem of motion due to the rotation of the boundary. 12. Forms of boundary to which the method applies.—The applicability of this method to solving the problems of motion due to translation and rotation depends upon the knowledge of a periodic conformal transformation which will make any particular form of boundary correspond to the real axis in the plane. That a considerable variety of such transformations and their corresponding boundaries is Phil. Mag. S. 6. Vol. 35. No. 205. Jan. 1918. K available is demonstrated in the writer's paper on the subject referred to above. In particular, mention may be made of polygonal boundaries (l. c. § 8); and it may be noticed that for a regular polygon of n sides the transformation is K being a constant; the latter expression, with accented letters, would be the first factor under the sign of integration in formula (25). In all cases where the periodic transformation is known the solution of the hydrodynamical problems is reduced to quadratures. In certain cases the integrations can be completed; this is noticeably the case when f(), and therefore also h2, is the sum of a finite number of terms harmonic in . The integration may be accurately effected by the method used for approximation in article 10 above. Of the terms arising from the multiplication of h2 into the series [22] and [23] there are only a finite number which do not yield zero result when integrated with respect to through a range λ, and each of these can be integrated separately with respect to n'. The simplest example is the ellipse, for which the transformation is so that h2 = z = ccosh {a-(2πi/λ)5}, 2(x+ η 22 {cosh 2 ( a +27) - cos()}.. (32) λ λ The working out of this case may be used to test the method, as the results are otherwise known. Another simple integrable case corresponds to a boundary whose polar equation is r = a + 2b cos 20, (a>2b). The transformation is (33) z = b exp(-2ig/λ) + a exp (2πig/λ) + b exp (6πig/X). (34) 12th November, 1917. XIII. On the Relation of the Audibility Factor of a Shunted Telephone to the Antenna Current as used in the Reception of Wireless Signals. By Prof. G. W. O. HOWE, D.Sc., M.I.E.E.* [See paper with the same title by M. van der Pol, vol. xxxiv. T p. 184.] HE audibility factor of a radio-telegraph signal is defined as the ratio of the actual sound-producing current in the telephone-receiver to the minimum value to which this current could be reduced for the signals to remain just readable. It is assumed that the wave-form of the telephone current, and therefore also the character of the sound, remain the same in the two cases. This ratio is usually determined by shunting the telephone-receivers with a noninductive resistance until the signals are only just readable. If there is any possibility of the total rectified current being affected by the decreased resistance of the detector circuit due to the addition of the shunt, a resistance should be inserted in series with the shunted receiver to maintain the total resistance of the detector circuit approximately constant. From the value of the shunt it is then necessary to calculate the ratio of the total or joint current to that through the receiver. It is not clear from Mr. van der Pol's paper how he determined the resistance of the receiver which he gives as 1240 ohms; but since nothing is stated to the contrary, it would appear that he has treated the receiver as a noninductive resistance equal in value to the actual resistance of the receiver to continuous current. If so, the results obtained will be in error for two reasons: firstly, because the effective resistance of a telephone-receiver at the frequency employed, viz. 467, is considerably greater than its resistance to continuous current; and secondly, because an alternating current divides between two alternative paths in a manner depending on the impedances and not on the resistances. As an example of the magnitude of the error thus introduced, the following figures may be quoted: a 3200-ohm receiver had an effective resistance R at 750 cycles per second of 6200 ohms and an impedance Z of 9320 ohms, whilst at a frequency of 1000 these values were increased to 7250 and 11,200 ohms respectively. Thus Z750=2·9 R。 and Communicated by the Author. Z1000=3.15 Ro. In the case of a 60-ohm receiver, it was found that Z637=4·27R。 and Z1115=6·35 Ro. As a rule at such frequencies the reactance is of the same order as the effective resistance, so that the current lags about 45° behind the terminal P.D.; this is, of course, merely a rough approximation. There may be some doubt as to the correctness of treating the pulsating telephone current as a simple alternating current; but in the opinion of the writer, the pulsating current of audible frequency produced by the detector as the result of the successive wave-trains may be regarded as a steady current with a fundamental alternating current and a number of harmonics superposed upon it, the fundamental giving the pitch, and the harmonics the character of the sound heard in the receiver. If the character of the note remain constant, it would appear sufficient to consider the amplitude of the fundamental, and to assume that this sinusoidal current divides between the receiver and the shunt in accordance with the ordinary laws of alternating-current circuits. The writer is well aware that references can be given to papers in which the ordinary continuous current-resistance of the receiver was apparently used in calculating the audibility factor, but in a recent paper Austin, who has done much experimental work on this subject, is careful to point out that the effective resistance of the receiver must be determined for the given frequency and telephone pulse form*. Since Mr. van der Pol refers to papers by Hogan and Love, both of whom refer to the impedance and not the resistance of the telephone-receiver, it is possible that he has also used the impedance, notwithstanding the statement in his paper. If so, the paper would be of greater value and interest were this definitely stated. If Mr. van der Pol did not take the precaution to keep the resistance of the detector circuit approximately constant, as mentioned above, the correctness of his experimental results is open to some doubt. In order to see in what direction his results would be modified by employing the impedance of the receiver instead of the resistance, it has been assumed in the following table that the impedance Z is equal to four times the resistance Ro, and that the telephone current lags 45 degrees behind the P.D. The values obtained from the simple vector diagram are as follows: *Proc. Inst. Radio-Engineers, 1917, v. p. 239. In view of the unavoidable lack of precision in all audi- Z+S as the The figure is a reproduction of fig. 2 in Mr. van der Pol's paper, with the addition of two dotted lines which represent log M plotted against log I instead of log Ro+S For the upper dotted line it has been assumed that Z=3Ro, whilst for RYD 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (total) log (receiver) 1.6 1.8 2.0 2.2 2.4 2.6 the lower Z=4R。 as in the table above. It is seen that if the |