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high school mathematics be required for graduation by all high schools in the state of Missouri.

Professor R. A. WELLS of Park College was elected Chairman and Mr. W. A. LUBY, of the Kansas City Polytechnic Institute, Vice-Chairman for the coming year. It was voted that the office of Secretary-Treasurer be considered permanent, and the present officer was reëlected.

It was voted that a committee be appointed by the new chairman to act on the reports of the National Committee on Mathematical Requirements. This committee has been appointed as follows: Professor E. R. Hedrick, University of Missouri, chairman; Professor W. H. Zeigel, State Normal School, Kirksville; Professor R. R. Fleet, William Jewell College; Miss Zoe Ferguson, Central High School and Junior College, St. Joseph; Mr. Alfred Davis, Soldan High School, St. Louis; Mr. Percival Robertson, The Principia, St. Louis.

At the joint meeting on Tuesday the following addresses were given:

I. "Recent advances in dynamics." Address of the retiring Vice-President of Section A of the A. A. A. S., PROFESSOR G. D. BIRKHOFF.

II. "Some recent developments in the calculus of variations."

Address of the

retiring chairman of the Chicago Section of the American Mathematical Society, PROFESSOR G. A. BLISS, University of Chicago.

III. "A suggestion for the utilization of atmospheric molecular energy." MR. H. H. PLATT, Philadelphia.

At the regular session of the Missouri Section the following seven papers were read:

(1) Opening address as President of the Association, PROFESSOR H. E. SLAUGHT, University of Chicago.

(2) "The determination of logarithmic formulas," PROFESSOR E. R. HEDRICK, University of Missouri.

(3) "A Simple Treatment of Fourier's Series," PROFESSOR LOUIS INGOLD, University of Missouri, and MR. T. W. JACKSON, Jamestown College, N. D. (4) "An Elementary Method of Quadrature," PROFESSOR OTTO DUNKEL, Washington University.

(5) "Plans of the National Committee on Mathematical Requirements," MR. CHARLES AMMERMAN, McKinley High School, St. Louis.

(6) "Preliminary report of the National Committee on Mathematical Requirements," PROFESSORS HEDRICK, ZEIGEL, and FLEET, MISS ZOE FERGUSON and MR. ALFRED Davis.

(7) "Geometric treatment of certain optical problems," (Illustrated by lantern views and models.) PROFESSOR WM. H. ROEVER, Washington University, chairman of the Section.

In the absence of the authors the paper by Professor Ingold and Mr. Jackson was read by Professor Hedrick. Abstracts of the papers, except (5) and (6), follow below, the numbers corresponding to the numbers in the list of titles above:

1 Published in Science, January 16, 1920, pp. 51-55.

(1) Professor Slaught's address dealt with the program of the Association's plans, especially as related to the work of the Sections.

(2) Given a phenomenon subject to a law of the form y= ax", if we make the usual transformation X = log x, Y = log y, A = log a, and then determine the line Y = A +nX by the usual theory of least squares, there is an obvious exaggeration of the errors for small values of x. To correct this, Professor Hedrick suggested that the errors in Y be loaded by a factor given by the law of the mean, from which we find approximately

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where (x, y) are the values of x and y given by experiment. Then the expression which is to be made a minimum becomes

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where (X, Y) are the values of X and Y corresponding to (xi, Yi).

The remainder of the usual work can be written down very easily by comparison with the usual theory. In work done upon actual experimental data, this formula has proved to be convenient and to give results that are more satisfactory than those given by the usual theory.

Similar changes in the theory of least squares can be made to counteract the distortion caused by any other transformation, in a similar manner.

Finally, the connection was pointed out between this subject and the question of approach of a variable curve to a fixed curve. An extension of the principle mentioned above gives a satisfactory measure of the distance between two curves under analogous circumstances.

(3) The object of the paper of Professor Ingold and Mr. Jackson was to give a relatively simple proof of the convergence of the sine or cosine series, whose coefficients are determined in the Fourier manner, for a large class of functions including the usual examples; and also to prove that these series actually represent the functions from which they are obtained.

Their final result may be stated thus: Any function which may be expressed as a definite integral from 0 to x of a bounded integrable function may be represented by a sine or cosine series of the Fourier type for values of x between 0 and T.

For this class of functions the proof of convergence and representability is much simpler than the usual proofs.

(4) In the presentation of the summation formula in the integral calculus it is desirable to give an example of the evaluation of an area by finding the common limit of the sums of the areas of the inscribed and circumscribed rectangles. The usual examples of the texts involve the summation of series, which is quite difficult and unfamiliar to the student. Professor Dunkel gave a method for finding the area under curves y=x", m = a positive integer, which follows rather closely the lines of presentation of the summation formula.

The series to be actually summed is such that the sum is obvious and the whole process involves only elementary algebra.

The method can be extended to negative values of m, m = 1 being excluded, and also to fractional values. Compare 1920, 116-117.

(7) The object of Professor Roever's paper was to illustrate a purely geometric method of treatment by its application to the solution of some problems in geometric optics. The problems treated in this way were certain problems on brilliant points which have already been treated analytically by the author in Annals of Mathematics, second series, vol. 3, April, 1902; Transactions of the American Mathematical Society, vol. 9, July, 1908; and AMERICAN MATHEMATICAL MONTHLY, vol. 20, December, 1913. An example of this geometric method is given in an article entitled: "Geometric Description of the Halo on the Dome of the St. Louis Cathedral," which will soon appear in a scientific number of the Washington University Studies.

PAUL R. RIDER, Secretary-Treasurer.

ENVELOPE ROSETTES.1

By WILLIAM F. RIGGE, Creighton University, Omaha, Neb.

One way of drawing a cardioid is to make a pen start in phase 90° from the center of a disk and move along a radius with simple harmonic motion, while the disk revolves with a uniform angular speed of the same period. If now, instead of a simple harmonic movement with the equation p = 1 cos 0, the amplitude being unity, we give the pen a double harmonic motion, and write the equation

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in which m differs from unity by some small aliquot fraction; we shall then get a series of harmonic curves which have one variable parameter, and which must therefore have a common envelope. The problem before us is to find this envelope.

The Inner Envelope a Cardioid.-The method of procedure, according to the textbooks, is to differentiate the above equation by regarding the variable parameter m as the only variable in it, and then to eliminate the parameter between these two equations. This will give us

sin me = 0,

1 Readers of this article are reminded of earlier articles by Professor Rigge in this MONTHLY ("Concerning a new method of tracing cardioids," 1919, 21-32; "Cuspidal rosettes," 1919, 332340) in which the discussion, with special reference to possibilities of his machine for tracing curves, is along similar lines. We have already referred (1920, 132) to the interesting illustrated account of this machine in the Scientific American Supplement, 1918, February 9 and 16 (partly reproduced in La Nature, 1919, September 27) EDITOR.

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There is therefore an inner envelope which is a cardioid, and an outer one in which the radius vector of this cardioid is increased by 2.

Fig. 1 has been drawn to represent these curves. This figure, like all those in this article, has the same position on the page that it had on the disk at the moment when it was completed and the drawing pen had returned to its initial position. If we can imagine the disk, which turns in a clockwise direction, to be now alone arrested, the pen will keep on moving up and down along the vertical line of the page through the cusp, that is, along what is generally denoted as the Y axis in figures, but which we may call here the mechanical axis of Y. The mathematical axis of + Y which is used in the equations just given and which convention directs to run always upward, runs to the right in this Fig. 1, so that the figure must be turned 90° in an anticlockwise direction in order to have it. oriented in the usual way. The reason for this departure from the customary mathematical practice was that, by presenting the mechanical aspect of the figures, the changes that come over them when the initial phase or position of the pen or the rotation frequency of the disk is altered, may be seen to better advantage. The mathematical axes must therefore be rotated to suit each figure in particular. This will present no great difficulty.

The motion of the pen in Fig. 1 was the resultant of two simple harmonic movements both of the same amplitude, one with a unit period and the other with a period m, 15/16 or 16/15 as long, while the disk had a period of either component. In practice component A had a wheel with 32 cogs which made 15 revolutions while component B with 30 cogs made 16, the disk in the meantime with a 30- or 32-cog wheel making 16 or 15 turns. In Fig. 1 a 32-cog wheel with 15 revolutions was used on the disk. A radius (through the cusp) may be seen to cut the compound curve in 15 points. Had a 30-cog wheel with 16 revolutions been employed, there would have been 16 such intersections.

The pen was placed at the center of the disk (at the cusp) when both of its components were in phase 90°. When set in motion the pen started to draw a

2 It is rather questionable to call m a variable parameter since only one value of m is considered at a time. There is a single curve with several lobes, giving the appearance of so many different curves all tangent to their envelope. Perhaps we might speak of it as the envelope of the lobes. It is true that this envelope happens to be the same for all values of m, at least for all rational values of m, and that its equation can be obtained by the usual process if we make m a variable parameter, but the envelope of a family of curves might be quite different from the "envelope" of the lobes of any one of them, and in the case of the curves represented by the given equation it might be difficult to apply the theory of envelopes to the variation of the given curve produced by a continuous variation of m.-EDITOR.

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