of the writer. Nevertheless I have too high an opinion of your character to imagine that any misunderstanding which might formerly have subsisted between you and any part of my family, in which I myself could have had no share, will not at all prejudice you against me and in any degree withhold you from lending me your advice and perhaps assistance upon the present occasion. I flatter myself I have some slight ground to go upon in this case, which I own I am most willing to catch at.

"I am married to your late ward, the eldest Miss Masters, and have now living with me her younger sister, still under age, and, of course, in a manner claiming your patronage, as well as their mother, the widow of your late friend. From this connection it is well known that I possess a very considerable property in the city of Philadelphia and its environs, besides two or three valuable estates of my own in the Province of Pennsylvania, a whole undivided Proprietary of New Jersey; yet with all this property, I have not been able for more than two years past to procure one shilling from that country, nor have during that time so much as received a line from my friend and agent, Mr. Tench Francis, who it is probable has at this time a handsome sum of money belonging to me in his hands. The purse I brought with me to England is nearly exhausted, tho' it has been managed with the strictest economy. I have not yet tried, nor would I willingly at present, what American security would produce in this country.

"I should think myself infinitely obliged to you if you could point out to me in what manner I could procure either from America, or in any other way, a temporary subsistence. I have not a doubt but that in time matters will turn out much to the advantage of everybody concerned and connected with that country.

"Let me entreat you to favor me with an answer to this letter under cover to my Bankers, Messrs. Barclay, Bevan & Co., No. 56 Lombard street, in doing which you will lay a lasting obligation upon one of the many who revere your character and admire your abilities.

"Give me leave to subscribe myself, Dear Sir,

"Your very sincere friend,

"RICHD. PENN."

When it is remembered that the hostility of the Penns to Frank. lin was so strong that Governor John Penn declined to be Patron

of the American Philosophical Society because it had chosen Franklin for its President, and that Richard Penn had been Lieutenant Governor (as Deputy for that uncle and his brother) from 1771 to 1773, it must have been difficult for Franklin not to feel that such a letter from such a man was indeed a tribute to his position, achieved solely by his own efforts.

From this mass of correspondence, I have selected some letters showing the state of public opinion in New England in 1774, and from London in 1775, including a characteristic letter from Priestley and from Charles Lee and Wayne in the field. Much more might be printed to show how well Franklin kept in touch with all that was of interest during his long and busy career. It is well that this venerable Society, so largely the result of his labors, should be made the custodian of the papers that follow almost his daily thoughts, and it is to be hoped that the preparation and publication of a Calendar showing their contents may be completed at no distant day, certainly by the two hundredth anniversary of the birth of our founder, and thus perpetuate his memory.

Franklin's legacy to the Philosophical Society was ninety-one volumes of the History of the Royal Academy of Sciences at Paris, thus helping that collection of publications of scientific societies that make so valuable a portion of its Library.

THE ORBIT OF THE DOUBLE STAR 518.

BY ERIC DOOLITTLE.

(Read April 3, 1903.).
INTRODUCTION.

It is well known to astronomers that many of the stars of the sky which to the naked eye appear to be but single stars are when viewed with the telescope seen to be made up of two or more stars very close together. About twenty thousand such double stars have been measured and catalogued, and the number is continually being added to through the discoveries of the great modern telescopes. There are scarcely fifty of these, however, of which a determination of the orbit is possible.

It was in the years 1802 and 1803 that the classic memoirs of Herschel appeared, in which it was shown for the first time that

the two stars of a binary system revolve in elliptic orbits about their common center of gravity. The first method for determining the orbit of the companion star about its primary was given by Savary in 1827, who applied his method to the binary Ursæ Majoris. This was thus the first double star of which an orbit was computed.

In the method of Savary, the elements of the orbit were derived from the least possible number of measures which would theoretically determine them. It was thus but very poorly adapted to secure good results, since all double star measures are liable to errors which are very large in proportion to the quantities to be determined from them. The method was improved by Encke, and other methods were subsequently devised by Sir John Herschel, Villarceau, Thiele and others; but in all of these the development was from the point of view of the pure mathematician, rather than from that of the practical astronomer.

The astronomer who essays to compute the orbit of a double star will find that he has at hand a great mass of measures, which, having been made by observers of varying experience and with instruments of all degrees of perfection, are more or less discordEach one of these measures consists of a determination at a given time of the distance and direction of the companion star from its primary.

ant.

If now these measures be plotted, by taking a point on the paper to represent the principal star and laying off from this point each measured distance and direction to the companion star, a series of other points will be obtained which will represent to the eye the path which the companion has pursued about its primary. Were the measures free from error, the points which indicate the position of the companion would lie accurately upon the perimeter of an ellipse; but practically they are very far from doing so, especially if the double star is very close and difficult of measurement.

The ellipse which the companion appears to describe does not represent the true orbit of the body in space, since the true orbit is viewed more or less obliquely. It is evidently the projection of the true orbit on a plane tangent to the celestial sphere at the point at which the double star is situated. While the true orbit in space is an ellipse of which the principal star occupies the focus, the apparent or projected orbit, though also necessarily an ellipse, will not have its focus at the principal star. Nevertheless, Kepler's

second Law, which states that the areas swept over by the radiusvector are proportional to the corresponding times, will evidently be true, provided that in the apparent orbit these radii-vectores are drawn from the principal star instead of from the focus.

Having plotted the series of measures as above described, the first step in the determination of a double star orbit is to draw the apparent ellipse in such a manner that it shall represent them reasonably well; the various sectorial areas are then measured with a planimeter, or otherwise, and the trial ellipse changed in shape and position until finally, after several trials, the measured positions and the law of areas are both approximately satisfied.

To fix the shape of the true orbit and its position in space, and to predict the future motion, there must next be determined the following seven elements :

(1) The Period, P. This can be measured directly from the apparent ellipse, since, by Kepler's Law, any sectorial area is to that of the whole ellipse as the time occupied in the description of the area is to the Period.

(2) The Time of Periastron Passage, T. This is the date at which the companion passes the nearer vertex of the true ellipse. It can evidently be found from the apparent ellipse by an application of Kepler's Law.

(3) The Eccentricity, e. This, since it is a ratio, can be obtained from the apparent ellipse.

(4) The Inclination, i, of the true orbit to the tangent plane.
(5) The Longitude, 2, of the intersection of two planes.
(6) The Longitude, λ, of periastron.

The last three elements are obtained by solving a spherical tri angle. The longitudes are measured from the hour circle passing through the star, from the north point in the direction of motion. (7.) The Semi-Major Axis, a.

The elements of the true orbit as thus obtained enable us to predict the direction and distance of the companion for any time. The next step of the computation is to obtain the computed distance and direction at the date of each observation. A comparison of the computed with the observed positions furnishes a basis for improving the elements by the principles of Least Squares. The same process is repeated with the improved elements, until a satisfactory agreement between the computed and observed positions is

obtained.

T

THE COMPUTATION.

There are available for this determination measures on 141 nights, as shown in the following table. In the first column will be found the date of observation; in the second, the measured distance; in the third, the measured angle; in the fourth, the number of nights on which the measures were made, and in the fifth, the name of the observer.