Cambridge Philosophical Society, 2S5 



III. If the numerator and the denominator of a proper fraction are 

 each a recurrent function of t, then the series, arising from the deve- 

 lopment of the fraction according to the positive integer powers of t, 

 will be the right arm of a self- repeating recurring series. 



By way of example, the author applies his first theorem to the 

 summation of the infinite series 1'' — 2' + 3^ — &c., and compares his. 

 process with the corresponding processes of Laplace and of Sir John 

 Herschel. The sum in question is given by Sir John Herschel (see 

 Jameson's Journal, January 1820) in terms of the differences of the 

 powers of 0, extending from A'O^ to A^O''. In the author's process, 

 the requisite differences extend from A'O^ only to A^O^, and the nu- 

 merical coefficients of these are of diminished magnitude, and of 

 very easy determination. 



The author makes other applications of his theorems; but on 

 these we forbear to enter. 



A paper was read by Professor Challis on the Determination of the 

 Longitude of the Cambridge Observatory by Galvanic Signals. 



The experiment of which this paper contains the details, was 

 made at the suggestion of the Astronomer Royal, and conducted 

 according to a scheme arranged by him for giving and receiving 

 the signals. A galvanic connexion having been established between 

 the Greenwich Observatory and the Cambridge Telegraph Office, 

 by means of the London central station of the Electric Telegraph 

 Company, signals were sent on the nights of May 17 and 18, 

 1853, between 11^ and 12^ mean time. The signals were made 

 by causing two needles, one at Greenwich, the other at Cam- 

 bridge, to start by completing the galvanic circuit at either place of 

 observation. The times of starting were noted at both places, and 

 reduced to the sidereal times of the respective observatories, to serve 

 by comparison for determinations of the difference of their longi- 

 tudes. On each night the signals were made alternately for a quarter 

 of an hour at one station, in batches containing an arbitrary number 

 of signals not exceeding nine, and then for a quarter of an hour at 

 the other station in a similar manner. On the first night the total 

 number of signals was 151, and on the second night 139. The two 

 observers, Mr. Dunkin of the Greenwich Observatory, and Mr. Todd 

 of the Cambridge Observatory, changed places in the interval between 

 the two nights' observations ; Mr. Todd observing at Greenwich, and 

 Mr. Dunkin at Cambridge, en the second night. Also it was arranged 

 that the two observers should observe identical stars on the two 

 nights, as well as the stars ordinarily used for clock errors, and that 

 the same apparent right ascensions of the stars should be employed 

 for reducing the signal-times at both observatories. The Cambridge 

 Observatory time was conveyed with the greatest care to the Tele- 

 graph Office at the Cambridge Railway Station by the transfer of 

 three chronometers. By a first calculation, the longitude of the 

 Cambridge Observatory was found to be 23®'03 east of Greenwich. 

 Professor Challis subsequently made another calculation, taking 

 into account the effect on the times of meridian transits of stars 

 produced by the forms of the transit-pivots, according to a method 



