and substituting these values in the expressions for G′ and G", we get G'*=E"i. G*Eli. where a'g" {c" (a"+g")+a" (g"+d") } {q′(a′ +g′)+a′(g' +d') }' a"g' +++ਕ+ਗ+G+ {c' (a' + g′) + a' (g' +d')} {q"(a'' +g′′)+a" (g"+d"}3 the expression of the current which produces the signals (single and duplex) through a perfect line, as was given in the first part of this investigation (vol. xlviii. p. 136). when we get G'=Elik. G" E'i. = a'g' {c"(a"+g'k)+a" (d" +g'k)} {q′(a' +g′)+a′(g' + d')}' الو الله {c' (g"+a'k)+a' (g" +d'k) } {q′′ (a" +g′′) +a" (d" +g") } Now it will be seen that G' has clearly a maximum with respect to g', while G" has a maximum with respect to g"; thus, if we take g as the only variable in G' (k constant) and differentiate with respect to g', we get and if we take g" as the only variable in G" and differentiate, we get These two equations must be fulfilled simultaneously in order to have the simultaneous maxima of the two currents in question. Executing the differentiation, and resubstituting for k its value 7, we get, after reduction, 9" a'a' (c' +d") (q'+d')—g'g" (a'+q′) (a" +c") —g'(a" +g") {q′ (a' +g') + a' (g'+d')} de" de Now the terms in the two equations which have and dg' dg" for factors become independently zero-the first for d=d', and the second for a"d"; and substituting these values for d' and d" in the other two terms, both become zero for a'a"-g'g"=0; whence it follows that is one of the simultaneous solutions of the two equations*. Thus, substituting for d' its value a', and for d" its value a", we get a'g" G'=E"i (c"+a'') (a" +g") (a' +g') (a' + q′) ' a"g G" Ei '(c' + a') (a' + g′) (a' +g") (a"+q′′) The first equation has clearly a maximum with respect to a', and the second with respect to a"; namely, Thus it follows generally that a=d=g represents a maximum of the currents; and this, in consequence of the immediate balance, gives at last a=d=g=f, the known regularity-condition, which thus has also to hold good in order to make the two currents G' and G" simultaneous maxima. The first problem for the bridge method has therefore now been generally solved; and the results are expressed by the following formulæ : When the insulation is perfect (i= ∞), the results revert to those originally obtained in the special solution, viz.:— *The other solutions, however, which are possible from a mathematical point of view, are impossible with respect to the physical problem; for the quantities, being all electrical resistances, must be taken with the same sign, say positive. It will be clear that the given solution fulfils the following conditions, which are necessary and sufficient to place duplex telegraphy on a par with single telegraphy. i. Any variation in the resistance of the line has the least possible disturbing effect on the receiving instrument. ii. Any disturbance can be eliminated by a single adjustment in the b branch without disturbing balance in the distant station. iii. Maximum magnetic moment of the receiving instrument. iv. Maximum current. There seems to me to be no other method that can fulfil all these conditions simultaneously; and the "double-balance"* method must therefore be pronounced perfect in every conceivable respect. I am convinced that if the general problem of duplex working were investigated by means of the variation calculus, the double-balance method would come out as the final and only solution †. Addendum-Historical. When reading this paper before the Asiatic Society (on the 4th I have called this method the "double-balance" method, since there are two balances to be fulfilled in each station-namely, balance in the b branch for the arriving current, and balance in the g branch for the outgoing current. + The double-balance method was introduced on one of the important Bombay-Calcutta main lines in June last. Since then this duplex method has been working so satisfactorily and with such regularity and speed, even during the worst time of the year (south-west monsoon), when necessarily the insulation as well as the inductive capacity of lines are so enormously variable, that about its thorough practicability no doubt can be entertained; and Colonel Robinson, Director-General of Telegraphs in India, has consequently decided to introduce this duplex method also on the other long main lines of India. At present the apparatus for the Bombay-Madras line (worked direct 800 miles) is almost finished, and the apparatus for Calcutta-Rangoon is under manufacture. The Calcutta-Bombay main line is worked duplicè with Jabalpur only in translation; distance between Calcutta and Jabalpur 850 miles; distance between Jabalpur and Bombay 640 miles. The wire is almost throughout No. 5 B. W. G. (diam. =5 millims.). This experiment, on such a large scale and made under the most unfavourable meteorological conditions, has proved to evidence the practicability of the double-balance method, which certainly will invariably succeed on any line where single telegraphy is possible. of February, 1874), and further, when editing the First Part for publication in the Journal of the Society, I was unacquainted with the fact that a most complete history of duplex telegraphy had been published by Dr. Karl Eduard Zetzsche* (Leipzig, 1865). According to Professor Zetzsche †, the bridge method of duplex telegraphy was already invented in 1863 by Maron, a Prussian telegraph-inspector; and Dr. Zetzsche very truly remarks that the bridge method would seem to be that least affected by variations in the resistance of the line. To this, from an historical point of view, most valuable book I refer the reader. It is to be hoped that an English translation of it may soon be published. [To be continued.] XVI. The Eruption of Etna on the 29th of August, 1874. By Professor ORAZIO SILVESTRI, of Catania‡. IN Na short account published by me on the 1st of July, 1874 (see No. 56 of the Gazzetta del Circolo dei Cittadini, Catania, July 12, 1874; also Bullettino del Vulcanismo Italiano, fascicolo 6, 7, Rome, July and August 1874; also Bollettino del R. Comitato Geologico d'Italia), of the eruptive phenomena which have been observed since last May in the interior of the great central crater of Etna, I expressed an opinion which attracted such general attention as to be quoted in the principal Italian and foreign journals, to the effect that an early outbreak on the Mongibello side, and a great eruption immediately after, were to be expected. This opinion has been fulfilled in less than two months, but fortunately (as I am about to relate) under such conditions as not to produce serious damage, except the consequences of the fear and terror which are generally excited by the more formidable phenomena of this mighty volcano. After an uninterrupted succession of eruptive phenomena in the central crater, which has in consequence undergone remarkable modifications, at 4 o'clock on the morning of Saturday, August 29, the rural population scattered round the base or on the slope of the mountain on its northern side, even beyond the villages as far as the cultivated belt reaches, in a semicircle of about 80 kilometres, were awakened by subterranean noises, which were speedily followed by two shocks of earthquake, threatening *Die Copiertelegraphen, die Typendrucktelegraphen, und die DoppelTelegraphie, ein Beitrag zur Geschichte der electrischen Telegraphie, von Dr. Karl Eduard Zetzsche: Leipzig, 1865. † Page 125 in the work quoted. Translated from a pamphlet published by Lorenzo Rizzo, Largo Spirito Santo, Catania (1874). Communicated by Professor H. E. Roscoe, F.R.S. |