Thus the two equations which determine the absolute magnitude of g' and g" respectively are from which g' and g" can be expressed—namely, g= − } q' + } √/q'(3L+q'), g′′=−}q′′+}√/q′′(3L"+q′′), and where q'=i+1', and q=i+l". Supposing now i∞, or the insulation perfect, we have L'=L"=L, and the former special solution. But so long as i is not infinite, L' and L" may be different from each other, and therefore also g' different from g"; and further, will be somewhat too large. These values, however, will represent a very close approximation in the case of any line in tolerably good electrical condition; and as a line worked duplicè represents two lines, it can always be afforded to select the best sections, when the above values for g and g" will be sufficiently correct for all practical purposes, especially if it be remembered that when once g' and g" have been fixed they cannot be easily altered, and that therefore I and L" must be invariably certain averages, either for the whole year or for certain seasons. This, however, belongs more to the practical application than to the theory of duplex telegraphy. The resistance of the b branch in each station can now be easily calculated from the balance equations and the values given for g' and g". The value of the b branch must be calculated in order to be able to ascertain that maximum part of b which will have to be made variable in increments for the purpose of adjusting balance; and to this interesting question we shall revert further on. The general solution of the problem might now be considered complete if it were not for the currents which produce the signals, of which we do not know as yet with certainty that we have the maxima in the solution given above. It must, however, be understood that this solution represents the only true one from our physical point of view, and that, if it should not be identical with that giving the maxima currents when considered generally by themselves from the beginning, the solution would not be thereby invalidated, but only the duplex method in question would prove to be not quite so perfect as could be desired. The sequel, however, will show that the relation a=d=g=f represents also the maxima currents that are possible under the circumstances. As this investigation is of great importance in forming a correct opinion of the value of the method, it will be fully gone into. Maxima Currents.-When considering the question of currents for any telegraphic circuit, the two conditions which invariably should be fulfilled are: First. Greatest possible constancy of current. Secondly. Maximum current. How far these two conditions can be fulfilled simultaneously depends clearly on the special circuit and the special arrangements adopted; but so much is certain, that, from a practical point of view, the first condition (constancy of current) will always be of far greater importance than the second, inasmuch as the required strength of currents can be obtained by employing cells efficient in kind, sufficient in number, and properly arranged to suit requirements. Thus in our case, when we consider the currents which produce the signals in duplex telegraphy, before going to the condition of maximum current we must ascertain first the condition of greatest possible constancy of current. Now it has been proved before that immediate balance in each station is requisite in order to make the effect of any disturbance on the receiving instrument as small as the circumstances will allow of. But as these disturbances were considered with respect to one and the same instrument (i. e. independently of the magnetic moment), these disturbances are then simply due to the disturbances in the signalling current; from which it follows at once that the fulfilment of the immediate balance condition is required also in order to have the greatest possible constancy in the signalling current. Thus, when investigating the question of maxima currents, we are justified in presupposing the rigid fulfilment of the immediate balance for both stations, i. e. ad-gf=0. Further, as it has been shown before that the fulfilment of the regularity-condition a=d=g=f for both stations does make the effect of the disturbances still smaller, we have only to investigate the current at balance, and to show that the condition of maximum current becomes identical with the regularity-condition, whence it would follow that the duplex method under consideration is perfect in every conceivable respect. The question to be solved stands, therefore, as follows: Two signalling currents, the expressions of which are known, have to be made simultaneous maxima, while the different variables are linked together by four condition equations. the current which produces single and duplex signals in station I.; b' G" E' the current which produces single and duplex signals in station II. a' d' —g' f' =0, (3) (4) immediate balance in both stations. Now is a function of p"; but, on account of equation (4), p" is independent of b"; thus d' is also independent of b". In the same way it follows that " is independent of b'; thus b' and b′′ can be explicitly expressed at once, and from the four condition equations we have The resistance of the b branch in each station can now be easily calculated from the balance equations and the values given for g' and g". The value of the b branch must be calculated in order to be able to ascertain that maximum part of b which will have to be made variable in increments for the purpose of adjusting balance; and to this interesting question we shall revert further on. The general solution of the problem might now be considered complete if it were not for the currents which produce the signals, of which we do not know as yet with certainty that we have the maxima in the solution given above. It must, however, be understood that this solution represents the only true one from our physical point of view, and that, if it should not be identical with that giving the maxima currents when considered generally by themselves from the beginning, the solution would not be thereby invalidated, but only the duplex method in question would prove to be not quite so perfect as could be desired. The sequel, however, will show that the relation a=d=g=f represents also the maxima currents that are possible under the circumstances. As this investigation is of great importance in forming a correct opinion of the value of the method, it will be fully gone into. Maxima Currents. When considering the question of currents for any telegraphic circuit, the two conditions which invariably should be fulfilled are :— First. Greatest possible constancy of current. Secondly. Maximum current. How far these two conditions can be fulfilled simultaneously depends clearly on the special circuit and the special arrangements adopted; but so much is certain, that, from a practical point of view, the first condition (constancy of current) will always be of far greater importance than the second, inasmuch as the required strength of currents can be obtained by employing cells efficient in kind, sufficient in number, and properly arranged to suit requirements. Thus in our case, when we consider the currents which produce the signals in duplex telegraphy, before going to the condition of maximum current we must ascertain first the condition of greatest possible constancy of current. Now it has been proved before that immediate balance in each station is requisite in order to make the effect of any disturbance on the receiving instrument as small as the circumstances will allow of. But as these disturbances were considered with respect to one and the same instrument (i. e. independently of the magnetic moment), these disturbances are then simply due to the disturbances in the signalling current; from which it follows ad-gf=0. Further, as it has been shown before that the fulfilment of the regularity-condition a=d=g=f for both stations does make the effect of the disturbances still smaller, we have only to investigate the current at balance, and to show that the condition of maximum current becomes identical with the regularity-condition, whence it would follow that the duplex method under consideration is perfect in every conceivable respect. The question to be solved stands, therefore, as follows: Two signalling currents, the expressions of which are known, have to be made simultaneous maxima, while the different variables are linked together by four condition equations. the current which produces single and duplex signals in station I.; G" b' the current which produces single and duplex signals in station II. a' d' —g'f' =0, immediate balance in both stations. Now d is a function of p"; but, on account of equation (4), p" is independent of b"; thus c' is also independent of b". In the same way it follows that d" is independent of b'; thus V and b can be explicitly expressed at once, and from the four condition equations we have |