ment of contact with the battery become approximately constant at any one part of the line-and, if there be no leakage, will attain the same strength at every part of the circuit, including the battery and receiving instrument. The second method, first introduced by Mr. Varley, and now in pretty general use on submarine lines, is somewhat different. The end of the coils of the receiving instrument, which in the first method is connected with the earth, is now joined to one armature or inductive surface of a so-called condenser, properly speaking an electrical accumulator, the other armature of which is to earth; or, which comes to the same thing, the condenser is placed between the line and the receiving instrument. As there is now no longer a complete conductive circuit, no permanent current can flow through the receiving instrument, or indeed through any part of the line, if the insulation be perfect. Imagine the condenser to be a continuation of the cable, in fact a length of cable having the same capacity as the condenser, insulated at its further extremity, and the receiving instrument connecting the main cable with its imaginary continuation, as shown in fig. 1; where f is the battery at the sending-end of the line, one pole of which is to earth, K a key for making contact between its other pole and the cable A, and e the receiving instrument placed between A and the continuation B. Then when contact is made at K it is evident that only so much electricity will pass through e as will charge B up to the potential of the further end of A. The current through e will therefore be transient, rising to a maximum and then dying away. This method of representation would be perfect if we could neglect the resistance of the conductor inside B; as, however, in practice the capacity of the condenser is only a fraction of that of the line, there will be little difference due to this cause. And if the capacity of B be very small, we may consider the flow of current through e to be strictly dependent on the rise or fall of potential of the end of A. To find an expression for the potential and the current at any point of a cable insulated at one end, at any time after contact is made with a battery at the other end, the only way, as far as I am aware, is to follow the method given by Sir William Thomson in 1855 (Proc. Roy. Soc.), making the necessary alterations to suit the changed conditions of the problem. It is to express the actual potential at any time as the difference of two functions, one being the known final distribution of potential, and the other the departure from the final potential, the latter being expressed by an infinite convergent series every term of which is of the form sin x.e-t. Let be the length of the line, k the electrical resistance of the conductor per unit of length, c its electrostatic capacity per unit of length, k, the resistance of the dielectric per unit of length to conduction in a radial direction, V the electromotive force of the battery, the resistance of which is neglected, v the potential, and C the current at any point x of the conductor, measured from the battery-end, at the time t from the moment of making contact. The differential equation of conduction in a telegraphic line is where h= √ k k1 ; and we must find a solution of this to satisfy the following conditions, which are given by the circumstances of the case. 1. v=V when x=0. dv 2. =0 when x=l. dx 3. v=0 when t=0, except when x=0. 4. v=f(x) when t= ∞. To find the function f(x) expressing the permanent distribu subject to the first and second conditions. We thus obtain dv dx In expanding (2) in a series of sines we must remember that =0 when x=l, and accordingly use the expansion When the insulation is perfect and h=0, this becomes We can employ (5) and (6) to determine the flow through the receiving instrument, by giving x a value something less than 7; but it is preferable to use the series for dv obtained from (3) and A unit of time of a very convenient magnitude for practical calculations is Employing this unit, we have the following series for v and dv dt t 1 40a 10 40a + 3 10 40u &c.). (9) 9t dt ckl -3.10 40a 5.10 ̄40a The "arrival-curve" for v, calculated from equation (9), is shown in fig. (2). 100 Fig. 2. (7) By comparison with the arrival-curve for the current at the remote end when to earth, we see that, broadly speaking, it takes about four times as long for the potential to nearly attain its maximum when the end is insulated as it takes for the current to nearly attain its maximum when the end is to earth. Thus, when the end is to earth, the current reaches 98 per cent. of its maximum strength in 20a; and when the end is insulated, the potential reaches 98 per cent. of its maximum in 80a. This relation does not hold good throughout the whole extent of the curves; but there is a general similarity. We may conclude that signalling by means of an electrometer connected with the insulated end of a cable would be much slower than the ordinary plan of a galvanometer or recording instrument worked by the current. dv Fig. 3 represents from t=0 to t=80 a calculated from equation (10), and is closely the same as the arrival-curve for the current in condenser signalling. It will be seen that dt reaches its maximum in 7a. dv The strength of the current will of course depend on the capacity of the condenser, and will be proportional thereto so long as it is small compared with the capacity of the line. As, however, an increase in the capacity is equivalent to lengthening the line, the maximum strength of the current will not be so soon reached with the larger capacity: although the signals will be stronger, they will be more retarded. Hence the best capacity to be used on any line, which should theoretically be as small as possible, must be determined by the delicacy of the instrument and the battery-power employed. When the capacity of the condenser is one seventeenth part of that of the line, the maximum strength of a signal is about one tenth of the per |