This result might, of course, have been anticipated from the special solution, since equation (VIII.) gives only a relation between the branches, quite independent of i. It remains now to determine the magnitude of one of the branches; and to this end we have to consider the magnetic moments of the receiving instruments.

Maxima Magnetic Moments.-By definition we have

S=P-Q

for both stations; and as it has been proved before quite generally that S=0 if A=0 (i. e. if rigid balance in the station for the outgoing current be established), we know at once that at or near balance the currents which in one and the same station produce single and duplex signals must be identical-and need therefore express the magnetic moment in each station for one current only, by presupposing balance in both the stations. The currents which at or near balance produce the signals are E" μ' 4 g"+c"

and

G'=

in station I.,

in station II.*

These expressions follow from the general formulæ by fulfilling the regularity equation (VIII.) for both stations, and, in addition, the balance-conditions.

*For balance in station II. the current passing through station I. is

But" on account of a=d=g=f in each station ;

But g"-b"c" on account of balance in station I.;

Multiplying now G' by g' and G" by ✔g", we get

the magnetic moments of the two instruments in Nos I. and II. stations respectively; and considering that

where Q=(g'+1') (g" +l")+i(g'+g" +l'+l"), we may write the two above expressions as

The first expression has clearly an absolute maximum with respect to g', and the second with respect to g"; but these two maxima cannot be simultaneously fulfilled, and do not therefore represent a solution in this particular case. But if we consider that during a duplex signal both the instruments g' and g" are in circuit, while during a single signal, though not both the instruments yet certainly their equivalent in resistances is in circuit, it will be clear why simultaneous maxima of the two single expressions are not possible. It represents simply the more general case to which the question belongs of making the magnetic moments of two instruments, connected up in the same single circuit, maxima. In this case it is well known we can do nothing more than make the sum of the magnetic moments a maximum; and here therefore we must do the very same. Adding, then, we get

which expression has a maximum with respect to both g' and g" considered as independent variables; and such indeed, according to the nature of the problem, they really are.

Thus, differentiating P with respect to g' and g", we get

*This can be easily shown by substituting for μ, μ", c', and c" their actual values.

But as the same kind of instruments are employed in both the stations, we require evidently also the same force in both to produce the signals, no matter what the state of the line

Thus we must put*

may

be.

or

PP",

Substituting these values in the above equations and reducing, and further dividing the first equation by '+i and the second by "+i, we get at last

ill

=L", measured conduction from station II.,

i+l

*This supposition in the case of a perfect line is fulfilled by itself, since then the two instruments are not only of the same kind, but absolutely identical.

Thus the two equations which determine the absolute magnitude of g' and g" respectively are

and

Ll+g-38 (1+7+7)=0,

L+g"-39" (1+)=0;

from which g' and g" can be expressed—namely, g′ =−}q'+z√/q'(3L′+q′),

g′′=−3 q′′+}√/q′′(3L′′+q′′),

and

where

q'=i+l',

and

q'=i+l".

Supposing now i∞, or the insulation perfect, we have L'=L"=L, and

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 L' 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.

and

dy"

E"

dQ

‚ = Q — 2 √g" { √g" + " √g} d=0.
:Q−2√g" {√g"+

But as the same kind of instruments are employed in both the stations, we require evidently also the same force in both to produce the signals, no matter what the state of the line may be. Thus we must put*

P'=P",

Substituting these values in the above equations and reducing, and further dividing the first equation by '+i and the second by "+i, we get at last

=L", measured conduction from station II.,

i+l'

This supposition in the case of a perfect line is fulfilled by itself, since then the two instruments are not only of the same kind, but absolutely identical.