616 PROFESSOR CHRYSTAL ON THE DIFFERENTIAL TELEPHONE. 
JL x,, satisfy a system of linear differential equations with constant 
coeflicients and certain equations of continuity.* 
Elimination from these equations gives for the current in any branch 
(a) +a,D +a,D?....+4,D’)x 
=A(Ay+A,D+....4+A,D")sinnt . é : ot) 
where D stands for = and the a’s and Q’s are functions of the coefficients of 
induction, resistances, and capacities of the circuits. 
Hence 
(Ay HAD +. ...+ AqD?) (ay —a,D-+a,D?-....) Sane, 
(a) + a,D°+ a,D! +....)?—(a,+a,D?+.. ~ PD? 
= | {(Ap —) n+. one »)(a@y — ayn? + tere + (A, —Agn” Joan )a,- agn?+. _ .)n?}. sinnt 
+ {—(Ap—Agn? +... .)(a,— ayn? +...) + Ay—Agn? +....)(ay — ayn? +....)n}cosnt \ 5), 2) 
whence 
— § Apa Agr st. P+, ——— oi 1:3 
a ea, | 2 +n7(a,—a,n* + sd calh sin(nt + €) : - (3) 
In (2) and (8) the letters a are the same for all the 2’s, but the d’s are differ- 
ent for different ~’s. Hence the expression for the difference of two a’s, say 
x,—a#,, takes the same form as the expression (3), the place of \,, 4, &c., being 
taken by the differences of the corresponding coefficients for », and 2,. 
The condition therefore that the current in any branch shall be zero, or 
the conditions that the currents in two assigned branches shall be equal, take 
the form 
Ny Agu + Ant—... .=0 
‘ oath) 
A, —Agn* +Azn*—. ee 40) 
For a given frequency of the simple harmonic variation, therefore, two con- 
ditions are in general necessary and sufficient to secure either that the current 
in any one branch of the system shall be zero, or that the currents in two 
named branches shall be equal. 
If this is to be the case for all frequencies, then all the \’s must vanish ; 
and q conditions must in general be satisfied. It is in general impossible, 
therefore, to get an absolute null method depending on one adjustment only, 
when the telephone is used as an indicutor. 
* It would be easy to work out the general theory with more detail, much in the way that a system 
of conductors is treated when induction is neglected. (See Maxwext’s “Electricity and Magnetism,” or 
article “Electricity,” Encyclopedia Britannica, vol. viii. p. 43). 
