384 J. Trowbridge—Gaugain’s Galvanometer. 
Let us suppose at first that R and D have been measured 
correctly. We shall then have 
sos a (2 +R? +D?)* tan 6, in which D= es 
Suppose now that, while R remains constant D differs from 
its true value by aD, in which @isa very small quantity. We 
shall then have, neglecting small quantities in Eq. 1: 
z 
(=e? +R?+(D+aD)?) *tan(é—m) 
in denoting the decrement of the angle of the needle. The 
difference in value of these two expressions for the strength of 
a current is 
i= So(C+R 4D?) tan dO—(?+R? +(D-+aDy)'tan(6—m.) 
Suppose now that D remains constant or R varies from its 
true value by the small quantity #D, in order to produce the 
same decrement as in the angle; we shall have 
F _T (?+ D?+(R+6R)?)? tan (6—m.) 
2 On (R+ BR)? 
jai, T [CAR EDY tand _(0-4D'+ (R4-AR)*)*tan(d—m) 
‘42s ~ 'R? (R+AR)? 
The difference in the values of the strengths of the currents 
are indicated by the final terms in the values of i—7, and?—t, 
which are 
(2?-+-R?-+-D? (i+-a@)?)% tan (6—m) 
R? 
(2D? +R? (1+ )2)8 tan (6—m) 
ce  RrUae 
Since sea we have by developing and neglecting the squares 
of a and £, 
5 R?a\3 5 ° 
(orto ey (ng anes 
R? R?(14+)? 
Neglecting a and 6 when they are additive to a large whole 
number, the differences in the strengths of the currents will be 
ak2 
2R2 and by Re” a is smaller from 
the nature of the case than 6. Hence 
4 ie 
man <2pRt. 
expressed by the terms 
