equal electrical resistances. 347 
Let us put 
A—B=2, A—C=y, A—D)=z. 
Then 
| B-C=y-a, B-D=z—-2, C-—D=2z-y. 
“Then the six observed differences A—B, A—C, &e. give us six 
equations for the determination of the three quantities «, y, 2. 
These equations are, however, not quite consistent and we employ 
the method of least squares to reduce the six equations to three, 
which when solved will give us the most probable values of 
ee, Y, 2. 
Denoting the six differences by A,, A,, ... A;, we have the 
following six equations 
4 = 
A, — a = Ay 
y=A, Saal +z=A 
A; 
The method of least squares directs us to multiply each one of 
these equations by the coefficient of in it and then to add the 
“six equations together to form a single equation. In our case the 
coefficients of « taken in order are 1, 0,0, -—1, —1,0. A second 
equation is formed by multiplying each one of the six equations 
by the coefficient of y in it and then adding the six equations 
together. A third equation is formed in like manner by adding 
together the six equations after each has been multiplied by the 
coefficient of z in it. When this is done in our case, we obtain 
_the following three equations : 
30—- y— z2=A,—A,—A;=m, 
—#+3y—-— z=A,+A,—A,=™m, 
—x— y+3z2=A,4+4,+4+ A,= 7. 
The values of x, y, z—say, X, Y, Z—derived from these last three 
equations are the most probable values. We obtain 
X= (2m 45 ese hs) 
VY=i( m+2m+ m3), 
4= 4( m+ m+ 2s). 
We can now determine the most probable values of A, 5, C, D in 
terms of M, the mean value of these four quantities. For 
Maa eB Cn p= A 1 (eyez). 
Thus  A=|M+i4(X4+V4+D=MP4Antmtm) 
BE ee OAV. | DA ae 
