ADJUSTMENT BY LEAST SQUARES 165 
can be solved by any of the methods appropriate to linear equations, 
including that of Gauss. This series is, however, obviously adapted to 
solution by gradual approximation. Every coefficient is unity ex- 
cept those for the key correlates such as m,, and these are integers 
not less than 3. 
As to the first approximation, we can put 
/ 
m, =m, + Am,, 
where 
m,’ = d,/N,. 
Similarly 
mM,’ = d2/Nz (3 
My’ = dy/ Ny y 
whence 
I I I 
Am, = — 2m, = — 2Xd,/n, + — ZAm,z. 
Ny Ny Ny 
Writing 
it 
my’ = —Xd,/nz, (4) 
n, 
we have m,’-++m,’’ as a second approximation to m,. 
Similarly we can find the values m,”’, m,’’, - - - and obtain a third 
approximation m,’+m,"+m,"’ to m, by writing 
I 
m,!"" = — Im". (5) 
ny 
The procedure can be continued as far as desired, and can be ter- 
minated at any stage when the added increments to m,’, m,’’, m,'"’, be- 
come sufficiently small. It is simple and lends itself to routine exe- 
cution. 
Although exact solutions for particular networks, and methods of 
more rapid approximation, will be discussed later, we may test this 
particular method for the case discussed by Roman in the paper previ- 
ously cited. 
Figure 2 illustrates the network solved by Roman. There are 
seven station points and six triangles. In Table I, column d gives the 
original excesses for each triangle. The successive approximations 
to the m’s are calculated from these d’s and the previously obtained 
approximations for the adjacent triangles given in the third column. 
The residuals do not disappear very rapidly, but the procedure is 
easy and rapid. 
825 
