FROM THE ORDNANCE TRIGONOMETRICAL SURVEY. 
613 
might be obtained that would satisfy all the geometrical relations of the triangula- 
tion. The question then is to determine that system which is the most probable, 
and the solution derived from the theory of probabilities is, that the most probable 
system of corrections xx' ... is that which makes the function '1(lvx'^)* a minimum. 
If n be the number of observed angles in a network of triangiilation, m the number 
of points, then 2{m—2) will be the number of angles absolutely required to fix all 
the points, consequently the geometrical figure must supply n—2m-{-4 equations of 
condition amongst the true angles or amongst the corrections to the observed angles : 
we have therefore w— 2-1-4 equations of the form 
0 '^(ii-\-hiX-\- c^x^ ..... 
0 = ^2 + -f- c^x' -f- d^x" -f- , 
which are to be determined so that the quantity 
U = IV x^ -{- w’x'^ -T w"x"‘‘ -f- 
shall be a minimum. Multiplying the equations of condition by unknown quantities 
X 2 X 3 ...., we obtain by the theory of maxima and minima of functions of many 
variables, 
wx .... 
w'x' =Ci?li-l-C 2 X 2 -l-C 3 X 3 -l-.... 
Substituting the values of xx'... in terms of XjXa.... as obtained from these equations, 
in the equations of condition we get a system of equations from which may be 
determined ; and having obtained the numerical values of these quantities, the last 
set of equations will give the numerical values of the required corrections xx' ... 
In order that the results of the triangulation as applied to the determination of the 
figure of the earth might have the greatest weight possible, the most probable system 
of corrections has been calculated according to Bessel’s method, shortly described 
above. The principal and only objection to the application of this method of obtain- 
ing the most trustworthy results, is the extremely voluminous and tedious nature of 
the calculations. The total number of equations of condition for the triangula- 
tion is 920; if therefore the whole were to be reduced in one mass, it would involve 
as a small part of the work the solution of an equation of 920 unknown quantities. 
The following method of approximation therefore was adopted : the triangulation 
was divided into twenty-one parts or figures, each affording a not unmanageable 
number of equations of condition ; four of these parts, not adjacent, were first adjusted 
by the method just explained. The corrections determined in these figui'es were 
substituted, so far as they entered, in the equations of condition of the adjacent 
figure, and the sum of the squares of the remaining corrections in that figure made 
a minimum. Thus each division of the triangulation, with the exception of the four 
specified above, is dependent upon one or more of the figures to which it is adjacent. 
* Where w is the weight of the observation corresponding to the error x. 
