154 
FIFTH REPORT— 1835. 
and by the weight ( w) of the determination which it represents, 
and adding the results, we have 
S (it a 2 ) x + S (w a h) y = S (w a c); (G) 
and, performing the same operation with respect to the coeffi¬ 
cient of the other unknown quantity, 
S (w a h) x + S [w b 2 ) y = S {w h c), (G) 
These are the two final equations which, by elimination, will 
furnish the most probable values of the quantities sought. 
Let the values of x and y, obtained from these equations, be 
A and B ; then substituting in (D), 
r cos u = A, r sin u B ; 
and, dividing, we have 
, B 
tan u = ~ ‘ 
A ? 
by which the direction of the isodvnamic line is determined. 
•> «r 
Again, squaring and adding, 
r = VIF+W-, (I) 
which gives the rate of increase of the force in the normal direction. 
The lines of absolute intensity, and of dip, will be obtained by 
a similar process, the only difference being in the values of the 
second members of the equations (F). 
Before we can apply these formulae to the investigation of the 
lines of horizontal intensity, it is necessary to assign the weights 
due to each equation of condition, or to the determination which 
it involves. We shall assume, accordingly, that the weights of 
the values of [h — A ; ), recorded in the preceding table, are mea¬ 
sured by the number of separate comparisons from which they 
have been deduced, and we shall have, on this principle, 
Limerick . . . weight =12, 
Markree. . . . _ = 3, 
Armagh .... - = 2, 
the weights of each of the other determinations being represented 
by unity. 
The values of a, b , and c being given in Tab. VII., we may 
now proceed to calculate the coefficients of the equations (G). 
The elements of this calculation are given in the following 
table. 
