210 ON determinations of the absolute value, secular change, 
months, indicates distinctly the existence of secular change. Pursuing therefore the 
same method of obtaining its mean value during the period of observation as was 
adopted in the case of the horizontal force, we obtain from the fifty-two conditional 
equations 2/=+0'’0741 as the monthly value of the secular change, equivalent to a 
mean annual increase of the inclination during the period of observation of 0'’89 ; and 
75 ° 16'‘09 as the mean inclination on the 1st of March, 1847, with a probable error 
reduced to 4:0'’17, the probable error of a single monthly determination being 
+ l''23; whence we may infer that the hypothesis of a secular increase in the in- 
clination of 0'’89 annually during the period of observation is more probable than the 
hypothesis of no secular change, in the proportion of r4 to 1. 
Total Force ; Hs mean value and secular change . — Having thus derived from the 
series of fifty-two months of observation the mean value of the horizontal force 
=3'53043, and of the inclination 75° 16''09, each for the epoch of the 1st of March, 
1847, we have for the value of the total force in absolute measure at the same epoch 
3‘53043x sec. 75° 16''09= 13‘8832. With reference to the inclination element of the 
result, we might safely regard this value as a final determination ; but we cannot 
quite do as much in respect to the element of the horizontal force, as it may yet have 
to receive the corrections already noticed (though they are likely to be extremely 
small), when the values of the constants of inertia and induction shall be finally 
ascertained on the return of the instruments to Woolwich. When these constants 
and their probable errors are known, the probable error of the finally corrected value 
of the total force will also be assignable. 
The elements from which we have to infer the secular change of the total force are 
the secular changes of the horizontal force and of the inclination derived from the 
observations ; these are an annual decrease of ’0042 in absolute measure of the hori- 
zontal force, and an annual increase of 0''89 of inclination. A secular change of the 
horizontal force may be produced, either by a secular change of the inclination affect- 
ing the horizontal component of the total force according to the known principles 
of the resolution of forces, or by a secular change in the total force itself; or, 
finally, it may be the joint production of both. An increase of the inclination causes 
a decrease of horizontal force, and vice versa ; so far therefore we may regard the 
annual decrease of the horizontal force at Toronto as attributable in part at least to 
the annual increase of the inclination. But an annual increase of 0''89 in the latter 
element is equivalent to an annual decrease in the horizontal component of the force 
of not more than *0035. There remains, therefore, an excess of *0007 in the secular 
decrease of the horizontal force, which is unaccounted for by the secular change of 
the inclination, and is indicative of the existence of a small annual decrease in the 
total force during the period of observation. The uncompensated portion of the 
horizontal force on which this inference is founded is indeed small in absolute 
amount, but its magnitude must be judged of in relation to the probable errors 
