OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
193 
But it only corrects the error so far as the motion along- r is concerned ; another 
correction (supposing still the rotation is neglected or forgotten) is necessary to be 
i d 
applied at right angles to r, namely, the imaginary force Thus the true 
and complete correcting force is the resultant of the two forces 
r»^,and ---jt 
It appears to me that the idea here suggested might be applied with great advantage 
to cases of motion on or near the earth’s surface. The beautiful pendulum expert 
ment whieh made so much noise last year, and the various investigations respecting 
it, give great interest to such cases of motion. I propose therefore to investigate 
here, by the aid of the Symbolic Forms, the proper symbol of the imaginary force 
which corrects completely for the earth’s rotation supposed to be neglected, By 
the aid of this symbol, it will be found that the greatest possible simplicity is intro- 
duced into investigations such as those relating to the pendulum experiment. It 
will enable the investigator to forget altogether the earth’s rotation in framing his 
equations of motion, and at the same time to correct his error hy the introduction of a 
simple term. 
(91.) Let a denote a line pointing in the direction of the earth’s polar axis (north 
suppose), and representing, by its length, the earth’s angular velocity. In other 
words, let u be the directrix of the earth's rotation, then (as will be shown in Sect. V.) 
it is easy to see, that, if u denote (symbolically) the distance of any point from the 
earth’s centre, the velocity communicated to it by the earth’s rotation (if it be fixed 
to the earth) is represented by the symbol* 
J}co.U. 
Now let d denote differentiation (of u) on the erroneous supposition that the 
earth is fixed, and S the true and complete differentiation ; then the true velocity of 
, . .8m 
the point u is 
the velocity {Tia.u) due to the rotation. Hence we have 
and this must be the resultant of the erroneous velocity and 
8m du 
dt dt 
.M. 
The effeetive accelerating force will be obtained by the true and complete differen- 
tiation of the correct velocity, that is. 
or 
dt 
8m 
’ dt 
(observing that is a constant). 
* For Dw .u denotes a line at right angles to both w and u, and its magnitude is nr sin 9 ; where n is the 
magnitude of w, r that of u, and fl the angle which u makes with w. Therefore Dw.m is manifestly the velocity 
caused in the point « by the rotation cy. 
MDCCCLII. 2 C 
