OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
197 
Now, here, the operation (D 7 .) is performed only on lines at right angles to 7 ; 
we may therefore put \/— 1 for (D 7 .) (see art. 55) ; and thus (15.) becomes 
^+2«sin/x/-l^-;M=0 (16.) 
The roots of the equation 
(1) +2’* S'" ($) -f =0 
are — nsin/\/ — — n^sia^l—p 
or {—n sin Z+^)\/ — 1 j 
if, for brevity, we put sin^ Z+|=m^ Hence the solution of (16.) is 
(17.) 
The constants A and B here denote two arbitrary lines in the horizontal plane. 
If the earth were fixed, the form of the solution would have been 
for then nf=-V 
Now indicates a uniform backward rotation of with an 
angular velocity n sin 1. Thus it appears that the apparent curve of motion of the 
pendulum will be the same form as if the earth were fixed, only there will be a slow 
angular regression of the whole about the vertical as axis at the rate wsinZ per 
second. 
I may observe, in passing, that the simplest interpretation of ( 17 .) is this ; that the 
motion of the point (u) results from the superposition of two motions, 
\,{m—ns,\Til)tiJ—\ t),— (m+»sini)fV— 1 . 
and these are two uniform circular* motions, the former that of the line A forward 
with an angular velocity {m—n sin 1 ) ; the latter that of B backward with an angular 
velocity m—n sin 1. 
(95.) As my object is simply to exemplify the application of my notation, I shall 
not proceed to a second approximation; which however is very easily effected by 
substituting for T in (13.) its complete value given by (14.), after having put for u in 
(14.) the value (17-) just obtained. The result is important, especially as regards 
motion near the equator. 
* Ae^V-i denotes A turned out of its position (round 7) through an angle d, and therefore M=Aen<V-i is 
an equation indicating that the motion of the point (a) results from rotation round the origin at the distance A, 
the angle nt being described in the time t. 
