9 SAI — f' uat 
o 
Substituting the previously derived value of Ó  , we obtain: 
o 0: Lf 2 ar — [Ba ar? — [twat 
Now this output is amplified and applied to the platform torquer in such a direc- 
tion as to cause this output to approach zero. Examining the above expression, 
we see that if the platform remains vertical, the angle through which it has 
moved, f «dt , must be equal to the double integral of the acceleration. In ad- 
dition, the deviation from local vertical, 8 , must be zero. 
The previously described system is a rather complicated one even in the 
simplified, non-rigorous presentation given above. It is not amenable to direct 
intuitive understanding. However, an analagous system can be set up, which is 
more directly understandable. Reference to Figure 3 will make it easier to es- 
tablish the analogue. Thus, if a pendulum is set up on a vehicle at A, it will 
hang down, the bob locating itself on a line from A to the center of the earth. 
Imagine that the earth offers no resistance to the passage of the bob into it as 
the length of the pendulum increases. If the vehicle moves suddenly to point B, 
the inertia of the bob causes it to remain on the line AO. The error in vertical 
indication created by the motion is seen to decrease as the length of the pendu- 
lum increases, becoming zero when the length is equal to the radius of the earth. 
But a simple pendulum of this length has a period given by: