266 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
and since 6^ = 0 when t = 0 , 
Similarly 
^2=^{<^osy 
2>r A , 1 
— ij-cos</.| 
For the instant at which the aeroplane arrives at we have 27 rt/T' = if}, 
and then 
27tv . , ^ 27rr . , • X 
— sin i/. ; ( 1 - cos . 
The above results may be obtained more simply as follows. At time 0 
the pivoted system is upright and there comes into existence a horizontal 
accelerating force, due to the turning 
of the aeroplane. This accelerating 
force is directed at each instant to- 
wards the centre of the path in which 
the aeroplane is turning. The mass m 
is constrained to move with the aero- 
plane, and hence there must be applied 
to it, at each instant, a force of amount 
^ j. , directed towards the centre of 
~jT 
the path. This force is applied at the 
pivots P 2 P 2 ’ applied is equiva- 
lent to an equal force applied to m, 
together with a couple of moment mv^h applied to the pivoted system 
about the pivots (fig. 7). This couple turns with the aeroplane. 
Considerations of symmetry show that after the aeroplane has turned 
through an angle ijj there has been applied to the pivoted system an 
integral couple about an axis which bisects the angle between the 
initial and final directions of P 2 P 2 • time t the pivots P 2 P 2 i^^he an 
27T . 
angle — with the direction of this axis, and the component I of the 
integral couple is therefore given by 
I = mv^,h cos 
And since ldt = Cnd6, where 6 is the deflection of the pivoted system, with 
respect to the vertical, at time t about an axis perpendicular to the 
