MR. G. T. WALKER O^v'' BOOMERANGS. 
O Q 
If all tlie terras be collected on the same side there residt three equations of the 
form 
-f" “b ” Oj 
ftoflj ~b ^3^^' ~ 
^ 3 "i ~b bgllo + c^w = 0 , 
in which the coefficients may contain the operator djdt, or circular functions of the 
time. 
The equations giving the motion under gravity of a boomerang with the two 
distortions will differ from these in three ways. 
First of all, if Vm'n be the direction cosines of a line drawn vertically downwards, 
the equations of translation will be 
m (U 4 u-n,) = rngV, 
m (— 4 U^'g) = nigm', 
m [u: — L fio) = Rq "b Ri ”b R3 ~b ^'>Tgn , 
Pi, R 2 , Qn Q 2 > ^3 bearing to Z^, Z 3 the same relations as Pq, Qq, Pq to Zq. 
From the second equation 
^ 3 - 1 ; ’ 
and from the first 
ij = gU 
Now ^3 in numerical value is comparable with and n with 30, so that the 
additional terms in (l)due to the consideration of dg will be of negligible magnitude. 
Again, V is small when the path is nearly in a horizontal plane ; hence, in considering 
the steady motion corresponding to a particular portion of the path, the change in U 
need not trouble us. 
When the forces due to the distortions—and these will not involve w —are 
introduced, we obtain 
4 4 — Pi “b R 35 
”b boHa 4 — Qi "b Q 35 
agifi 4 63O3 4 = Ri + P3 — cos 6 , 
where 6 is the angle between the axis of revolution and the upward vertical, and Ojn 
being small, we shall treat d in the third equation as constant. 
We next regard the right-hand sides of these equations as expanded in the form 
VOL. CXC.—A. F 
