MR. G. T. WALKER ON BOOMERANGS. 
31 
The couple about the axis 2 is the integral of 
(lu^ -f + (-^ + 5 cos 4)) A. iUi + 1 ^ 1 ^)], 
leading to 
Gq = S?a \_KU + XcDg ['2k.;^v + {k^ + K^) wg}] — Sco^wg {kk^^ + 2\K^hi) 
— SoJ^X [k.2 4- + 2k^OJ^V + + Kq) Wg^}, 
G, = — f — KWg {m'^v + {m ^ + m.^) wg} + \u {u^ + v^) -f 2CO3U 
P 
+ ^Wg" (Wg^ + mg'*) — 2 (m/ + m'g*) Wg^}], 
G.7 = — ■“ {IlUV + ih^ 4" 24 ^) Wg^^} — \{u^ 4" 'P^) {f'2'^ 4" "4 
" T T 
4 - 2^3 {l^V^ + l-^K?') + (3^7* + 2^8'*' 4- h^) <^3^ + hl^z\- 
The equations of angular motion are 
Awi — (B — C) waojg = F, 
Bojg — (C — A) wgWj = G, 
Cwg - (A - B) <^1^2 == d. 
Neglecting the product w^ojo, vve see that Wg may be replaced by n, a constant. 
Also, for a thin flat body, C is sensibly equal to A + B, and if m be the mass per unit 
area, we have 
A = Sm^Cj^, B = Sm/fo^, 
so that our equations become 
y^UlK^ (ojj + nwj) = F, 
Sm/c.g^ (ftjg — no)^ — G. 
We shall first of all discuss the motion of an uudistorted boomerang free from the 
action of gravity. If we revert to our former axes OX, OY, OZ, of which OX is 
the projection on the primary [daiie of the direction of motion, we shall obtain as the 
equations of translation, 
U -|“ = 0 , 
'—■ WVl^ -f" U^g — 0 , 
m {iv — 1102 ) = . 
O 
Hence, neglecting squares, U is constant, and t^g, the angular velocity of the axes, • 
