MR. G. T. WALKER ON BOOMERANGS. 
29 
X ['w^ + ^ 2 (a? +- s cos (f)) voj^ — 2 {y — s sin </>) uo) 2 , 
■f {oc^^ + + 2.S* (a? cos (^ — ;?/ sin <^) 4- <^3^] 
- -1" cos (J) — {y — s sin 4>) 0 )^] — (-h “) sin {v + (a; + s cos (f>) oi^\ 
-\- ID — {x -{- _s cos <^) Wg + (y — s sin (f)) Wj 
This may be regarded as the sum of three forces : ( 1 ) Zq the force which is exerted 
on a boomerang without distortion, (2) due to the rounding, (3) Z^ due to the 
twisting. 
Denoting by brackets ( ) the operation of taking the mean value over the area of 
the boomerang, we shall adopt the notation 
(x^) = K./, (^2) = (x3) = /C3^ (x/) - k/, 
(x^) z= (y4) _ ^^4^ 
{y sin (f)) = ?!, {xy sin (p) = 1^, (/ cos (/>) = 
{x^y sin 4>) = {xy- cos 4>) — 4 ®, {y^ sin <^) = / 3 ^ 
{x^y sin (^) = {xSf cos cf)) = /g^, (a:iy3 sin (^) — /g'^, {y"^ cos </>) — 
{x'^y sin </> + cos (f) + x^y^ sin <f) + xy‘^ cos (/>) = /n^, 
(s 2 cos^ (^) = {s^ sin 2 (^) = 
{s^x cos 2 (^) = m^, {s^x sin 2 (p) = {s^y sin cp cos (p) — mg'", _ 
{s^x^ cos^ (p) — {sh:~ sin^ (/>) = {s^xy sin <p cos = m/, 
(s 2 y 2 cogS g;j ^2 
(.s^a:;^ gjj ^2 _ ^,^ 5 ^ (s^aj^y sin (p cos </>) = cos^ <p) — rn^^, 
{d^xy'^ sin^ <p) = di {s^y^ sin (p cos (p) — 
Then for a plane surface we find at once 
Zo = — XS [(w® + v^) w — 2 {k^uo}-^ 4 - K^vo)^) W 3 4- + xi) — {xi 4- «■/) " 2 } " 3 "] 
where S is the area, and terms in have been omitted since they are multiplied by 
the small terms w, (o^. 
The rounding produces a force 
Zj = “ 2x('cZS ~ {{11 — y(^z) cos (p ~ {v xco^) sin (p] [uo)^ sin (p + '^<>>3 cos (p 
4 - {x cos (p — y sin (p) <J 3 ^} 
= — Swoja {(w'o® — mQ^)v 4- Wg}. 
P 
