32 
MR. G. T. WALKER ON BOOMERANGS. 
is zero. Thus we are justified in replacing by U sin nt, U cos nt. 
then have 
ojj flj sin nt — 122 cos nt 
ftjg = flj cos nt + fta sin nt 
We -shall 
( 0 , 
and on multiplying the former rotation equations by 
sin nt cos nt 
2 > 2 ^ 
and adding, we get 
, • , sin 7it . ^ cos nt 
m (fli + 2^0.2) = + Go — Pq say, 
S/^i^ 
.• . cos nt sin nt . 
ni (t22 ‘Znil^j — c 0 ^ ”1 Gq Go* 
m {n, - Un,) = s = R„. 
Now 
K , K ^ 
P„ = 2X„CJ» - ^ 
+ X 
+ K 
4 .-2 
4- 
KTh / 
nhv cos nt + 
^ ) i(;U sin ^^nt 
^2 «l‘ 
4 — (^1 
t K}nV 
X 
\K-\ ' 
/I . 1 \ 
Sin "int I -- d-s 1 
V/ 
+ 
„ /dJ [12^ (3 cos nt P cos ?>nt) + Vl^ (sin nt + sin 3n?)] 
( /cd + n..^ + 
\ '^1 
(12^ cos 27^^ + Og sin 27ii). 
Similarly 
Q„ = - xu^n, + ^ ^ ..n. - 
A 
) 
f2j (cos nt — cos 2>)it^ (ii j 3 ) ~^ ^3 ^ 3 — ~k^- 
/ I . 1 
K / /Co' 
y 
Key' 
\ /^JlV I ^5* + 
V^ + 
n~n., 
together with terms whose coetficients involve circular functions of nt. 
Also 
Ro = ~ ^ {U^ + (/c^^ fi- /Co") n^ \ 'W + X (/Cj^ + kq^) 
— X — /cg^) 7rU (n^ cos 2nt + ftj sin 2nt) 
(f2j cos nt + P 2 ^^0- 
