35 
:\[R. G. T. WALKER ON BOOMERANGS. 
Hence the substitution of the series for Oj, Og, iv gives as the equations foi 
"2. “3 
XUi^ai + 2m-^na., — 2hiUr^ = Ai 
1 
k\J Cl.. 
— 2nq')/a-j -j- XIJ ^ i f” 
/c~ ^ j 
— X [k.^ + K.v) nXJ«| — -|- = -^3 ,J 
■ ( 2 ). 
where/'n ^^^'6 linear functions of a, jS, y . . . in which the constant coefficients all 
contain X or k, but not the numerically more inq)ortant quantity m, as a factor. 
In order, then, to obtain a steady motion about which minute oscillations are going 
on, wm neglect the terms/i,/g,/g in our first approximation. This is equivalent to 
taking two points on the path at an interval corresponding to a number of complete 
revolutions (say twelve), and asserting that the angular change in the axis of lotation 
is that due to the non-periodic portion or mean of the couples lu action duinig that 
period. 
Slalnlity. 
For steady motion to be possible it is necessary that the values of Ho, w, gneii 
by the equations 
inOi + XUpHi + 2mpna,j — 2\n\Jw = 0 
, -r-r o /cUlC 
— /npinj -f- wHo -j- XXJpHj _,o — h 
fC 
— X {ki -h nUnj — mHOg d- mw -f- XUg^^c = 0 
shall be always small. 
On inserting numerical values, it appears that this condition 
of n to U be large enough to give the determinant 
is satisfied if the ratio 
XUf, — 2XnU 
0 TT . 
— 2 mp?, XUp, ” 
K 
— X (kp 4* x^) — wU, xup 
a positive value. 
If 2a = TT — ^ where ^ is small, the motion is unstable unless with actual values, 
n > 270. 
When 2a = 120° the critical value of n is 26, and when the arms are at rigid 
angles, stability is secured when n = 22. 
These values are rather larger than those found necessary in practice, but their 
mutual relations are correct. The first time that a beginner attempts it, he can make 
a boomerang whose arms are at right angles travel steadily, but the more ontuse the 
F 2 
