184 
PROF. H. F. BAKER ON CERTAIN LINEAR 
and so, to this approximation, 
to — cti + ot-2 — —Aft-+X . —-—— • 
n n—l 
The characteristic factor is then A” +2<z)i , the differential equation being 
72 
—A + (n 2 + AnXh + 8ftA&j cos 2 1 + 8 n\ 2 b, cos 4£ +...) x — 0, 
dt 2 
Thus q is always real, when A is small enough, provided lz x is not zero, even if h be 
zero. The result agrees with that found in § 6 for n = 3, if allowance be made for 
the change of notation. 
[.December 1 , 1915. — -Consider the differential equation differing from that just 
preceding only by the substitution of H for A h in the term inXh of the coefficient of 
x, where H is supposed to be of the form Xh 1 + X 2 h 2 +X 3 h 3 + — The computation of q 2 
proceeds then exactly as before. The formulae for a x + a 2 + a 3 , yi + y 2 + y 3 , given above, 
p. 178, substituting H for A h, show that, for n = 1, q~ is then of the form 
(H — aj) (H — a 2 ) Q, wherein Q is a power series in H, A^ 1? A 2 k 2 , ..., reducing to 1 when 
H = 0, A = 0, and 
ai = -4-» 2 + (W-hh) x 3 +..., 
a 2 - ^a-PA 2 - (W-hh) x 3 + ••• • 
The value of cf is positive, and the motion represented by the differential equation 
is stable, so long as H does not lie between these values. Similarly for n = 2, from 
the formulae at the bottom of p. 181, the range in which q 2 is negative is when H lies 
between 
-(PC-Oa 2 and {W-h) A 2 , 
these being accurate as far as A 3 . Unless f h 2 < h 2 < X^-h 2 , these limits are of 
opposite sign, and include H = 0. This is the result given on p. 142 (save for a slight 
difference of notation). For n = 3, an analogous computation shows that q 2 is positive 
except when H is between 
PA 2 -PA 3 and pA 2 + PU, 
where 
P = fU 3 —3U4 +A 
and this range does not include H = 0 unless = 0. It would appear, from the 
formula above (p. 184), that the corresponding interval for greater integer values of n 
is between two quantities of the forms 
2 n 
n~ 
&A 2 +Pa 3 , 
9/ 
n 
n 2 — 1 
A 2 a 2 +Qa''. 
