178 
PROF. H. F. BAKER ON CERTAIN LINEAR 
Hence 
— (a x + a 2 + a 3 + y x + y 2 + y 3 ) 
= \(h + k l ) + \ a (h a + hk 1 +%k*)+\*{-2h*-2h%+^hk 1 a +i§k 1 3 -$hb 2 -bk l k a ) 
and 
— («! + 0-2 + a 3) +71 + 72 + 73 
= A (h-hy) + A 2 (— 37^ + 3 h^+n*) +A^10A 3 -10^ 1 -- 2 #M 1 2 +|M 2 +f|^ 1 3 -^ 1 4). 
The product of these gives the value of q\ namely, 
(f = A 2 (/z 2 -^ 2 ) -A% (2A 8 -3^ 8 ) +A 4 [5 (A 8 -^ 3 ) 2 - W-2^4]. 
This agrees with the value found above by a quite different method (§ 4). 
The matrix of coefficients of r, after £ has been replaced by 1, is of the form 
and its square is (a 2 —y 2 ) times the matrix unity. The matrix Q 0 W (u) of § 14 is thus 
or 
Q 0 w (u) = 1 + A 0 w+^q 2 w 2 + ~ q 2 A 0 w z + ~ gV+ 
O ! 4 1 
/ C + aS, —yS \, 
yS, C —aS 
where C = ch(qw), S = - sh ( qw). From this it is easily seen that for the calculation of 
q the method we have followed is less laborious than to use the equation 
| Qo" (^) — p | = 0. 
The differential equation from which we have started is, to terms in h 2 , if we 
suppose A = 1, 
,72,,. 
+ (l+4/?. + 8&! cos 2£ + 8 Jc 2 cos 4 t)x = 0. 
If we compare this w r ith the form considered by Hill (‘ Coll. Works,’ I., pp. 246, 
268), we have, with his numerical values, 
h = 0-03971 09848 99146, 
/q = -0-01426 10046 86726, 
l 2 = 0-00009 58094 99389. 
