150 
PROF. H. F. BAKER ON CERTAIN LINEAR 
If we now write 
= A^K n 2 +2iB n K n , 
which are equivalent with 
A _ K n^ ) n n 
n 0,k 2 -4) 
the equations may be replaced by 
Qn = — 2?.A B K n + B n K,‘\ 
2iP n + K n Qn t 
i K n— 4) 
AP„_i + a re P7t+^ n Q 7l + AP n + 1 — 0, 
•^Qn-l + C nPn + ( 4Qn+ AQ n + 1 = 0, 
• • (A) 
wherein 
a n — 1 + 
P 
> K = 
— 2ip 
<n— 4 * K n (fC n 2 — 4) 
_ 2iq 
K n( K n~ 4) 
d n — 1 + 
q 
4 
so that 
A (1 _ hc - KS-Kn+W } 
n n n n *.>(^-4) 
it being remembered that y> + g = 3, g>g = jm 2 . 
When we eliminate P n _ l5 Q n+1 from the equations (A), we obtain an infinite 
determinant, which, leaving aside questions of convergence, we may denote by 
A 
«_! A 
A c_j d_ i . A 
A . a b A 
A 
d 
A 
A 
eq A 
A Cj dj A 
= 0. 
The product of the diagonal determinants a n d n —b n c n is here 
Sin IT ( K — O']) . Sin 7T ( K — IT 2 ) • SID 7 T (k — (T3) . Sill 7T (/C— CT 4 ) 
Sin 4 7TK 
where <r u cr 2 , cr 3 , <x 4 are the four roots of cP—cP + \nP — 0, previously considered. In 
using this determinant to obtain a further approximation to k it would seem 
