114 Proceedings of the Koyal Society of Edinburgh. [Sess. 
(a - 3)x + by x 
(3 -a-b)y (a-2)x + (b-l)y 2x 
-(3-a-b)y (3 — a — b)y (a — \)x + (b — 2)y 3x 
(3 -a—b)y —(3 — a—b)y (3 — a—b)y ax + (b—3)y , 
where now the diagonal term is invariant to the interchange of a , x with 
b , y, and so likewise the coefficient of y in the elements outside the 
diagonal. The latter coefficient in the case of the n th order is n—l — (a + b), 
and may be conveniently replaced by a single letter. 
(8) If we try to simplify the form of § 7 by increasing the first column 
by the second, the second by the third, and so on, the diagonal elements 
return to their original awkward law of formation, the result being 
(a—2)x + by x 
(a-2)(x-y) ax + (b-\)y 2x 
. (a— \){x-y) (a + 2)x+(b — 2)y 3x 
a(x — y ) ax + (b—3)y . 
This, however, is not valueless, when we observe that it would be 
K 4 (a— 2 , x,b , y) save for the occurrence of ax in its last element instead 
0 f Changing therefore ax into (a + 4<)x — 4x, we have the quasi 
recurrence-formula 
K 4 (a-3,x,b,y) = K 4 (a - 2 , x , b , y) - 4zK 3 (a- 2 , x , b , y), 
and generally 
K n (a-n+l,x,b,y) = K n (ct -n + 2 ,x,b ,y) - nxK^a - n + 2 , x , 6 , y). 
As it is only the first of the four variables that changes in it, the 
equality may be viewed as giving the increment of K n due to a receiving 
the increment 1. 
(9) There is a generalisation of the foregoing which, merely for its 
own sake, deserves to be noted in passing. If the n-line continuant 
whose diagonals are 
b 2b 3b .... 
a ct-\-b-\-z cl + 2b + 2z ct-\-3b-\-3z .... 
c c + z c + 2z .... 
be denoted by 
then 
H Ja 
b b 
c z. 
„ / b b 
Hn ft 
\ c z 
b b 
H n ( a + b J ^ — rcfrHn-, (a-vb 
\ c + z zj \ 
b b 
c + z z, 
The mode of proof is quite similar to that of § 8. 
