of Edinburgh, Session 1873 - 74 . 231 
where of course h<p<n. An important particular case is that 
for which p = n-l and h = 2. 
7. Another result which is easily proved by induction is 
:( 1 1 )={-iyK( b *-') . (Y.) 
\-«l, -«./ ' V«l«- • • ' ' 
8. In any continuant 
K f • • • . K-i \ 
a„ j 
we may call a n _ x a n the ' main diagonal , and bfi . 2 ..... b n _ x 
the minor diagonal ; a y , a 2 , , b lt & 2 , . . . . being known as 
elements. When each element of the minor diagonal is unity, the 
continuant may be called simple , and in writing such continuants 
we may agree to omit the minor diagonal, putting, for example, 
K (« 1 a 2 a 3 .... for K 
' 11 .. 
A « a « 3 - 
9. If the elements of the first column of the determinant 
K(1 a x al . . . a n ) be subtracted from the corresponding elements of 
the second column, it will be seen that 
K (1, a x , a 2 , . . . a n ) = K (a L + 1 , a 2 , . . . a H ) . (VI.) 
10. From (II.) it is clear that 
K (0 , a 2 , a 3 , . . . a J = K (a 3 . . . a H ) , 
thence, with the help of (III.), we can show that 
K (. . . a, b, c, 0, e j, g, , . . ) = K ( . . . a, b, c + e,f, g, . . . ) (VII.), 
and from this that 
K ( . . . a, b, c, 0, 0, 0, e,/, . ..) = K(. . . a, b, c + e,/, . . .) 
and so, generally, when the number of consecutive zero elements is 
odd. 
11. Similarly, from (II.) 
K (0, 0, a. 6 , a 4 , . . . a n ) - K (a. i9 a 49 . . . a n ) , 
2 o 
VOL. VIII. 
