668 Proceedings of Royal Society of Edinburgh. [sess. 
consisting of twenty-two very short paragraphs, almost every one 
of which contains a concisely stated property, accompanied, where 
necessary, by an indication of the mode of proof. 
‘ Continuant ’ is formally defined as being a determinant which 
has the elements lying outside the principal diagonal and the two 
bordering minor diagonals each equal to zero, and which has the 
element of one of these minor diagonals each equal to negative 
unity. The continuant 
«i \ 
— 1 ^2 ^2 
~ 1 <h 
( b-2 \ 
is denoted by Kvoq a 2 a 3 ... / , a Y , a 2 , a 3 , . . . being spoken 
of as the main diagonal, and b x , b 2 , . . . as the minor diagonal. 
When the b’s are each equal to unity, the continuant is called a 
simple continuant, and denoted by K(a : , a 2 , a 3 , . . . ). 
Of results previously formulated it may be worth noting that the 
most important, viz. 
K| 
( V- 
• bn- 1 \ 
. . . . a n ) 
• k( 
\a h . . . 
. . b p _ x \ 
) where h <p < n 
= K| 
{ V-. 
V«i • • • • 
■ Vi \ 
• • • aj 
• K(" bh •' 
\a h . . . 
• K _i \ 
+(-y- h+i b h -A . 
( 
■ • b h _ 3 \ 
• bp+z • • 
\«i • • • 
\a p+2 • • • • 
. . . a n J 
is not obtained in the same manner as in Thiele’s paper of 1870, 
but from a general theorem regarding the product of any deter- 
minant by one of its own minors.* 
Leaving aside other results of this kind, we have remaining on 
the first three pages — 
* This theorem was not published until five years afterwards. See paper 
entitled “ General Theorems on Determinants” in Trans. R. Soc. Edinburgh , 
xxix. pp. 47-54. 
