230 
Proceedings of the Royal Society 
3. By the cyclical transposition of rows and thereafter of 
columns, we establish a first law of continuants, viz. : — 
k( b ‘ b — 1 )Ak( J ”- 1 . (I.) 
\a x a 2 . . . . a n _ 1 aj \a n a n _ 1 . ... a 2 aj v ' 
4. By expansion of the continuant in terms of its principal 
minors we have 
K 
f b ' J — 1 ) = a^( b * 6 -' ) + bfi( l * ' ' ■ b ) (II.) 
• • • <WV 1 W* a • • • <WV 1 V«3 • • • v 7 
5. From this we see how to evaluate a continuant for special 
values of its elements, and also to change a continuant into the 
ordinary notation, i.e.> to free it of determinant forms. Thus, 
K / 4 6 S 9 7 \ 
K \7 2 3 1 4 5J 
would be evaluated by first evaluating K ^ > thence K ^ , 
thence K 
/ W) 7 \ 
\3 1 4 5/ 
and so on. 
6. By means of Laplace’s expansion-theorem we can establish a 
result which includes (II.) viz., 
K 
(' h ®*-i ^ = kY ^ “ ^ -1 ^ K ( ^ +1 * * * ^ 
\a L a 2 ...a p ... a n _ x aj \a L a 2 a p ) \a p+1 a n ) 
+ IJl( bl b r+f- b n-i 
Vh. a p-\J vh>+2 a n) 
(in.); 
and, using instead the present author’s extension of Laplace’s 
theorem, we arrive at a still more general proposition, viz., 
kY • * * ^-1 \k( ‘ ‘ V- 1 \ 
W*a W V 
= kY ^eY ^ 
V«i <v w °v 
(iv.), 
